A111662 Expansion of x^2*(1-x)*(x^2+x+1)*(x^6+x^3+1)/((2*x-1)*(2*x^9-x^6+x^3-1)).

0, 0, 1, 2, 4, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4610, 9220, 18440, 36882, 73764, 147528, 295056, 590112, 1180224, 2360450, 4720900, 9441800, 18883606, 37767212, 75534424, 151068852, 302137704, 604275408, 1208550818, 2417101636

Offset: 0

Author

Creighton Dement, Aug 14 2005

Keywords

• nonn
• easy

Comments

Initial terms factored: [0,0,1,2,(2)^2,(3)^2,(2) (3)^2,(2)^2 (3)^2,(2)^3 (3)^2,(2)^4 (3)^2,(2)^5 (3)^2,(2)^6 (3)^2,(2)^7 (3)^2,(2)^8 (3)^2,(2) (5) (461),(2)^2 (5) (461),(2)^3 (5) (461),(2) (3)^3 (683),(2)^2 (3)^3 (683),(2)^3 (3)^3 (683),(2)^4 (3)^3 (683),(2)^5 (3)^3 (683),(2)^6 (3)^3 (683),(2) (5)^2 (17) (2777),(2)^2 (5)^2 (17) (2777),(2)^3 (5)^2 (17) (2777),(2) (7) (19) (70991),(2)^2 (7) (19) (70991),(2)^3 (7) (19) (70991),(2)^2 (3)^2 (11) (381487)]

Note that for each 3 terms in a row the sequence doubles: a(3*n+1) = 2*a(3*n) = 4*a(3*n-1). Andrew Howroyd, Mar 09 2024

Floretion Algebra Multiplication Program, FAMP Code: 2ibaseksumseq[.5'i + .5i' + .5'ii' + .5'jj' + .5'kk' + .5e], sumtype: sum[(Y[0], Y[1], Y[2]),mod(3)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2,0,-1,2,0,2,-4).

Crossrefs

Cf. A111663.

Programs

Mathematica

CoefficientList[Series[x^2*(1 - x)*(x^2 + x + 1)*(x^6 + x^3 + 1)/((2*x - 1)*(2*x^9 - x^6 + x^3 - 1)), {x, 0, 50}], x] (* G. C. Greubel,Jun 09 2017 *)
LinearRecurrence[{2,0,1,-2,0,-1,2,0,2,-4},{0,0,1,2,4,9,18,36,72,144,288,576},40] (* Harvey P. Dale, Mar 26 2024 *)

PARI

Vec(x^2*(1-x)*(x^2+x+1)*(x^6+x^3+1)/((2*x-1)*(2*x^9-x^6+x^3-1))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = 2*a(n-1) + a(n-3) - 2*a(n-4) - a(n-6) + 2*a(n-7) + 2*a(n-9) - 4*a(n-10) for n>11. - Colin Barker, May 11 2019