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

0,2

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).

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

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 *)

Prog

(Magma) [2^(n-1)*(4^n-1): n in [0..30]]; // G. C. Greubel, Dec 27 2022
(SageMath)
A120689=BinaryRecurrenceSequence(10, -16, 0, 3)
[A120689(n) for n in range(31)] # G. C. Greubel, Dec 27 2022

Crossrefs

Cf. A016131, A081342.

Sequence in context: A332426 A043030 A178015 * A359614 A180426 A246246

Adjacent sequences: A120686 A120687 A120688 * A120690 A120691 A120692

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 25 2006

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

Status

approved