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

Author

Robert G. Wilson v, Jan 11 2001

Keywords

• nonn

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

• Michael Somos, Somos Polynomials
• Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Crossrefs

Cf. A001840.

Programs

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 *)

PARI

{a(n) = binomial(n-3, 2)\3}; /* Michael Somos, Oct 18 2018 */

Sage

[floor(binomial(n,2)/3) for n in range(-2,59)] # Zerinvary Lajos, Dec 01 2009

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

Extensions

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