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 A332936 Number of blue nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and 1 green node. 1
 7, 51, 387, 2943, 22383, 170235, 1294731, 9847143, 74892951, 569602179, 4332138579, 32948302095, 250590001023, 1905875101899, 14495230812123, 110244221191287, 838468077093927, 6377011953177555, 48500691394138659, 368874495293576607, 2805493888166196879, 21337327619448845211 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. LINKS George Strand Vajagich, Youtube video explaining graph W multiplication, YouTube video. Index entries for linear recurrences with constant coefficients, signature (8,-3). FORMULA 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). From Colin Barker, Mar 03 2020: (Start) G.f.: (1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2). a(n) = 8*a(n-1) - 3*a(n-2) for n > 1. (End) From Stefano Spezia, Mar 03 2020: (Start) a(n) = ((4 - sqrt(13))^n*(-23 + 7*sqrt(13)) + (4 + sqrt(13))^n*(23 + 7*sqrt(13)))/(2*sqrt(13)). E.g.f.: exp(4*x)*(91*cosh(sqrt(13)*x) + 23*sqrt(13)*sinh(sqrt(13)*x))/13. (End) a(n) = 7*A190976(n+1) -5*A190976(n). - R. J. Mathar, Apr 30 2020 EXAMPLE For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 7 to get 357 add 30 to get 387. For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 7 to get 774 add 234 to get 2943. PROG (Python) g=1 b=7 sg=0 sb=0 bl=[] gl=[] for int in range(1, 20):   sg=g*1+b*2   sb=b*7+g*2   g=sg   b=sb   gl.append(g)   bl.append(b) print(bl) (PARI) Vec((1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2) + O(x^40)) \\ Colin Barker, Mar 03 2020 CROSSREFS Cf. A331211. Sequence in context: A285880 A147958 A104454 * A222849 A273055 A019472 Adjacent sequences:  A332933 A332934 A332935 * A332937 A332938 A332939 KEYWORD nonn,easy AUTHOR George Strand Vajagich, Mar 02 2020 STATUS approved

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Last modified June 12 09:24 EDT 2021. Contains 344946 sequences. (Running on oeis4.)