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A331211 Number of green nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and one green node.

%I A331211 #42 Apr 26 2020 15:17:17
%S A331211 1,15,117,891,6777,51543,392013,2981475,22675761,172461663,1311666021,
%T A331211 9975943179,75872547369,577052549415,4388802753213,33379264377459,
%U A331211 253867706760033,1930803860947887,14684827767302997,111686210555580315,849435201142733529,6460422977475127287
%N A331211 Number of green nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and one green node.
%H A331211 Colin Barker, <a href="/A331211/b331211.txt">Table of n, a(n) for n = 0..1000</a>
%H A331211 Thezebraherd user, <a href="https://youtu.be/o7_L_Mo-xpU">Graph W multiplication</a>, Youtube video.
%H A331211 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-3).
%F A331211 a(n) = a(n-1) + 2*b(n-1), b(n) = 2*a(n-1) + 7*b(n-1) with a(0) = 1 and b(0) = 7 where b(n) = A332936(n).
%F A331211 From _Colin Barker_, Mar 03 2020: (Start)
%F A331211 G.f.: (1 + 7*x) / (1 - 8*x + 3*x^2).
%F A331211 a(n) = 8*a(n-1) - 3*a(n-2) for n>1.
%F A331211 (End)
%F A331211 From _Stefano Spezia_, Mar 03 2020: (Start)
%F A331211 a(n) = ((4 - sqrt(13))^n*(-11 + sqrt(13)) + (4 + sqrt(13))^n*(11 + sqrt(13)))/(2*sqrt(13)).
%F A331211 E.g.f.: exp(4*x)*cosh(sqrt(13)*x) + (11*exp(4*x)*sinh(sqrt(13)*x))/sqrt(13).
%F A331211 (End)
%e A331211 For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 2 to get 102 add 15 to get 117.
%e A331211 For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 2 to get 774 add 177 to get 891.
%o A331211 (Python)
%o A331211 g=1
%o A331211 b=7
%o A331211 sg=0
%o A331211 sb=0
%o A331211 bl=[]
%o A331211 gl=[]
%o A331211 for int in range(1,20):
%o A331211   sg=g*1+b*2
%o A331211   sb=b*7+g*2
%o A331211   g=sg
%o A331211   b=sb
%o A331211   gl.append(g)
%o A331211   bl.append(b)
%o A331211 print(gl)
%o A331211 (PARI) Vec((1 + 7*x) / (1 - 8*x + 3*x^2) + O(x^20)) \\ _Colin Barker_, Mar 03 2020
%Y A331211 Cf. A332936 (number of blue nodes).
%Y A331211 Similar sequences with a cycle size 3..6 are: A007483, A048876, A189274(n+1), A054490.
%K A331211 nonn,easy
%O A331211 0,2
%A A331211 _George Strand Vajagich_, Mar 01 2020
%E A331211 a(14)-a(21) from _Stefano Spezia_, Mar 03 2020
%E A331211 Typo in a(14) fixed by _Colin Barker_, Apr 26 2020