OFFSET
0,2
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).
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)
MATHEMATICA
Table[Sum[Binomial[n+7-j, n-2*j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)
PROG
(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
(Magma) [(&+[Binomial(n+7-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Mar 04 2000
EXTENSIONS
Terms a(28) onward added by G. C. Greubel, May 24 2018
STATUS
approved