A000973 Fermat coefficients.
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
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).
Programs
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Haskell
a000973 n = a258708 n (n - 8) -- Reinhard Zumkeller, Jun 23 2015
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Magma
[Binomial(2*n-8, 7)/8: n in [8..40]]; // Vincenzo Librandi, Apr 10 2012
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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
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Mathematica
CoefficientList[Series[(1+7*x+7*x^2+x^3)/(1-x)^8,{x,0,40}],x] (* Vincenzo Librandi, Apr 10 2012 *)
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
Extensions
More terms from David W. Wilson, Oct 11 2000
Comments