A120689 a(n) = 10*a(n-1) - 16*a(n-2), with a(0) = 0 and a(1) = 3.

0, 3, 30, 252, 2040, 16368, 131040, 1048512, 8388480, 67108608, 536870400, 4294966272, 34359736320, 274877902848, 2199023247360, 17592186028032, 140737488322560, 1125899906777088, 9007199254609920, 72057594037665792

Offset: 0

Author

Gary W. Adamson, Jun 25 2006

Keywords

• nonn
• easy

Comments

a(n) is a leg in a Pythagorean triangle along with A081342(n) (the hypotenuse) and 4^n. Example: a(4) = 2040, A081342(4) = 2056; then sqrt(2056^2 - 2040^2) = 256 = 4^4. Characteristic polynomial of M = x^2 -10*x + 16.

Order of modular group of degree 2^(n-1)+1. - Artur Jasinski, Aug 04 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• E. Mathieu, Mémoire sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses variables de toutes les manières possibles, Journ. de math. pures et appliquées (2) 5 (1860), 9-42 (see p. 39).
• Index entries for linear recurrences with constant coefficients, signature (10,-16).

Crossrefs

Cf. A016131, A081342.

Programs

Magma

[2^(n-1)*(4^n-1): n in [0..30]]; // G. C. Greubel, Dec 27 2022

Maple

a[0]:=0: a[1]:=3; for n from 2 to 20 do a[n]:=10*a[n-1]-16*a[n-2] end do: seq(a[n], n = 0 .. 20); # Emeric Deutsch, Aug 16 2007
seq(binomial(2^n,2)*(2^n + 1),n=0..19); # Zerinvary Lajos, Jan 07 2008

Mathematica

Table[2^(n-1) (4^n-1), {n,0,20}] (* Artur Jasinski, Aug 04 2007 *)

SageMath

A120689=BinaryRecurrenceSequence(10,-16,0,3)
[A120689(n) for n in range(31)] # G. C. Greubel, Dec 27 2022

Formula

a(n) = 8^n - A081342(n).

Given M = 2 X 2 matrix [5,3; 3,5]; M^n * [1,0] = [A081342(a), a(n)]. E.g. a(4) = 2040, right term in = M^4 * [1,0] = [2056, 2040] = [A081342(4), a(4)].

a(n) = 2^(n-1)*(4^n - 1). - Artur Jasinski, Aug 04 2007

From R. J. Mathar, Feb 16 2011: (Start)

a(n) = 3*A016131(n-1).

G.f.: 3*x / ( (1-2*x)*(1-8*x) ). (End)

E.g.f.: (1/2)*(exp(8*x) - exp(2*x)). - G. C. Greubel, Dec 27 2022

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 13 2007

More terms from Emeric Deutsch, Aug 16 2007