A000973 - OEIS

%I M4976 N2137 #59 Jun 24 2023 21:50:34

%S 1,15,99,429,1430,3978,9690,21318,43263,82225,148005,254475,420732,

%T 672452,1043460,1577532,2330445,3372291,4790071,6690585,9203634,

%U 12485550,16723070,22137570,28989675,37584261,48275865,61474519

%N Fermat coefficients.

%C a(n) = A258708(n,n-8). - _Reinhard Zumkeller_, Jun 23 2015

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

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

%H Vincenzo Librandi, <a href="/A000973/b000973.txt">Table of n, a(n) for n = 8..1000</a>

%H R. P. Loh, A. G. Shannon, A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980.

%H P. A. Piza, <a href="http://www.jstor.org/stable/3029103">Fermat coefficients</a>, Math. Mag., 27 (1954), 141-146.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -28, 56, -70, 56, -28, 8, -1).

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

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

%F 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;

%F 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

%p A000973:=(z+1)*(z**2+6*z+1)/(z-1)**8; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%p A000973:=n->binomial(2*n-8, 7)/8; seq(A000973(n), n=8..40); # _Wesley Ivan Hurt_, Apr 16 2014

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

%o (Magma) [Binomial(2*n-8, 7)/8: n in [8..40]]; // _Vincenzo Librandi_, Apr 10 2012

%o (Haskell)

%o a000973 n = a258708 n (n - 8)  -- _Reinhard Zumkeller_, Jun 23 2015

%Y Cf. A053129.

%Y Cf. A258708.

%K nonn,easy

%O 8,2

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_, Oct 11 2000