+ i׊^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t H t ^RI Ht^RIHt^R IHt^R IHtHt^R IHt^R IHtHtHt^R IHt^RIHt^RI H!t!H"t"^RI#H$t$H%t%^RI&H't'^RI(H)t)H*t*H+t+!RR]4t,!RR],]R7t-!RR],4t.!RR].4t/!RR].4t0!RR],4t1R(R!lt2!R"R#],4t3!R$R%]34t4!R&R']34t5R #)))Basic)cacheit)Tuple)call_highest_priority)global_parameters) AppliedUndefexpandMul)Integer)Eq)S Singleton)ordered)DummySymbolWildsympify)Matrix)lcmfactor)Interval Intersection)Idx)flatten is_sequenceiterablecba]tRt^toRtRt^t]R4tRt ] R4t ] R4t ] R4t ] R4t] R 4t] R 4t] R 4t]R 4tR tRtRtRtRtRt]!R4R4tRt]!R4R4tRtRt]!R4R4t Rt!Rt"R!Rlt#R t$Vt%R#)"SeqBasezBase class for sequencesTc dVPpV# \d\PpT#i;i)zKReturn start (if possible) else S.Infinity. adapted from Set._infimum_key )startNotImplementedErrorrInfinity)exprr"s& v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/series/sequences.py _start_keySeqBase._start_key s6  JJE # JJE  s //c r\VPVP4pVPVP3#)zDReturns start and stop. Takes intersection over the two intervals. )rintervalinfsup)selfotherr*s&& r&_intersect_intervalSeqBase._intersect_interval,s+   u~~>||X\\))c &\RV,4h)z&Returns the generator for the sequencez(%s).genr#r-s&r&gen SeqBase.gen4s"*t"344r1c &\RV,4h)z-The interval on which the sequence is definedz (%s).intervalr3r4s&r&r*SeqBase.interval9s"/D"899r1c &\RV,4h):The starting point of the sequence. This point is includedz (%s).startr3r4s&r&r" SeqBase.start>s","566r1c &\RV,4h)z8The ending point of the sequence. This point is includedz (%s).stopr3r4s&r&stop SeqBase.stopCs"+"455r1c &\RV,4h)zLength of the sequencez (%s).lengthr3r4s&r&lengthSeqBase.lengthHs"-$"677r1c R#)z-Returns a tuple of variables that are boundedrCr4s&r& variablesSeqBase.variablesMs  r1c VPUUu0uF/qPPVP4Fq"kK K1 upp#uuppi)z This method returns the symbols in the object, excluding those that take on a specific value (i.e. the dummy symbols). Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n, m >>> SeqFormula(m*n**2, (n, 0, 5)).free_symbols {m} )args free_symbols differencerD)r-ijs& r&rHSeqBase.free_symbolsRsH!II0Iq~~Jt~~.0/!0/I0 10s5A c WP8gWP8d\RV: RVP: 24hVP V4#)z#Returns the coefficient at point ptzIndex z out of bounds )r"r= IndexErrorr* _eval_coeffr-pts&&r&coeff SeqBase.coeffcs:  ?b99nB NO O##r1c:\RVP,4h)zhThe _eval_coeff method should be added to%s to return coefficient so it is availablewhen coeff calls it.)r#funcrPs&&r&rOSeqBase._eval_coeffjs"!#9%)II#./ /r1c VP\PJdVPpM VPpVP\PJdRpM^pW!V,,#)aReturns the i'th point of a sequence. Explanation =========== If start point is negative infinity, point is returned from the end. Assumes the first point to be indexed zero. Examples ========= >>> from sympy import oo >>> from sympy.series.sequences import SeqPer bounded >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(0) -10 >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(5) -5 End is at infinity >>> SeqPer((1, 2, 3), (0, oo))._ith_point(5) 5 Starts at negative infinity >>> SeqPer((1, 2, 3), (-oo, 0))._ith_point(5) -5 )r"rNegativeInfinityr=)r-rJinitialsteps&& r& _ith_pointSeqBase._ith_pointpsR@ ::++ +iiGjjG ::++ +DD4r1c R#)a  Should only be used internally. Explanation =========== self._add(other) returns a new, term-wise added sequence if self knows how to add with other, otherwise it returns ``None``. ``other`` should only be a sequence object. Used within :class:`SeqAdd` class. NrCr-r.s&&r&_add SeqBase._addr1c R#)a Should only be used internally. Explanation =========== self._mul(other) returns a new, term-wise multiplied sequence if self knows how to multiply with other, otherwise it returns ``None``. ``other`` should only be a sequence object. Used within :class:`SeqMul` class. NrCr_s&&r&_mul SeqBase._mulrbr1c \W4#)a= Should be used when ``other`` is not a sequence. Should be defined to define custom behaviour. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2).coeff_mul(2) SeqFormula(2*n**2, (n, 0, oo)) Notes ===== '*' defines multiplication of sequences with sequences only. r r_s&&r& coeff_mulSeqBase.coeff_muls$4r1c z\V\4'g\R\V4,4h\ W4#)zReturns the term-wise addition of 'self' and 'other'. ``other`` should be a sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) + SeqFormula(n**3) SeqFormula(n**3 + n**2, (n, 0, oo)) zcannot add sequence and %s isinstancer TypeErrortypeSeqAddr_s&&r&__add__SeqBase.__add__s1%))84;FG Gd""r1rocW,#NrCr_s&&r&__radd__SeqBase.__radd__ |r1c |\V\4'g\R\V4,4h\ W)4#)zReturns the term-wise subtraction of ``self`` and ``other``. ``other`` should be a sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) - (SeqFormula(n)) SeqFormula(n**2 - n, (n, 0, oo)) zcannot subtract sequence and %srjr_s&&r&__sub__SeqBase.__sub__s3%))=U KL LdF##r1rwcV)V,#rrrCr_s&&r&__rsub__SeqBase.__rsub__sr1c $VPR4#)zNegates the sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> -SeqFormula(n**2) SeqFormula(-n**2, (n, 0, oo)) rX)rgr4s&r&__neg__SeqBase.__neg__s~~b!!r1c z\V\4'g\R\V4,4h\ W4#)a3Returns the term-wise multiplication of 'self' and 'other'. ``other`` should be a sequence. For ``other`` not being a sequence see :func:`coeff_mul` method. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) * (SeqFormula(n)) SeqFormula(n**3, (n, 0, oo)) zcannot multiply sequence and %s)rkr rlrmSeqMulr_s&&r&__mul__SeqBase.__mul__s1%))=U KL Ld""r1rcW,#rrrCr_s&&r&__rmul__SeqBase.__rmul__rur1c#"\VP4F'pVPV4pVPV4xK) R#5irr)ranger@r\rR)r-rJrQs& r&__iter__SeqBase.__iter__s4t{{#A#B**R. $sAAc\V\4'd#VPV4pVPV4#\V\4'd{VP VP r2Vf^pVf VPp\Y#VP;'g^4Uu.uF"q@PVPV44NK$ up#R#uupirr) rkintr\rRslicer"r=r@rr[)r-indexr"r=rJs&& r& __getitem__SeqBase.__getitem__"s eS ! !OOE*E::e$ $ u % %++uzz4}|{{%uzzQ79789JJtq1279 9 & 9s(C Nc  b^RIHpVRVUu.uFqT!\V44NK pp\V4pVf V^,pM\ W'^,4p.p \ ^V^,4EFp ^V ,p .p \ V 4Fp V P WmW,4K \V 4pVP4^8wgK\V!VP\WjV 444pW{8Xd\VRRR1,4p Mr.p \ WV , 4Fp V P WmW,4K \V 4pW,\WkR48XgK\VRRR1,4p M VfV #\V 4p V ^8Xd.R3#Wj^, ,W:^, ,,^W^, ,W:,,, r!\ V ^, 4FpWV,VV,,, p\ V V, ^, 4F<pWV,VV,,VVV,^,,,,pK> W)V,VV^,,,,pK W!\V4\V4, 43#uupi)a Finds the shortest linear recurrence that satisfies the first n terms of sequence of order `\leq` ``n/2`` if possible. If ``d`` is specified, find shortest linear recurrence of order `\leq` min(d, n/2) if possible. Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...`` Returns ``[]`` if no recurrence is found. If gfvar is specified, also returns ordinary generating function as a function of gfvar. Examples ======== >>> from sympy import sequence, sqrt, oo, lucas >>> from sympy.abc import n, x, y >>> sequence(n**2).find_linear_recurrence(10, 2) [] >>> sequence(n**2).find_linear_recurrence(10) [3, -3, 1] >>> sequence(2**n).find_linear_recurrence(10) [2] >>> sequence(23*n**4+91*n**2).find_linear_recurrence(10) [5, -10, 10, -5, 1] >>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10) [1, 1] >>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30) [1/2, 1/2] >>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x) ([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1))) >>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x) ([1, 1], (x - 2)/(x**2 + x - 1)) )simplifyNrX) sympy.simplifyrr lenminrappendrdetLUsolverr)r-ndgfvarrtxlxrcoeffsll2mlistkmyrJrKs&&&& r&find_linear_recurrenceSeqBase.find_linear_recurrence/s(D ,*.r( 3(QXfQi ( 3 V 9AAA!e Aq!A#A1BE1X QX&u Auuw!|QYYva"g788$QttW-FqAALLQS*'5M3&3.($QttW-F#$ =MF AAv4x1vecl*As EH0D,D1qsA1eQh&A"1Q3q5\AYqt^EAaCEN::*51Q3<//A $ xq &)(;<<>> from sympy import EmptySequence, SeqPer >>> from sympy.abc import x >>> EmptySequence EmptySequence >>> SeqPer((1, 2), (x, 0, 10)) + EmptySequence SeqPer((1, 2), (x, 0, 10)) >>> SeqPer((1, 2)) * EmptySequence EmptySequence >>> EmptySequence.coeff_mul(-1) EmptySequence c"\P#rr)rEmptySetr4s&r&r*EmptySequence.intervals zzr1c"\P#rr)rZeror4s&r&r@EmptySequence.lengths vv r1c V#z"See docstring of SeqBase.coeff_mulrC)r-rRs&&r&rgEmptySequence.coeff_muls r1c\.4#rr)iterr4s&r&rEmptySequence.__iter__s Bxr1rCN) rrrrrrr*r@rgrrrrs@r&rrzsA(r1r) metaclassca]tRtRtoRt]R4t]R4t]R4t]R4t ]R4t ]R4t R t Vt R #) SeqExpriaSequence expression class. Various sequences should inherit from this class. Examples ======== >>> from sympy.series.sequences import SeqExpr >>> from sympy.abc import x >>> from sympy import Tuple >>> s = SeqExpr(Tuple(1, 2, 3), Tuple(x, 0, 10)) >>> s.gen (1, 2, 3) >>> s.interval Interval(0, 10) >>> s.length 11 See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula c(VP^,#rrGr4s&r&r5 SeqExpr.gensyy|r1cz\VP^,^,VP^,^,4#)rrGr4s&r&r*SeqExpr.intervals' ! Q1a99r1c.VPP#rrr*r+r4s&r&r" SeqExpr.start}}   r1c.VPP#rrr*r,r4s&r&r= SeqExpr.stoprr1cJVPVP, ^,#rr=r"r4s&r&r@SeqExpr.lengthyy4::%))r1c8VP^,^,3#rrr4s&r&rDSeqExpr.variabless ! Q!!r1rCN)rrrrrrr5r*r"r=r@rDrrrs@r&rrs2::!!!!**""r1rcfa]tRtRtoRtR Rlt]R4t]R4tRt Rt R t R t R t VtR#) SeqPeriaj Represents a periodic sequence. The elements are repeated after a given period. Examples ======== >>> from sympy import SeqPer, oo >>> from sympy.abc import k >>> s = SeqPer((1, 2, 3), (0, 5)) >>> s.periodical (1, 2, 3) >>> s.period 3 For value at a particular point >>> s.coeff(3) 1 supports slicing >>> s[:] [1, 2, 3, 1, 2, 3] iterable >>> list(s) [1, 2, 3, 1, 2, 3] sequence starts from negative infinity >>> SeqPer((1, 2, 3), (-oo, 0))[0:6] [1, 2, 3, 1, 2, 3] Periodic formulas >>> SeqPer((k, k**2, k**3), (k, 0, oo))[0:6] [0, 1, 8, 3, 16, 125] See Also ======== sympy.series.sequences.SeqFormula Nc"\V4pRpRRRrepVfV!V4^\Prep\V\4'd3\ V4^8XdVwrEpM\ V4^8Xd V!V4pVwrV\ V\\34'd VeVf\R\V4,4hV\PJd V\PJd \R4h\WEV34p\V\4'd\\\V444pM\RV,4h\V^,V^,4\PJd\P #\"P$!WV4#)cVPp\VP4^8XdVP4#\R4#)rr)rHrpopr) periodicalfrees& r&_find_xSeqPer.__new__.._find_xs6**D:**+q0xxz!Sz!r1NInvalid limits given: %sz/Both the start and end valuecannot be unboundedz6invalid period %s should be something like e.g (1, 2) )rrr$rrrrkrr ValueErrorstrrYtuplerrrrr__new__)clsrlimitsrrr"r=s&&& r&rSeqPer.__new__sSZ(  "tT$ >$Z0!QZZdA vu % %6{a!'$V!J'$ !fc]++u} 7#f+EF F A&& &41::+= "788!D)* z5 ) ) wz':!; > F1Ivay )QZZ 7?? "}}Sf55r1c,\VP4#rr)rr5r4s&r&period SeqPer.period+s488}r1cVP#rrr5r4s&r&rSeqPer.periodical/ xxr1c6VP\PJd&VPV, VP,pM#WP, VP,pVP V,P VP^,V4#r)r"rrYr=rrsubsrD)r-rQidxs&& r&rOSeqPer._eval_coeff3sc ::++ +99r>T[[0C ?dkk1Cs#(():B??r1c \V\4'dVPVPr2VPVPrT\ W54p.p\ V4F8pW(V,,p WHV,,p VP W,4K: VPV4wr\WpP^,W34#R#zSee docstring of SeqBase._addN rkrrrrrrr/rD r-r.per1lper1per2lper2 per_lengthnew_perrele1ele2r"r=s && r&r` SeqPer._add: eV $ $//4;;%**ELL%U*JG:&IIt{+' 2259KE'NN1$5u#CD D %r1c \V\4'dVPVPr2VPVPrT\ W54p.p\ V4F8pW(V,,p WHV,,p VP W,4K: VPV4wr\WpP^,W34#R#zSee docstring of SeqBase._mulNrrs && r&rd SeqPer._mulKrr1c \V4pVPUu.uF q"V,NK pp\W0P^,4#uupir)rrrrG)r-rRrpers&& r&rgSeqPer.coeff_mul\s?"&//2/Q5yy/2c99Q<((3sA rCrr)rrrrrrrrrrOr`rdrgrrrs@r&rrsU.`&6P@E"E"))r1rc\a]tRtRtoRtR Rlt]R4tRtRt Rt R t R t R t VtR#) SeqFormulaica Represents sequence based on a formula. Elements are generated using a formula. Examples ======== >>> from sympy import SeqFormula, oo, Symbol >>> n = Symbol('n') >>> s = SeqFormula(n**2, (n, 0, 5)) >>> s.formula n**2 For value at a particular point >>> s.coeff(3) 9 supports slicing >>> s[:] [0, 1, 4, 9, 16, 25] iterable >>> list(s) [0, 1, 4, 9, 16, 25] sequence starts from negative infinity >>> SeqFormula(n**2, (-oo, 0))[0:6] [0, 1, 4, 9, 16, 25] See Also ======== sympy.series.sequences.SeqPer Nc\V4pRpRRRrepVfV!V4^\Prep\V\4'd3\ V4^8XdVwrEpM\ V4^8Xd V!V4pVwrV\ V\\34'd VeVf\R\V4,4hV\PJd V\PJd \R4h\WEV34p\V^,V^,4\PJd\P#\P !WV4#)cVPp\V4^8XdVP4#V'g \R4#\ RV,4h)rrz specify dummy variables for %s. If the formula contains more than one free symbol, a dummy variable should be supplied explicitly e.g., SeqFormula(m*n**2, (n, 0, 5)))rHrrrr)formulars& r&r#SeqFormula.__new__.._find_xsN''D4yA~xxz!Sz! Or1Nrz0Both the start and end value cannot be unbounded)rrr$rrrrkrrrrrYrrrrr)rrrrrr"r=s&&& r&rSeqFormula.__new__s'" tT$ >$W-q!**dA vu % %6{a!'$V!G$$ !fc]++u} 7#f+EF F A&& &41::+= "788!D)* F1Ivay )QZZ 7?? "}}S622r1cVP#rrrr4s&r&rSeqFormula.formularr1c^VP^,pVPPW!4#r)rDrr)r-rQrs&& r&rOSeqFormula._eval_coeffs% NN1 ||  ''r1c \V\4'dtVPVP^,r2VPVP^,rTW$P WS4,pVP V4wrx\WcWx34#R#rrkr rrDrr/ r-r.form1v1form2v2rr"r=s && r&r`SeqFormula._addo eZ ( ( dnnQ&72 uq'92jj00G2259KEgE'89 9 )r1c \V\4'dtVPVP^,r2VPVP^,rTW$P WS4,pVP V4wrx\WcWx34#R#rrrs && r&rdSeqFormula._mulrr1c v\V4pVPV,p\W P^,4#r)rrr rG)r-rRrs&& r&rgSeqFormula.coeff_muls,,,&'99Q<00r1cj\\VP.VO5/VBVP^,4#r)r r rrG)r-rGkwargss&*,r&r SeqFormula.expands*&???1NNr1rCrr)rrrrrrrrrOr`rdrgr rrrs@r&r r csE&P%3N(::1 OOr1r ca]tRtRtoRtRRlt]R4t]R4t]R4t ]R4t ]R 4t ]R 4t ]R 4t ]R 4t]R 4tRtRtRtVtR#) RecursiveSeqia A finite degree recursive sequence. Explanation =========== That is, a sequence a(n) that depends on a fixed, finite number of its previous values. The general form is a(n) = f(a(n - 1), a(n - 2), ..., a(n - d)) for some fixed, positive integer d, where f is some function defined by a SymPy expression. Parameters ========== recurrence : SymPy expression defining recurrence This is *not* an equality, only the expression that the nth term is equal to. For example, if :code:`a(n) = f(a(n - 1), ..., a(n - d))`, then the expression should be :code:`f(a(n - 1), ..., a(n - d))`. yn : applied undefined function Represents the nth term of the sequence as e.g. :code:`y(n)` where :code:`y` is an undefined function and `n` is the sequence index. n : symbolic argument The name of the variable that the recurrence is in, e.g., :code:`n` if the recurrence function is :code:`y(n)`. initial : iterable with length equal to the degree of the recurrence The initial values of the recurrence. start : start value of sequence (inclusive) Examples ======== >>> from sympy import Function, symbols >>> from sympy.series.sequences import RecursiveSeq >>> y = Function("y") >>> n = symbols("n") >>> fib = RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, [0, 1]) >>> fib.coeff(3) # Value at a particular point 2 >>> fib[:6] # supports slicing [0, 1, 1, 2, 3, 5] >>> fib.recurrence # inspect recurrence Eq(y(n), y(n - 2) + y(n - 1)) >>> fib.degree # automatically determine degree 2 >>> for x in zip(range(10), fib): # supports iteration ... print(x) (0, 0) (1, 1) (2, 1) (3, 2) (4, 3) (5, 5) (6, 8) (7, 13) (8, 21) (9, 34) See Also ======== sympy.series.sequences.SeqFormula Nc\V\4'g\RPV44h\V\4'dVP 'g\RPV44hVP V38wd \R4hVPp\RV3R7p^pVPV4p V Fp \V P 4^8wd \R4hV P ^,PW7,4V,p V P4'dV P'dV ^8g\RPV 44hV )V8gKV )pK V'g3\V4Uu.uFp\RPV44NK pp\V4V8wd \!R4h\#V4p\%V4p\'R V4!p\P(!WW#WE4p \+V4UU u/uFwr}V!WW,4V bK up pV nWnV #uupiuup pi) zErecurrence sequence must be an applied undefined function, found `{}`z0recurrence variable must be a symbol, found `{}`z)recurrence sequence does not match symbolr)excludez)Recurrence should be in a single variablezDRecurrence should have constant, negative, integer shifts (found {})zc_{}z)Number of initial terms must equal degreec38"TFp\V4xK R#5irrr).0rs& r& 'RecursiveSeq.__new__..Os6g'!**g)rkrrlformatr is_symbolrGrUrfindrmatch is_constant is_integerrrrr rrr enumeratecachedegree)r recurrenceynrrZr"rrr7prev_ysprev_yshiftseqinits&&&&&& r&rRecursiveSeq.__new__#s"l++++16":7 7!U##1;;;++16!96 6 77qd?GH H GG qd # //!$F6;;1$ KLLKKN((/2E%%''E,<,<,<!..4fVn>>v8=f F 1uV]]1-. GF w<6 !HI I6g67mmCRGC7@7IJ7IGAQuy\4'7IJ   GKs #H<Ic (VP^,#zEquation defining recurrence.rr4s&r& _recurrenceRecursiveSeq._recurrenceXyy|r1c P\VPVP^,4#rA)r r9rGr4s&r&r8RecursiveSeq.recurrence]s$''499Q<((r1c (VP^,#)z*Applied function representing the nth termrr4s&r&r9RecursiveSeq.ynbrDr1c .VPP#)z3Undefined function for the nth term of the sequence)r9rUr4s&r&rRecursiveSeq.ygsww||r1c (VP^,#)zSequence index symbolrr4s&r&rRecursiveSeq.nlrDr1c (VP^,#)z"The initial values of the sequencerr4s&r&rZRecursiveSeq.initialqrDr1c (VP^,#)r:rr4s&r&r"RecursiveSeq.startvrDr1c "\P#)z&The ending point of the sequence. (oo))rr$r4s&r&r=RecursiveSeq.stop{szzr1c :VP\P3#)z&Interval on which sequence is defined.)r"rr$r4s&r&r*RecursiveSeq.intervals AJJ''r1c>WP, \VP48d#VPVPV4,#\ \VP4V^,4FupVPV,pVP P VPV/4pVP VP4pWPPVPV4&Kw VPVPVPX,4,#r)r"rr6rrrBxreplacer)r-rcurrent seq_indexcurrent_recurrencenew_terms&& r&rORecursiveSeq._eval_coeffs :: DJJ /::dffUm, ,S_eai8G W,I!%!1!1!:!:DFFI;N!O )224::>H,4JJtvvi( )9zz$&&g!5677r1c#`"VPpVPV4xV^, pK5i)T)r"rO)r-rs& r&rRecursiveSeq.__iter__s+ ""5) ) QJEs,.rC)Nr)rrrrrrrrBr8r9rrrZr"r=r*rOrrrrs@r&r'r'sJX3j))(( 8r1r'Ncp\V4p\V\4'd \W4#\ W4#)a Returns appropriate sequence object. Explanation =========== If ``seq`` is a SymPy sequence, returns :class:`SeqPer` object otherwise returns :class:`SeqFormula` object. Examples ======== >>> from sympy import sequence >>> from sympy.abc import n >>> sequence(n**2, (n, 0, 5)) SeqFormula(n**2, (n, 0, 5)) >>> sequence((1, 2, 3), (n, 0, 5)) SeqPer((1, 2, 3), (n, 0, 5)) See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula )rrrrr )r=rs&&r&sequencer_s04 #,C3c""#&&r1ca]tRtRtoRt]R4t]R4t]R4t]R4t ]R4t ]R4t R t Vt R #) SeqExprOpia Base class for operations on sequences. Examples ======== >>> from sympy.series.sequences import SeqExprOp, sequence >>> from sympy.abc import n >>> s1 = sequence(n**2, (n, 0, 10)) >>> s2 = sequence((1, 2, 3), (n, 5, 10)) >>> s = SeqExprOp(s1, s2) >>> s.gen (n**2, (1, 2, 3)) >>> s.interval Interval(5, 10) >>> s.length 6 See Also ======== sympy.series.sequences.SeqAdd sympy.series.sequences.SeqMul c \;QJd.RVP4FNK 5#!RVP44#)zZGenerator for the sequence. returns a tuple of generators of all the argument sequences. c38"TFqPxK R#5irrrr+as& r&r, SeqExprOp.gen..s.IqUUIr.)rrGr4s&r&r5 SeqExprOp.gens. u.DII.u.u.DII...r1c 6\RVP4!#)zUSequence is defined on the intersection of all the intervals of respective sequences c38"TFqPxK R#5irrr*rds& r&r,%SeqExprOp.interval..s<)Qjj)r.)rrGr4s&r&r*SeqExprOp.intervals <$))<==r1c.VPP#rrrr4s&r&r"SeqExprOp.startrr1c.VPP#rrrr4s&r&r=SeqExprOp.stoprr1c z\\VPUu.uFqPNK up44#uupi)z%Cumulative of all the bound variables)rrrGrD)r-res& r&rDSeqExprOp.variabless,W499=9akk9=>??=s8 cJVPVP, ^,#rrr4s&r&r@SeqExprOp.lengthrr1rCN)rrrrrrr5r*r"r=rDr@rrrs@r&raras0//>> !!!!@@**r1rac@a]tRtRtoRtRt]R4tRtRt Vt R#)rniaFRepresents term-wise addition of sequences. Rules: * The interval on which sequence is defined is the intersection of respective intervals of sequences. * Anything + :class:`EmptySequence` remains unchanged. * Other rules are defined in ``_add`` methods of sequence classes. Examples ======== >>> from sympy import EmptySequence, oo, SeqAdd, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), EmptySequence) SeqPer((1, 2), (n, 0, oo)) >>> SeqAdd(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) EmptySequence >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2, (n, 0, oo))) SeqAdd(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) >>> SeqAdd(SeqFormula(n**3), SeqFormula(n**2)) SeqFormula(n**3 + n**2, (n, 0, oo)) See Also ======== sympy.series.sequences.SeqMul caVPR\P4p\V4pV3RloS!V4pVUu.uFqD\P JgKVNK ppV'g\P #\ RV4!\PJd\P #V'd\PV4#\\V\P44p\P!V.VO5!#uupi)evaluatec<\V\4'd:\V\4'd!\\ SVP 4.4#V.#\ V4'd\\ SV4.4#\R4hz2Input must be Sequences or iterables of Sequences)rkr rnsummaprGrrlarg_flattens&r&r~ SeqAdd.__new__.._flatten sk#w''c6**s8SXX6;;5L}}3x-r2267 7r1c38"TFqPxK R#5irrrjrds& r&r,!SeqAdd.__new__..23d**dr.)getrrwlistrrrrrnreducerr r'rr)rrGr$rwrer~s&*, @r&rSeqAdd.__new__s::j*;*D*DEDz 7~<4aAOO#;4<?? " 3d3 4 B?? " ==& &GD'"4"456}}S(4((=s DDc RpV'd\V4Frwr#Rp\V4FOwrEW$8XdK VPV4pVfK$VUu.uFqwW539gK VNK ppVPV4M V'gKoTpK K\V4^8XdVP 4#\ VRR7#uupi)zSimplify :class:`SeqAdd` using known rules. Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent. Notes ===== adapted from ``Union.reduce`` TFrw)r5r`rrrrnrGnew_argsid1sid2rnew_seqres& r&r SeqAdd.reduce=s#D/ 'oFCz ffQiG*/3#Gt!AAt#G 0.8#D* t9>88: $/ /$H  B;B;c Ba\V3RlVP44#)z9adds up the coefficients of all the sequences at point ptc3D<"TFqPS4xK R#5irr)rR)r+rerQs& r&r,%SeqAdd._eval_coeff..cs2 1772;; s )rzrGrPs&fr&rOSeqAdd._eval_coeffas2 222r1rCN rrrrrrrrrOrrrs@r&rnrns/8")H!0!0F33r1rnc@a]tRtRtoRtRt]R4tRtRt Vt R#)rifaRepresents term-wise multiplication of sequences. Explanation =========== Handles multiplication of sequences only. For multiplication with other objects see :func:`SeqBase.coeff_mul`. Rules: * The interval on which sequence is defined is the intersection of respective intervals of sequences. * Anything \* :class:`EmptySequence` returns :class:`EmptySequence`. * Other rules are defined in ``_mul`` methods of sequence classes. Examples ======== >>> from sympy import EmptySequence, oo, SeqMul, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), EmptySequence) EmptySequence >>> SeqMul(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) EmptySequence >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2)) SeqMul(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) >>> SeqMul(SeqFormula(n**3), SeqFormula(n**2)) SeqFormula(n**5, (n, 0, oo)) See Also ======== sympy.series.sequences.SeqAdd caVPR\P4p\V4pV3RloS!V4pV'g\P #\ RV4!\PJd\P #V'd\PV4#\\V\P44p\P!V.VO5!#)rwc<\V\4'd:\V\4'd!\\ SVP 4.4#V.#\ V4'd\\ SV4.4#\R4hry)rkr rrzr{rGrrlr|s&r&r~ SeqMul.__new__.._flattensk#w''c6**s8SXX6;;5L#3x-r2267 7r1c38"TFqPxK R#5irrrjrds& r&r,!SeqMul.__new__..rr.)rrrwrrrrrrrrr r'rr)rrGr$rwr~s&*, @r&rSeqMul.__new__s::j*;*D*DEDz 7~?? " 3d3 4 B?? " ==& &GD'"4"456}}S(4((r1c RpV'd\V4Frwr#Rp\V4FOwrEW$8XdK VPV4pVfK$VUu.uFqwW539gK VNK ppVPV4M V'gKoTpK K\V4^8XdVP 4#\ VRR7#uupi)zSimplify a :class:`SeqMul` using known rules. Explanation =========== Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent. Notes ===== adapted from ``Union.reduce`` TFr)r5rdrrrrrs& r&r SeqMul.reduces #D/ 'oFCz ffQiG*/3#Gt!AAt#G 0.8#D* t9>88: $/ /$Hrc ^^pVPFpW#PV4,pK V#)zrs"$'734&$-&11&!#2$DD^=e^=@ "Gy"J0"g0"fN)WN)bqOqOfB7BJ'N7*7*tg3Yg3TqYqr1