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A003099
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a(n) = Sum_{k=0..n} binomial(n,k^2).
(Formerly M0576)
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18
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1, 2, 3, 4, 6, 11, 22, 43, 79, 137, 231, 397, 728, 1444, 3018, 6386, 13278, 26725, 51852, 97243, 177671, 320286, 579371, 1071226, 2053626, 4098627, 8451288, 17742649, 37352435, 77926452, 159899767, 321468048, 632531039, 1219295320, 2308910353, 4314168202
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n)*sqrt(n)/2^n is bounded: lim sup a(n)*sqrt(n)/2^n = 0.82... and lim inf a(n)*sqrt(n)/2^n = 0.58... - Benoit Cloitre, Nov 14 2003 [These constants are sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = 0.827112271364145742... and sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = 0.587247586271786487... - Vaclav Kotesovec, Jan 15 2023]
Binomial transform of the characteristic function of squares A010052. - Carl Najafi, Sep 09 2011
G.f.: (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^(k^2). - Ilya Gutkovskiy, Jan 22 2024
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MATHEMATICA
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Table[Sum[Binomial[n, k^2], {k, 0, Sqrt[n]}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)
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PROG
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(Magma) [(&+[Binomial(n, j^2): j in [0..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022
(SageMath)
def A003099(n): return sum( binomial(n, k^2) for k in range(isqrt(n)+1))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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