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A258121
Number of vertices of degree n in all Lucas cubes.
1
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
OFFSET
0,1
COMMENTS
Column sums of A245960.
LINKS
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.
FORMULA
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
MAPLE
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);
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 23 2015
STATUS
approved