A000195 -id:A000195 - cp's OEIS frontend

A108171 Tribonacci version of A076662 using beta positive real Pisot root of x^3 - x^2 - x - 1.

4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 3, 3

Offset: 0

Roger L. Bagula, Jun 13 2005

Three part composition of sequence based on the Fibonacci substitution twos order.

Cf. A007066, A076662, A000195.

NSolve[x^3 - x^2 - x - 1 = 0, x] beta = 1.8392867552141612 a[n_] = 1 + Ceiling[(n - 1)*beta^2] (* A007066 like*) aa = Table[a[n], {n, 1, 100}] (* A076662-like *) b = Table[a[n] - a[n - 1], {n, 2, Length[aa]}] F[1] = 2; F[n_] := F[n] = F[n - 1] + b[[n]] (* A000195-like *) c = Table[F[n], {n, 1, Length[b] - 1}]

b(n) = 1 + ceiling((n-1)*beta); a(n) = b(n) - b(n-1).

A113160 Table read by antidiagonals of continued fractions for log(n).

Original entry on oeis.org

0, 1, 1, 2, 10, 1, 3, 7, 2, 1, 1, 9, 1, 1, 1, 6, 2, 1, 1, 1, 1, 3, 2, 2, 1, 3, 1, 2, 1, 1, 3, 1, 1, 17, 12, 2, 1, 3, 7, 3, 4, 2, 1, 5, 2, 2, 1, 6, 1, 18, 19, 1, 14, 3, 2, 1, 32, 4, 1, 2, 1, 2, 4, 3, 2, 2, 1, 2, 1, 1, 330, 11, 2, 1, 3, 1, 2, 2, 1, 17, 1, 3, 3, 2, 1, 2, 1, 1, 16, 1, 2, 1, 1, 21, 4, 1, 1, 8, 2

Offset: 2

Franklin T. Adams-Watters, Jan 05 2006

			The table starts:
0,1,2,3,1,6,...
1,10,7,9,2,2,...
1,2,1,1,2,3,...

G. Xiao, Contfrac
Index entries for continued fractions for constants

Rows A016730 to A016740, A016441 to A016528; columns A000195, A057603.

A371672 a(n) = floor(log_phi(n)) with phi = A001622.

Original entry on oeis.org

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

Offset: 1

Aleksandar V. Petojevic, Apr 02 2024

nonn

Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.

Cf. A000195, A000523, A001622, A004216, A362692.

Table[IntegerPart[Log[n]/Log[GoldenRatio]],{n,1,130}]

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A108171 Tribonacci version of A076662 using beta positive real Pisot root of x^3 - x^2 - x - 1.

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A113160 Table read by antidiagonals of continued fractions for log(n).

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A371672 a(n) = floor(log_phi(n)) with phi = A001622.

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