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A119633 a(n) = (A046717(n))^3. 2
1, 125, 2197, 68921, 1771561, 48627125, 1305751357, 35319837041, 953054410321, 25737699078125, 694870802988517, 18761935323400361, 506568440928284281, 13677382220238009125, 369289011109685057677, 9970806079491650694881, 269211739130501631841441 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Square root of M1 = A, the 8 X 8 matrix with the cyclotomic third roots of unity, mapped in a Gray code format. The 3 cyclotomic third roots of unity are (u, 1/u and 1), where u = (-1/2 + (sqrt(3)/2)i), = (1 Angle 120 deg.); and 1/u = (-1/2, -(sqrt(3)/2)i), = (1 Angle -120 deg.). Third root = 1. Thus A^2 = M1. A =

  [ 1,   u,  1/u,  u,  1/u,  1,  1/u,  u;

    u,   1,   u,  1/u,  1,  1/u,  u,  1/u;

   1/u,  u,   1,   u,  1/u,  u,  1/u,  1;

    u,  1/u,  u,   1,   u,  1/u,  1,  1/u;

   1/u,  1,  1/u,  u,   1,   u,  1/u,  u;

    1,  1/u,  u,  1/u,  u,   1,   u,  1/u;

   1/u,  u,  1/u,  1,  1/u,  u,   1,   u;

    u,  1/u,  1,  1/u,  u,  1/u,  u,   1].

A046717 can be generated from the analogous 2 X 2 matrix: P = [ -1,2; 2,-1 ], (which has the square root, [ u,1/u; 1/u;u ]). Then left term of P^n * [ 1,0 ], (unsigned) = 1, 5, 13, 41, 121...(where A046717 begins 1, 1, 5, 13, ...).

Pascal's triangle squared: (1; 2,1; 4,4,1; 8,12,6,1; ...) rows can be generated by taking the dot product of the distinct terms (..., 4, 2, 1) in rows or columns of the analogous "M" matrices and their frequency: e.g., row 1 of the 8 X 8 matrix (unsigned) = [1, 2, 4, 2, 4, 8, 4, 2] with a frequency for (8, 4, 2, 1) being (1, 3, 3, 1). Dot product = the (8, 12, 6, 1) row of Pascal's Triangle squared.

Third powers of A046717: (deleting the first "1": (1, 5, 13, 41, 121, 365, ...)).

Leftmost term (unsigned) of M1^n * [1,0,0,0,0,0,0,0]; where M1 = 8 X 8 matrix:

  [ -1,  2, -4,  2, -4,  8, -4,  2;

     2, -1,  2, -4,  8, -4,  2, -4;

    -4,  2, -1,  2, -4,  2, -4,  8;

     2, -4,  2, -1,  2, -4,  8, -4;

    -4,  8, -4,  2, -1,  2, -4,  2;

     8, -4,  2, -4,  2, -1,  2, -4;

    -4,  2, -4,  8, -4,  2, -1,  2;

     2, -4,  8, -4,  2, -4,  2, -1].

LINKS

Colin Barker, Table of n, a(n) for n = 1..600

Index entries for linear recurrences with constant coefficients, signature (20,210,-540,-729).

FORMULA

G.f.: x*(1 + 105*x - 513*x^2 - 729*x^3) / ((1 + x)*(1 - 3*x)*(1 + 9*x)*(1 - 27*x)). - R. J. Mathar, Sep 09 2008

a(n) = ((-1)^n + 3^(1+n) + (-1)^n*3^(1+2*n) + 27^n) / 8 for n>0. - Colin Barker, Dec 23 2017

EXAMPLE

a(3) = 2197 = 13^3 = (A046717(a))^3.

a(4) = 68921 = 41^3 = leftmost term of M1^n * [1,0,0,0,0,0,0,0].

MATHEMATICA

Rest@ Nest[Append[#, 2 #[[-1]] + 3 #[[-2]]] &, {1, 1}, 15]^3 (* or *)

Rest@ CoefficientList[Series[x (1 + 105 x - 513 x^2 - 729 x^3)/((1 + 9 x) (1 - 3 x) (1 - 27 x) (1 + x)), {x, 0, 16}], x] (* Michael De Vlieger, Dec 22 2017 *)

PROG

(PARI) Vec(x*(1 + 105*x - 513*x^2 - 729*x^3) / ((1 + x)*(1 - 3*x)*(1 + 9*x)*(1 - 27*x)) + O(x^40)) \\ Colin Barker, Dec 23 2017

CROSSREFS

Cf. A046717, A120096.

Sequence in context: A102061 A080169 A017127 * A017223 A265470 A017331

Adjacent sequences:  A119630 A119631 A119632 * A119634 A119635 A119636

KEYWORD

nonn,easy,uned

AUTHOR

Gary W. Adamson, Jun 09 2006

STATUS

approved

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Last modified November 13 11:09 EST 2018. Contains 317133 sequences. (Running on oeis4.)