A136425 a(n) = floor((x^n-(1-x)^n)/sqrt(7)+1/2) where x = (sqrt(7)+1)/2.

1, 1, 3, 4, 8, 14, 25, 46, 84, 153, 279, 509, 927, 1691, 3082, 5618, 10241, 18667, 34028, 62029, 113070, 206113, 375719, 684889, 1248467, 2275800, 4148501, 7562201, 13784953, 25128255, 45805684, 83498067, 152206593, 277453693, 505763582

Offset: 1

Author

Cino Hilliard, Apr 01 2008

Keywords

• nonn

Comments

This is analogous to the formula for the n-th Fibonacci number. Even before truncation, these numbers are rational and the decimal part always ends in 5. For x = (sqrt(7)+1)/2, a(n)/a(n-1) -> x. The general form of x is (sqrt(r)+1)/2, r=1,2,3..

Programs

PARI

g(n,r) = for(m=1,n,print1(fib(m,r)",")) fib(n,r) = x=(sqrt(r)+1)/2;floor((x^n-(1-x)^n)/sqrt(r)+.5)

Formula

Asymptotically a(n) ~ A083099(n)/2^(n-1). - R. J. Mathar, Apr 20 2008

a(n) = floor(b(n)/2^n) where b(n) = 2*A083099(n)+2^(n-1). - R. J. Mathar, Sep 10 2016

Extensions

Definition corrected by R. J. Mathar, Apr 20 2008