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A180676 The Gi3 sums of the Pell-Jacobsthal triangle A013609.

%I A180676 #12 Sep 08 2022 08:45:54
%S A180676 1,1,1,1,17,337,6641,130865,2578785,50816737,1001378849,19732860833,
%T A180676 388849631729,7662550168241,150995835638929,2975477077435217,
%U A180676 58633827885912001,1155420016046016193,22768348266078953793
%N A180676 The Gi3 sums of the Pell-Jacobsthal triangle A013609.
%C A180676 The a(n+3) represent the Gi3 sums of the Pell-Jacobsthal triangle A013609. See A180662 for information about these giraffe and other chess sums.
%H A180676 G. C. Greubel, <a href="/A180676/b180676.txt">Table of n, a(n) for n = 0..750</a>
%H A180676 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (20,-6,4,-1).
%F A180676 a(n) = 20*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1, a(1)=1, a(2)=1 and a(3)=1.
%F A180676 a(n+3) = Sum_{k=0..n} A013609(n+3*k,4*k).
%F A180676 G.f.: (1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3+x^4).
%p A180676 nmax:=18: a(0):=1: a(1):=1: a(2):=1: a(3):=1: for n from 4 to nmax do a(n) := 20*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) od: seq(a(n),n=0..nmax);
%t A180676 LinearRecurrence[{20,-6,4,-1},{1,1,1,1},20] (* _Harvey P. Dale_, Jun 07 2015 *)
%o A180676 (PARI) my(x='x+O('x^30)); Vec((1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3 +x^4)) \\ _G. C. Greubel_, Jun 11 2019
%o A180676 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3+x^4) )); // _G. C. Greubel_, Jun 11 2019
%o A180676 (Sage) ((1-19*x-13*x^2-17*x^3)/(1-20*x+6*x^2-4*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 11 2019
%o A180676 (GAP) a:=[1,1,1,1];; for n in [5..30] do a[n]:=20*a[n-1]-6*a[n-2] + 4*a[n-3]-a[n-4]; od; a; # _G. C. Greubel_, Jun 11 2019
%Y A180676 Cf. A052942 (Gi1), A008999 (Gi2), this sequence (Gi3), A180677 (Gi4).
%Y A180676 Cf. A013609, A180662.
%K A180676 easy,nonn
%O A180676 0,5
%A A180676 _Johannes W. Meijer_, Sep 21 2010