A136424 a(n) = floor((x^n - (1-x)^n) / (2x-1) +.5) where x = (sqrt(6)+1)/2 (and hence 2x-1 = sqrt(6)).

1, 1, 2, 4, 6, 11, 19, 32, 55, 95, 164, 283, 488, 842, 1451, 2503, 4318, 7447, 12844, 22152, 38207, 65898, 113657, 196029, 338101, 583137, 1005763, 1734685, 2991888, 5160244, 8900104, 15350410, 26475540, 45663552, 78757977, 135837417

Offset

1,3

Comments

This is analogous to the closed form of 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(6)+1)/2, a(n)/a(n-1) -> x.

Links

Table of n, a(n) for n=1..36.

Formula

The general form of x is (sqrt(r)+1)/2, r=1,2,3..

a(n) = floor(b(n)/2^n) where b(n) = 2*A002532(n)+2^(n-1). - R. J. Mathar, Sep 10 2016

Prog

(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)

Crossrefs

Sequence in context: A224957 A115992 A115993 * A116732 A367736 A048239

Adjacent sequences: A136421 A136422 A136423 * A136425 A136426 A136427

Keyword

nonn

Author

Cino Hilliard, Apr 01 2008

Extensions

Definition corrected by Frederic van der Plancke (fplancke(AT)hotmail.com), May 08 2009

Status

approved