A000967 Sum of Fermat coefficients. (Formerly M1148 N0437)

1, 2, 4, 8, 18, 40, 91, 210, 492, 1165, 2786, 6710, 16267, 39650, 97108, 238824, 589521, 1459960, 3626213, 9030450, 22542396, 56393792, 141358274, 354975429, 892893120, 2249412290, 5674891000, 14335757256, 36259245522, 91815545800

Offset

1,2

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (corrected by Sean A. Irvine, April 17, 2019)

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.

Formula

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

a(n) = rounded(sum(A258993(n,k)/(2*k+1)): k = 0..n-1). - Reinhard Zumkeller, Jun 22 2015

Example

n...Sum_{c=1..n} (n:c).....a(n)
--------------------------------
.1........1.................1
.2........2.................2
.3........4.................4
.4........8+1/3.............8

Maple

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

Mathematica

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

Prog

(Haskell)
import Data.Function (on)
a000967 n = round $ sum $
            zipWith ((/) `on` fromIntegral) (a258993_row n) [1, 3 ..]
-- Reinhard Zumkeller, Jun 22 2015
(PARI) {a(n) = round(sum(k=0, n-1, binomial(n+k, n-k)/(2*k+1)))}; \\ G. C. Greubel, Apr 16 2019
(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
(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

Crossrefs

Cf. A143858, A258708.

Sequence in context: A329971 A052910 A367659 * A288309 A096813 A058387

Adjacent sequences: A000964 A000965 A000966 * A000968 A000969 A000970

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved