A110046 Expansion of (1+4*x-12*x^2-16*x^3)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)).
1, 8, 28, 144, 656, 3200, 15296, 73984, 356608, 1722368, 8313856, 40144896, 193826816, 935886848, 4518821888, 21818834944, 105350496256, 508677324800, 2456110759936, 11859152338944, 57261050298368, 276480810549248
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Robert Munafo, Sequences Related to Floretions
- Index entries for linear recurrences with constant coefficients, signature (4, 8, -16, -16).
Programs
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Maple
seriestolist(series((1+4*x-12*x^2-16*x^3)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'
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Mathematica
CoefficientList[Series[(1+4x-12x^2-16x^3)/((2x+1)(2x-1)(4x^2+4x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{4,8,-16,-16},{1,8,28,144},40] (* Harvey P. Dale, Jun 12 2016 *) -
PARI
Vec((1 + 4*x - 12*x^2 - 16*x^3) / ((1 - 2*x)*(1 + 2*x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, May 01 2019
Formula
From Colin Barker, May 01 2019: (Start)
a(n) = ((2 - 2*sqrt(2))^(1+n) + 2*(-(-2)^n + 2^n + 2^n*(1+sqrt(2))^(1+n))) / 4.
a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4) for n>3.
(End)