A159612 - OEIS

%I #40 Feb 08 2022 23:14:02

%S 1,4,8,24,56,152,376,984,2488,6424,16376,42072,107576,275864,706168,

%T 1809624,4634296,11872792,30409976,77901144,199541048,511145624,

%U 1309309816,3353892312,8591131576,22006700824,56371227128,144398030424,369882938936,947475060632,2427006816376

%N INVERT transform of (1, 3, 1, 3, 1, ...).

%C 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

%H Colin Barker, <a href="/A159612/b159612.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1, 4).

%F G.f.: x*(1+3*x)/(1-x-4*x^2). - _Philippe Deléham_, Mar 01 2012

%F a(n) = a(n-1) + 4*a(n-2), a(1)=1, a(2)=4. - _Vincenzo Librandi_, Mar 11 2011

%F a(n+1) = Sum_{k=0..n} A119473(n,k)*3^k. - _Philippe Deléham_, Oct 05 2012

%F 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

%e a(4) = 24 = (1, 3, 1, 3) dot (8, 4, 1, 1) = (8 + 12, + 1 + 3).

%t LinearRecurrence[{1, 4}, {1, 4}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Jul 17 2011 *)

%o (PARI) Vec(x*(1+3*x)/(1-x-4*x^2) + O(x^40)) \\ _Colin Barker_, Dec 22 2016

%Y Cf. A119473.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Apr 17 2009