A010868 -id:A010868 - cp's OEIS frontend

A195819 Multiples of 29.

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 870, 899, 928, 957, 986, 1015, 1044, 1073, 1102, 1131, 1160, 1189, 1218, 1247, 1276, 1305, 1334

Offset: 0

Omar E. Pol, Oct 12 2011

Length of hypotenuses on the main diagonal of the Pythagorean spiral whose edges have length A195033 and whose vertices are the numbers A195034, if n >= 1.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004922, A004942, A008605, A010868, A135631, A195033, A195034.

29Range[0,50] (* Harvey P. Dale, Oct 25 2011 *)

a(n) = 29*n.
From Elmo R. Oliveira, Mar 21 2024: (Start)
G.f.: 29*x/(x-1)^2.
E.g.f.: 29*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

Original entry on oeis.org

46, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

More generally, the ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2, k = 1, 2, 3,... is floor(phi^(2*k + 1))/(1 - x), and for the continued fraction expansion of phi^(2*k) is (floor(phi^(2*k)) + x - x^2)/(1 - x^2).

			phi^8 = (47 + 21*sqrt(5))/2 = 46 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/...)))))).

Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden ratio
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A001622.

Cf. continued fraction expansion of phi^k: A000012 (k = 1), A054977 (k = 2), A010709 (k = 3), A176260 (k = 4, for n>0), A010850 (k = 5), A040071 (k = 6, for n>0), A010868 (k = 7), this sequence (k = 8).

[46] cat &cat [[1, 45]^^50]; // Vincenzo Librandi, Jan 13 2016

ContinuedFraction[(47 + 21 Sqrt[5])/2, 82]

G.f.: (46 + x - x^2)/(1 - x^2).

a(n) = 23 + 22*(-1)^n for n>0. - Bruno Berselli, Jan 18 2016

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A195819 Multiples of 29.

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