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A180167 a(0) = 1, a(1) = 7; a(n)= 6*a(n-1) + 6*a(n-2) for n>1.

%I A180167 #40 Mar 17 2025 22:12:19
%S A180167 1,7,48,330,2268,15588,107136,736344,5060880,34783344,239065344,
%T A180167 1643092128,11292944832,77616221760,533454999552,3666427327872,
%U A180167 25199293964544,173194327754496,1190361730314240,8181336348412416,56230188472359936,386469148924634112
%N A180167 a(0) = 1, a(1) = 7; a(n)= 6*a(n-1) + 6*a(n-2) for n>1.
%H A180167 Colin Barker, <a href="/A180167/b180167.txt">Table of n, a(n) for n = 0..1000</a>
%H A180167 Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, <a href="https://arxiv.org/abs/2401.13524">Combinatorics on words and generating Dirichlet series of automatic sequences</a>, arXiv:2401.13524 [math.CO], 2025. See p. 14.
%H A180167 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, J. Int. Seq. 18 (2015) # 15.4.7.
%H A180167 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,6).
%F A180167 G.f.: (1 + x)/(1 - 6*x - 6*x^2); = INVERT transform of A180033
%F A180167 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
%F A180167 a(n) = A057089(n) + A057089(n-1). - _R. J. Mathar_, Apr 04 2012
%F A180167 E.g.f.: (4*sqrt(15)*sinh(sqrt(15)*x) + 15*cosh(sqrt(15)*x))*exp(3*x)/15. - _Ilya Gutkovskiy_, May 13 2016
%e A180167 a(4) = 2268 = 6*a(3) + 6*a(2) = 6*330 + 6*48.
%e A180167 Using the INVERT transform operation, a(3) = 330 = (205, 35, 6, 1) dot
%e A180167 (1, 1, 7, 48) = (205 + 35 + 42 + 48), where (1, 6, 35, 205, 1200, ...) = A180033.
%e A180167 G.f. = 1 + 7*x + 48*x^2 + 330*x^3 + 2268*x^4 + 15588*x^5 + 107136*x^6 + ...
%t A180167 CoefficientList[Series[(1 + x)/(1 - 6 x - 6 x^2), {x, 0, 21}], x] (* _Michael De Vlieger_, Dec 16 2021 *)
%o A180167 (PARI) Vec((1 + x)/(1 - 6*x - 6*x^2) + O(x^50)) \\ _Colin Barker_, May 13 2016
%Y A180167 Cf. A057089, A180033.
%K A180167 nonn,easy
%O A180167 0,2
%A A180167 _Gary W. Adamson_, Aug 14 2010