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 A006456 Number of compositions (ordered partitions) of n into squares. (Formerly M0528) 53
 1, 1, 1, 1, 2, 3, 4, 5, 7, 11, 16, 22, 30, 43, 62, 88, 124, 175, 249, 354, 502, 710, 1006, 1427, 2024, 2870, 4068, 5767, 8176, 11593, 16436, 23301, 33033, 46832, 66398, 94137, 133462, 189211, 268252, 380315, 539192, 764433, 1083764, 1536498, 2178364 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 0..500 from T. D. Noe) Jan Bohman, Carl-Erik FrÃ¶berg, Hans Riesel, Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. (Annotated scanned copy) N. Robbins, On compositions whose parts are polygonal numbers, Annales Univ. Sci. Budapest., Sect. Comp. 43 (2014) 239-243. See p. 242. FORMULA a(n) = 1, if n = 0;  a(n)=Sum(1 <= k^2 <= n, a(n-k^2)), if n > 0. - David W. Wilson G.f.: 1/(1-x-x^4-x^9-....) - Jon Perry, Jul 04 2004 a(n) ~ c * d^n, where d is the root of the equation EllipticTheta(3, 0, 1/d) = 3, d = 1.41774254618138831428829091099000662953179532057717725688..., c = 0.46542113389379672452973940263069782869244877335179331541... - Vaclav Kotesovec, May 01 2014, updated Jan 05 2017 G.f.: 2/(3 - theta_3(q)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018 MATHEMATICA a[n_]:=a[n]=If[n==0, 1, Sum[a[n - k], {k, Select[Range[n], IntegerQ[Sqrt[#]] &]}]]; Table[a[n], {n, 0,  100}] (* Indranil Ghosh, Jul 28 2017, after David W. Wilson's formula *) PROG (PARI) N=66;  x='x+O('x^N); Vec( 1/( 1 - sum(k=1, 1+sqrtint(N), x^(k^2) ) ) ) /* Joerg Arndt, Sep 30 2012 */ (Python) from gmpy2 import is_square class Memoize:     def __init__(self, func):         self.func=func         self.cache={}     def __call__(self, arg):         if arg not in self.cache:             self.cache[arg] = self.func(arg)         return self.cache[arg] @Memoize def a(n): return 1 if n==0 else sum([a(n - k) for k in range(1, n + 1) if is_square(k)]) print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 28 2017, after David W. Wilson's formula CROSSREFS Cf. A280542. Row sums of A337165. Sequence in context: A259466 A046420 A108318 * A018134 A245823 A143284 Adjacent sequences:  A006453 A006454 A006455 * A006457 A006458 A006459 KEYWORD nonn,easy AUTHOR EXTENSIONS Name corrected by Bob Selcoe, Feb 12 2014 STATUS approved

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Last modified August 14 18:13 EDT 2022. Contains 356122 sequences. (Running on oeis4.)