A058937 Maximal exponent of x in all terms of Somos polynomial of order n.

1, 0, 0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551

Offset

1,7

Comments

This sequence differs from A001840 only in four terms preceding it. That is, A001840(n) = a(n+5).

Let b(n) = 2^a(n+1). Then b(1)=b(2)=b(3)=b(4)=1 and b(n)*b(n-4) = b(n-1)*b(n-3) + c(n)*b(n-2)^2, c(3*n)=2, c(3*n+1)=c(3*n+2)=1 for all n in Z. - Michael Somos, Oct 18 2018

Links

Table of n, a(n) for n=1..61.

Michael Somos, Somos Polynomials

Formula

a(n) = 1 + a(n-1) + a(n-3) - a(n-4) for all n in Z.

G.f.: x*(1-2*x+x^2-x^3+2*x^4)/((1+x+x^2)* (1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

a(n) = a(7-n) for all n in Z. - Michael Somos, Oct 18 2018

Mathematica

e[1] = 1; e[2] = e[3] = e[4] = e[5] = 0; e[n_] := e[n] = 1 + e[n - 1] + e[n - 3] - e[n - 4]; Table[e[n], {n, 1, 70}]
a[ n_] := Quotient[ Binomial[n - 3, 2], 3]; (* Michael Somos, Oct 18 2018 *)

Prog

(Sage) [floor(binomial(n, 2)/3) for n in range(-2, 59)] # Zerinvary Lajos, Dec 01 2009
(PARI) {a(n) = binomial(n-3, 2)\3}; /* Michael Somos, Oct 18 2018 */

Crossrefs

Cf. A001840.

Sequence in context: A211004 A062781 A145919 * A130518 A001840 A022794

Adjacent sequences: A058934 A058935 A058936 * A058938 A058939 A058940

Keyword

nonn

Author

Robert G. Wilson v, Jan 11 2001

Extensions

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009

Status

approved