OFFSET
0,2
COMMENTS
a(n) is the generalized Euler number of an (n+2)-dimensional hypercube: (number of vertices) - (number of edges) + (number of faces) = A000079(n+2) - A001787(n+2) + A001788(n+1). - Amiram Eldar, Nov 08 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jonathan F. Mason and Richard H. Hudson, A Generalization of Euler's Formula and its Connection to Fibonacci Numbers, in: Frederic T. Howard (ed.), Applications of Fibonacci Numbers, Volume 9: Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications, Springer, Dordrecht, 2004, pp. 177-185, alternative link.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
FORMULA
For the sequence 1, 1, 1, 2, 8, 32, ... we have a(n) = 2^n*(n^2-5n+8)/8. - Paul Barry, Jun 26 2003
From R. J. Mathar, Jan 04 2011:
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).
G.f.: (1-4*x+8*x^2)/(1-2*x)^3. (End)
E.g.f.: (1 + 2*x^2)*exp(2*x). - G. C. Greubel, May 16 2019
MATHEMATICA
Table[2^n (Binomial[n, 2]+1), {n, 0, 30}] (* Vincenzo Librandi, Dec 22 2016 *)
LinearRecurrence[{6, -12, 8}, {1, 2, 8}, 30] (* Harvey P. Dale, May 16 2019 *)
PROG
(Magma) [2^n*(Binomial(n, 2)+1): n in [0..30]]; // Vincenzo Librandi, Dec 22 2016
(PARI) {a(n) = 2^(n-1)*(n^2-n+2)}; \\ G. C. Greubel, May 16 2019
(Sage) [2^(n-1)*(n^2-n+2) for n in (0..30)] # G. C. Greubel, May 16 2019
(GAP) List([0..30], n-> 2^(n-1)*(n^2-n+2)); # G. C. Greubel, May 16 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 16 2000
STATUS
approved