A053296 Partial sums of A053295.

1, 8, 37, 129, 376, 967, 2267, 4950, 10220, 20175, 38403, 70954, 127921, 226007, 392688, 672959, 1140260, 1914166, 3189022, 5280288, 8699540, 14275838, 23352118, 38102976, 62048869, 100888126, 163843187, 265838881, 431026972, 698489013, 1131463777, 1832277574, 2966502032, 4802042229

Offset: 0

Author

Barry E. Williams, Mar 04 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 (8,-27,49,-49,21,7,-13,6,-1).

Crossrefs

Cf. A000045, A053739, A014166, A136431, A228074.

Right-hand column 14 of triangle A011794.

Programs

Magma

[(&+[Binomial(n+7-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+7-j, n-2*j], {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+7-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

Formula

a(n) = Sum_{i=0..floor(n/2)} C(n+7-i, n-2i), n >= 0.

a(n) = a(n-1) + a(n-2) + C(n+6,6); n >= 0, with a(-1) = 0.

From G. C. Greubel, Oct 21 2024: (Start)

a(n) = Fibonacci(n+15) - Sum_{j=0..6} Fibonacci(14-2*j)*binomial(n+j,j).

a(n) = Fibonacci(n+15) - (1/6!)*(n^6 + 39*n^5 + 685*n^4 + 7185*n^3 + 48994*n^2 + 209496*n + 438480).

G.f.: 1/((1-x)^7*(1 - x - x^2)). (End)

Extensions

Terms a(28) onward added by G. C. Greubel, May 24 2018