A296184 -id:A296184 - cp's OEIS frontend

A368649 a(n) = round(n*rho), where rho = (5+sqrt(5))/2.

4, 7, 11, 14, 18, 22, 25, 29, 33, 36, 40, 43, 47, 51, 54, 58, 62, 65, 69, 72, 76, 80, 83, 87, 90, 94, 98, 101, 105, 109, 112, 116, 119, 123, 127, 130, 134, 137, 141, 145, 148, 152, 156, 159, 163, 166, 170, 174, 177, 181, 185, 188, 192, 195, 199, 203, 206, 210

Offset: 1

Jeffrey Shallit, Jan 02 2024

nonn

Also, the positive integers that do not appear in A118287.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Benoit Cloitre and Jeffrey Shallit, Some Fibonacci-Related Sequences, arXiv:2312.11706 [math.CO], 2023-2024.

Cf. A003231, A118287, A296184.

Round[Range[100](5+Sqrt[5])/2] (* Paolo Xausa, Jan 03 2024 *)

a(n) = floor(n*(5+sqrt(5))/2 + 1/2).

A296183 Decimal expansion of (1/2)*sqrt(7 + phi), with the golden section from A001622.

Original entry on oeis.org

1, 4, 6, 7, 8, 2, 4, 4, 0, 9, 5, 2, 1, 6, 1, 3, 6, 2, 8, 0, 9, 8, 1, 6, 3, 7, 2, 6, 4, 6, 7, 1, 2, 1, 3, 3, 7, 5, 4, 2, 5, 6, 5, 5, 5, 9, 8, 8, 8, 4, 2, 0, 0, 2, 0, 5, 1, 0, 2, 9, 9, 2, 9, 7, 5, 2, 3, 2, 9, 4, 3, 8, 3, 3, 9, 9, 6, 9, 5, 4, 4, 9, 3, 8, 2, 1, 4, 5, 9, 9, 3, 8, 1, 8, 3, 4, 2, 7

Offset: 1

Wolfdieter Lang, Jan 08 2018

In a regular pentagon inscribed in a unit circle this equals the second largest distance between a vertex and a midpoint of a side. The shortest such distance is (1/2)*sqrt(3 - phi) = (1/2)*A182007 = 0.58778525229..., and the longest 1 + phi/2 = (1/2)*(2 + phi) = (1/2)*A296184 = 1.80901699437...

			1.467824409521613628098163726467121337542565559888420020510299297523294383...

Cf. A001622, A182007, A296184.

First@ RealDigits[Sqrt[7 + GoldenRatio]/2, 10, 98] (* Michael De Vlieger, Jan 13 2018 *)

(1/2)*sqrt(7 + phi). From the comment on the pentagon above this results from sqrt((5/4)^2 + (sqrt(3 - phi)/2 + sqrt(7 - 4*phi)/4)^2).

A308148 Number of length-n binary words avoiding (5+sqrt(5))/2-powers.

Original entry on oeis.org

1, 2, 4, 8, 14, 26, 48, 88, 160, 292, 532, 966, 1756, 3194, 5810, 10552, 19182, 34868, 63376, 115172, 209316, 380422, 691384, 1256538, 2283666, 4150402, 7542974, 13708740

Offset: 0

Jeffrey Shallit, May 14 2019

An e-power, where e is a real number, is a word of length n and period p such that n/p >= e. To avoid an e-power means that no subword (contiguous block) is an e-power.

			For n = 4, all length-4 binary words avoid (5+sqrt(5))/2 = 3.618... powers except 0000 and 1111.

Cf. A296184 ((5+sqrt(5))/2).

cp's OEIS Frontend

A368649 a(n) = round(n*rho), where rho = (5+sqrt(5))/2.

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A296183 Decimal expansion of (1/2)*sqrt(7 + phi), with the golden section from A001622.

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Programs

Mathematica

Formula

A308148 Number of length-n binary words avoiding (5+sqrt(5))/2-powers.

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