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
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