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A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).
(Formerly M0576)
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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]
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
Partial sums of A103198.
Sequence in context: A114412 A352819 A016038 * A061941 A029505 A185092
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Carl Najafi, Sep 09 2011
STATUS
approved

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Last modified March 3 19:39 EST 2024. Contains 370512 sequences. (Running on oeis4.)