A003099 a(n) = Sum_{k=0..n} binomial(n,k^2). (Formerly M0576)

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

Offset

0,2

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..3000

Henry W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]

Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.

Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.

Formula

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

Mathematica

Table[Sum[Binomial[n, k^2], {k, 0, Sqrt[n]}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)

Prog

(PARI) a(n)=sum(k=0, sqrtint(n), binomial(n, k^2)) \\ Charles R Greathouse IV, Mar 26 2013
(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))
[A003099(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

Crossrefs

Cf. A010052, A206849.

Partial sums of A103198.

Sequence in context: A114412 A352819 A016038 * A061941 A029505 A185092

Adjacent sequences: A003096 A003097 A003098 * A003100 A003101 A003102

Keyword

nonn,easy

Author

N. J. A. Sloane, Henry W. Gould

Extensions

More terms from Carl Najafi, Sep 09 2011

Status

approved