A193600 Indices n such that Padovan(n) < r^n/(2*r+3) where r is the real root of the polynomial x^3-x-1.

1, 2, 4, 7, 9, 12, 14, 15, 17, 19, 20, 22, 25, 27, 30, 32, 33, 35, 37, 38, 40, 43, 45, 48, 50, 51, 53, 56, 58, 61, 63, 64, 66, 68, 69, 71, 74, 76, 79, 81, 82, 84, 86, 87, 89, 92, 94, 97, 99, 100, 102, 104, 105, 107, 110, 112, 113, 115, 117, 118, 120, 123

Offset: 1

Author

Francesco Daddi, Jul 31 2011

Keywords

• nonn

Comments

R is the so-called plastic number (A060006). Padovan(n) = (r^n)/(2r+3) + (s^n)/(2s+3) + (t^n)/(2t+3) where r (real), s, t are the three roots of x^3-x-1. Also Padovan(n) is asymptotic to r^n / (2*r+3).

Examples

			For n=25, Padovan(25) = A000931(25) = 200 < 200.023... = r^25/(2*r+3).
		

Crossrefs

Cf. A000931, A060006.

Programs

Mathematica

lim=200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n/(2*R + 3)], {n, lim}]; p = Rest[CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]]; Select[Range[lim], p[[#]] <= powers[[#]] &] (* T. D. Noe, Aug 01 2011 *)