A136421 - OEIS

%I #14 Sep 08 2022 08:45:32

%S 1,1,1,2,2,2,3,3,4,5,6,7,8,10,12,14,17,21,25,31,37,44,54,65,78,94,114,

%T 138,166,200,242,292,352,425,514,620,748,903,1090,1316,1589,1918,2315,

%U 2794,3373,4072,4915,5933,7162,8645,10436,12597,15206,18355,22156,26745

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

%C 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(2)+1)/2, a(n)/a(n-1) -> x.

%H G. C. Greubel, <a href="/A136421/b136421.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = floor(b(n)/2^n) where b(n) = A052542(n) + 2^(n-1) = 4*b(n-1) - 3*b(n-2) - 2*b(n-3). - _R. J. Mathar_, Sep 10 2016

%t Table[Floor[Fibonacci[n, 2]/2^(n-1) +1/2], {n,1,50}] (* _G. C. Greubel_, Oct 02 2018 *)

%o (PARI) fib(n,r) = x=(sqrt(r)+1)/2;floor((x^n-(1-x)^n)/sqrt(r)+.5);

%o g(n,r) = for(m=1,n,print1(fib(m,r)", "));

%o g(30,2)

%o (Magma) [Floor(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2^n*Sqrt(2))+ 1/2): n in [2..50]]; // _G. C. Greubel_, Oct 02 2018

%K nonn

%O 1,4

%A _Cino Hilliard_, Apr 01 2008