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A159612 INVERT transform of (1, 3, 1, 3, 1, ...).

%I A159612 #52 Jun 08 2025 03:33:07
%S A159612 1,4,8,24,56,152,376,984,2488,6424,16376,42072,107576,275864,706168,
%T A159612 1809624,4634296,11872792,30409976,77901144,199541048,511145624,
%U A159612 1309309816,3353892312,8591131576,22006700824,56371227128,144398030424,369882938936,947475060632,2427006816376
%N A159612 INVERT transform of (1, 3, 1, 3, 1, ...).
%C A159612 The sequence 1,1,4,8,24,... is an eigensequence of the sequence triangle of 1,3,1,3,1,3,1,..., which is the Riordan array ((1+3x)/(1-x^2),x). - _Paul Barry_, Feb 10 2011
%C A159612 From _Sean A. Irvine_, Jun 07 2025: (Start)
%C A159612 Also, the number of walks of length n-1 starting at vertex 1 in the following graph:
%C A159612   0   2
%C A159612   |\ /|
%C A159612   | 1 |
%C A159612   |/ \|
%C A159612   4   3. (End)
%H A159612 Colin Barker, <a href="/A159612/b159612.txt">Table of n, a(n) for n = 1..1000</a>
%H A159612 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A159612 Silvana Ramaj, <a href="https://digitalcommons.georgiasouthern.edu/cgi/viewcontent.cgi?article=3464&amp;context=etd">New Results on Cyclic Compositions and Multicompositions</a>, Master's Thesis, Georgia Southern Univ., 2021. See p. 33.
%H A159612 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,4).
%F A159612 G.f.: x*(1+3*x)/(1-x-4*x^2). - _Philippe Deléham_, Mar 01 2012
%F A159612 a(n) = a(n-1) + 4*a(n-2), a(1)=1, a(2)=4. - _Vincenzo Librandi_, Mar 11 2011
%F A159612 a(n+1) = Sum_{k=0..n} A119473(n,k)*3^k. - _Philippe Deléham_, Oct 05 2012
%F A159612 a(n) = 2^(-3-n)*((1-sqrt(17))^n*(-5+3*sqrt(17)) + (1+sqrt(17))^n*(5+3*sqrt(17))) / sqrt(17) for n > 0. - _Colin Barker_, Dec 22 2016
%F A159612 a(n) = A006131(n)+3*A006131(n-1). - _R. J. Mathar_, Jun 07 2025
%F A159612 E.g.f.: (exp(x/2)*(51*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2)) - 51)/68. - _Stefano Spezia_, Jun 07 2025
%e A159612 a(4) = 24 = (1, 3, 1, 3) dot (8, 4, 1, 1) = (8 + 12, + 1 + 3).
%t A159612 LinearRecurrence[{1, 4}, {1, 4}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Jul 17 2011 *)
%o A159612 (PARI) Vec(x*(1+3*x)/(1-x-4*x^2) + O(x^40)) \\ _Colin Barker_, Dec 22 2016
%Y A159612 Cf. A006131, A119473.
%Y A159612 Cf. A026597 (vertices 0, 2, 3, 4), A384604 (missing edge {0,4}).
%K A159612 nonn,easy
%O A159612 1,2
%A A159612 _Gary W. Adamson_, Apr 17 2009