A180167 a(0) = 1, a(1) = 7; a(n)= 6*a(n-1) + 6*a(n-2) for n>1.
1, 7, 48, 330, 2268, 15588, 107136, 736344, 5060880, 34783344, 239065344, 1643092128, 11292944832, 77616221760, 533454999552, 3666427327872, 25199293964544, 173194327754496, 1190361730314240, 8181336348412416, 56230188472359936, 386469148924634112
Offset: 0
Examples
a(4) = 2268 = 6*a(3) + 6*a(2) = 6*330 + 6*48. Using the INVERT transform operation, a(3) = 330 = (205, 35, 6, 1) dot (1, 1, 7, 48) = (205 + 35 + 42 + 48), where (1, 6, 35, 205, 1200, ...) = A180033. G.f. = 1 + 7*x + 48*x^2 + 330*x^3 + 2268*x^4 + 15588*x^5 + 107136*x^6 + ...
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
- Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
- Index entries for linear recurrences with constant coefficients, signature (6,6).
Programs
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Mathematica
CoefficientList[Series[(1 + x)/(1 - 6 x - 6 x^2), {x, 0, 21}], x] (* Michael De Vlieger, Dec 16 2021 *) -
PARI
Vec((1 + x)/(1 - 6*x - 6*x^2) + O(x^50)) \\ Colin Barker, May 13 2016
Formula
G.f.: (1 + x)/(1 - 6*x - 6*x^2); = INVERT transform of A180033
a(n) = ((3-sqrt(15))^n*(-4+sqrt(15))+(3+sqrt(15))^n*(4+sqrt(15)))/(2*sqrt(15)). - Alexander R. Povolotsky, Aug 22 2010, corrected by Colin Barker, May 13 2016
E.g.f.: (4*sqrt(15)*sinh(sqrt(15)*x) + 15*cosh(sqrt(15)*x))*exp(3*x)/15. - Ilya Gutkovskiy, May 13 2016