A000967 - OEIS

%I M1148 N0437 #33 Sep 08 2022 08:44:28

%S 1,2,4,8,18,40,91,210,492,1165,2786,6710,16267,39650,97108,238824,

%T 589521,1459960,3626213,9030450,22542396,56393792,141358274,354975429,

%U 892893120,2249412290,5674891000,14335757256,36259245522,91815545800

%N Sum of Fermat coefficients.

%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 Reinhard Zumkeller, <a href="/A000967/b000967.txt">Table of n, a(n) for n = 1..1000</a> (corrected by Sean A. Irvine, April 17, 2019)

%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.

%F Following Piza's definition for the Fermat coefficients: (n:c)=binomial(2n-c, c-1)/c, a(n)= Round( sum_ {c=1..n} (n:c) ). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005

%F a(n) = rounded(sum(A258993(n,k)/(2*k+1)): k = 0..n-1). - _Reinhard Zumkeller_, Jun 22 2015

%e n...Sum_{c=1..n} (n:c).....a(n)

%e --------------------------------

%e .1........1.................1

%e .2........2.................2

%e .3........4.................4

%e .4........8+1/3.............8

%p FermatCoeff:=(n,c)->binomial(2*n-c,c-1)/c:seq(round(add(FermatCoeff(n,c),c=1..n)),n=1..40); # Pab Ter, Oct 13 2005

%t Table[Round[Sum[Binomial[n+k, n-k]/(2*k+1), {k, 0, n-1}]], {n,1,35}] (* _G. C. Greubel_, Apr 16 2019 *)

%o (Haskell)

%o import Data.Function (on)

%o a000967 n = round $ sum $

%o             zipWith ((/) `on` fromIntegral) (a258993_row n) [1, 3 ..]

%o -- _Reinhard Zumkeller_, Jun 22 2015

%o (PARI) {a(n) = round(sum(k=0,n-1, binomial(n+k,n-k)/(2*k+1)))}; \\ _G. C. Greubel_, Apr 16 2019

%o (Magma) [Round((&+[Binomial(n+k,n-k)/(2*k+1): k in [0..n-1]])): n in [1..35]]; // _G. C. Greubel_, Apr 16 2019

%o (Sage) [round(sum(binomial(n+k,n-k)/(2*k+1) for k in (0..n-1))) for n in (1..35)] # _G. C. Greubel_, Apr 16 2019

%Y Cf. A143858, A258708.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005