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 A136425 a(n) = floor((x^n-(1-x)^n)/sqrt(7)+1/2) where x = (sqrt(7)+1)/2. 0

%I #7 Jan 16 2022 01:25:41

%S 1,1,3,4,8,14,25,46,84,153,279,509,927,1691,3082,5618,10241,18667,

%T 34028,62029,113070,206113,375719,684889,1248467,2275800,4148501,

%U 7562201,13784953,25128255,45805684,83498067,152206593,277453693,505763582

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

%C 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..

%F Asymptotically a(n) ~ A083099(n)/2^(n-1). - _R. J. Mathar_, Apr 20 2008

%F a(n) = floor(b(n)/2^n) where b(n) = 2*A083099(n)+2^(n-1). - _R. J. Mathar_, Sep 10 2016

%o (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)

%K nonn

%O 1,3

%A _Cino Hilliard_, Apr 01 2008

%E Definition corrected by _R. J. Mathar_, Apr 20 2008

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Last modified October 4 19:37 EDT 2023. Contains 365888 sequences. (Running on oeis4.)