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A258121
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Number of vertices of degree n in all Lucas cubes.
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1
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2, 5, 15, 39, 102, 267, 699, 1830, 4791, 12543, 32838, 85971, 225075, 589254, 1542687, 4038807, 10573734, 27682395, 72473451, 189737958, 496740423, 1300483311, 3404709510, 8913645219, 23336226147, 61095033222, 159948873519, 418751587335, 1096305888486, 2870166078123
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OFFSET
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0,1
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COMMENTS
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Column sums of A245960.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Sandi Klavzar, Michel Mollard and Marko Petkovsek, The degree sequence of Fibonacci and Lucas cubes, Discrete Math., Vol. 311, No. 14 (2011), pp. 1310-1322.
Index entries for linear recurrences with constant coefficients, signature (3,-1).
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FORMULA
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G.f.: (2-x)*(1+x^2)/(1-3*x+x^2).
a(n) = 3*F(2n+1) = 3*A001519(n+1) = A022086(2n+1) for n>=2; F(n) = A000045(n) are the Fibonacci numbers.
a(n) = F(n-1)^2 + F(n)^2 + F(n+1)^2 + F(n+2)^2 for n > 1, where F(n) is the n-th Fibonacci number (A000045). - Amiram Eldar, Jan 11 2022
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MAPLE
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g := (2-x)*(1+x^2)/(1-3*x+x^2): gser := series(g, x = 0, 35): seq(coeff(gser, x, n), n = 0 .. 32);
with(combinat): 2, 5, seq(3*fibonacci(2*n+1), n = 2 .. 32);
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MATHEMATICA
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CoefficientList[Series[(2 - x)*(1 + x^2)/(1 - 3 x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 19 2017 *)
Join[{2, 5}, LinearRecurrence[{3, -1}, {15, 39}, 30]] (* Vincenzo Librandi, Oct 19 2017 *)
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PROG
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(PARI) my(x='x+O('x^50)); Vec((2-x)*(1+x^2)/(1-3x+x^2)) \\ G. C. Greubel, Oct 19 2017
(Magma) I:=[2, 5, 15, 39]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 19 2017
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CROSSREFS
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Cf. A000045, A001519, A022086, A245960.
Sequence in context: A148339 A148340 A148341 * A242823 A059840 A280064
Adjacent sequences: A258118 A258119 A258120 * A258122 A258123 A258124
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KEYWORD
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nonn,easy
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AUTHOR
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Emeric Deutsch, Jun 23 2015
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STATUS
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approved
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