A139333 Floor of entry (2,1) of [0,1; 1,phi]^n.

1, 1, 3, 7, 15, 32, 68, 144, 302, 633, 1328, 2783, 5832, 12219, 25604, 53648, 112408, 235529, 493503, 1034034, 2166605, 4539674, 9511953, 19930339, 41759920, 87499309, 183336777, 384144447, 804895549, 1686492803, 3533698227, 7404136642, 15513842972, 32506061868

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, Apr 13 2008

Keywords

• nonn

Comments

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).

Examples

			a(5) = 15 since [0,1, 1,phi]^5 = [7.472...,15.708...; 15.708...,32.888...].
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...
		

Crossrefs

Cf. A001622, A112576, A136319, A192232.

Programs

Mathematica

a[n_] := Floor[MatrixPower[{{0, 1}, {1, GoldenRatio}}, n][[1, 2]]]; Array[a, 35] (* Amiram Eldar, Jun 06 2025 *)

Formula

a(n)/a(n-1) tends to 2.095293... = exp(arcsinh(phi/2)) (A136319).

a(n) = floor(A192232(n) + A112576(n)*phi), where phi is the golden ratio (A001622). - Amiram Eldar, Jun 06 2025

Extensions

a(22) corrected and more terms added by Amiram Eldar, Jun 06 2025