A000973 Fermat coefficients. (Formerly M4976 N2137)

1, 15, 99, 429, 1430, 3978, 9690, 21318, 43263, 82225, 148005, 254475, 420732, 672452, 1043460, 1577532, 2330445, 3372291, 4790071, 6690585, 9203634, 12485550, 16723070, 22137570, 28989675, 37584261, 48275865, 61474519

Offset

8,2

Comments

a(n) = A258708(n,n-8). - Reinhard Zumkeller, Jun 23 2015

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 8..1000

R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.

P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.

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 (8, -28, 56, -70, 56, -28, 8, -1).

Formula

a(n) = binomial(2*n-8, 7)/8.

G.f.: (x^8)*(1+7*x+7*x^2+x^3)/(1-x)^8.

G.f.: A(x)= (1+7*x+7*x^2+x^3)/(x-1)^8 = 1 + 45*x/(G(0)-45*x), |x|<1; if |x|>1, G(0)=45*x;

G(k) = (k+1)*(2*k+3) + x*(k+5)*(2*k+9) - x*(k+1)*(k+6)*(2*k+3)*(2*k+11)/G(k+1); (continued fraction Euler's 1st kind, 1-step ). - Sergei N. Gladkovskii, Jun 15 2012

Maple

A000973:=(z+1)*(z**2+6*z+1)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation
A000973:=n->binomial(2*n-8, 7)/8; seq(A000973(n), n=8..40); # Wesley Ivan Hurt, Apr 16 2014

Mathematica

CoefficientList[Series[(1+7*x+7*x^2+x^3)/(1-x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 10 2012 *)

Prog

(Magma) [Binomial(2*n-8, 7)/8: n in [8..40]]; // Vincenzo Librandi, Apr 10 2012
(Haskell)
a000973 n = a258708 n (n - 8)  -- Reinhard Zumkeller, Jun 23 2015

Crossrefs

Cf. A053129.

Cf. A258708.

Sequence in context: A174383 A341396 A307158 * A034266 A087661 A319777

Adjacent sequences: A000970 A000971 A000972 * A000974 A000975 A000976

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, Oct 11 2000

Status

approved