OFFSET
1,3
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).
FORMULA
a(n+1) = 3*(3^n - 2*2^n + 1)/2 + 113*(4^n - 3*3^n + 3*2^n - 1)/6 + 625*(5^n - 4*4^n + 6*3^n - 4*2^n + 1)/24. - formula fitted by John W. Layman
a(n) = (125/24) * 5^n - (64/3) * 4^n + (135/4)*3^n - (76/3) * 2^n + 209/24 proven in McMorris and Zaslavsky, matches Layman's formula with an offset of 1. - Sean A. Irvine, Apr 12 2016
E.g.f.: (1/24)*exp(x)*(-1 + exp(x))^2*(209 - 798*exp(x) + 625*exp(2*x)). - Ilya Gutkovskiy, Apr 12 2016
G.f.: x^3*(3 + 86*x + 120*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)). - Andrew Howroyd, Mar 28 2025
MAPLE
A005175:=-z**2*(3+86*z+120*z**2)/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(125/24) 5^n - (64/3) 4^n + (135/4) 3^n - (76/3) 2^n + 209/24, {n, 20}] (* Michael De Vlieger, Apr 12 2016 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Name clarified by Andrew Howroyd, Mar 28 2025
STATUS
approved