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%I #24 Apr 30 2020 10:05:20
%S 7,51,387,2943,22383,170235,1294731,9847143,74892951,569602179,
%T 4332138579,32948302095,250590001023,1905875101899,14495230812123,
%U 110244221191287,838468077093927,6377011953177555,48500691394138659,368874495293576607,2805493888166196879,21337327619448845211
%N Number of blue nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and 1 green node.
%C The series of green nodes in n-th power W exponentiation for all n<6 n blue 1 green, 2 edge per node graphs already corresponds with an existing OEIS sequence (empirical). For example the number of blue nodes in n-th power W exponentiation of a square containing 3 blue nodes and 1 green node corresponds to A163063.
%H George Strand Vajagich, <a href="https://youtu.be/o7_L_Mo-xpU">Youtube video explaining graph W multiplication</a>, YouTube video.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-3).
%F g(n) = g(n-1) + 2*a(n-1), a(n) = 2*g(n-1) + 7*a(n-1) with g(0) = 1 and b(0) = 7, where g(n) = A332211(n).
%F From _Colin Barker_, Mar 03 2020: (Start)
%F G.f.: (1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2).
%F a(n) = 8*a(n-1) - 3*a(n-2) for n > 1.
%F (End)
%F From _Stefano Spezia_, Mar 03 2020: (Start)
%F a(n) = ((4 - sqrt(13))^n*(-23 + 7*sqrt(13)) + (4 + sqrt(13))^n*(23 + 7*sqrt(13)))/(2*sqrt(13)).
%F E.g.f.: exp(4*x)*(91*cosh(sqrt(13)*x) + 23*sqrt(13)*sinh(sqrt(13)*x))/13.
%F (End)
%F a(n) = 7*A190976(n+1) -5*A190976(n). - _R. J. Mathar_, Apr 30 2020
%e For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 7 to get 357 add 30 to get 387.
%e For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 7 to get 774 add 234 to get 2943.
%o (Python)
%o g=1
%o b=7
%o sg=0
%o sb=0
%o bl=[]
%o gl=[]
%o for int in range(1,20):
%o sg=g*1+b*2
%o sb=b*7+g*2
%o g=sg
%o b=sb
%o gl.append(g)
%o bl.append(b)
%o print(bl)
%o (PARI) Vec((1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2) + O(x^40)) \\ _Colin Barker_, Mar 03 2020
%Y Cf. A331211.
%K nonn,easy
%O 0,1
%A _George Strand Vajagich_, Mar 02 2020
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