This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A139333 #12 Jun 06 2025 08:34:35
%S A139333 1,1,3,7,15,32,68,144,302,633,1328,2783,5832,12219,25604,53648,112408,
%T A139333 235529,493503,1034034,2166605,4539674,9511953,19930339,41759920,
%U A139333 87499309,183336777,384144447,804895549,1686492803,3533698227,7404136642,15513842972,32506061868
%N A139333 Floor of entry (2,1) of [0,1; 1,phi]^n.
%C A139333 Floor of entry (2,1) and (1,2) of [0,1; 1,phi]^n. Floor numerators of barover[phi] = [phi, phi, phi,...] where phi = 1.618033989... (A001622).
%F A139333 a(n)/a(n-1) tends to 2.095293... = exp(arcsinh(phi/2)) (A136319).
%F A139333 a(n) = floor(A192232(n) + A112576(n)*phi), where phi is the golden ratio (A001622). - _Amiram Eldar_, Jun 06 2025
%e A139333 a(5) = 15 since [0,1, 1,phi]^5 = [7.472...,15.708...; 15.708...,32.888...].
%e A139333 a(5) = 15 = floor 15.708203..., since the first five numerators of continued fraction[phi, phi, phi, phi, phi,...] = [1, phi, (phi^2 + 1), (phi^3 + 2*phi), (phi^4 + 3*phi^2 + 1), where (phi^4 + 3*phi^2 + 1) = 15.708203...
%t A139333 a[n_] := Floor[MatrixPower[{{0, 1}, {1, GoldenRatio}}, n][[1, 2]]]; Array[a, 35] (* _Amiram Eldar_, Jun 06 2025 *)
%Y A139333 Cf. A001622, A112576, A136319, A192232.
%K A139333 nonn
%O A139333 1,3
%A A139333 _Gary W. Adamson_ and _Roger L. Bagula_, Apr 13 2008
%E A139333 a(22) corrected and more terms added by _Amiram Eldar_, Jun 06 2025