A053309 Partial sums of A053308.

1, 10, 56, 231, 782, 2300, 6085, 14820, 33775, 72905, 150438, 298925, 575333, 1077748, 1972851, 3540913, 6249235, 10871723, 18683233, 31775031, 53566369, 89633545, 149052839, 246575109, 406146248, 666605513, 1090907965

Offset: 0

Author

Barry E. Williams, Mar 06 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-44,111,-174,168,-84,-6,39,-26,8,-1).

Crossrefs

Cf. A053296, A053295, A136431.

Cf. A228074.

Programs

Magma

[(&+[Binomial(n+9-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018

Mathematica

Table[Sum[Binomial[n+9-j, n-2j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)

PARI

for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+9-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

Formula

a(n) = Sum_{i=0..floor(n/2)} C(n+9-i, n-2i), n >= 0.

a(n) = a(n-1) + a(n-2) + C(n+8,8); n >= 0; a(-1)=0.

G.f.: 1/((x^2 + x - 1)*(x-1)^9). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009