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

1,3

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).

Links

Table of n, a(n) for n=1..34.

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

Example

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...

Mathematica

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

Crossrefs

Cf. A001622, A112576, A136319, A192232.

Sequence in context: A368346 A117079 A026745 * A099444 A374678 A132402

Adjacent sequences: A139330 A139331 A139332 * A139334 A139335 A139336

Keyword

nonn

Author

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

Extensions

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

Status

approved