A220838 Tropical version of Somos-4 sequence A006720.

-1, 0, 0, 0, 1, 1, 2, 3, 3, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 22, 25, 27, 30, 33, 35, 39, 42, 45, 49, 52, 56, 60, 63, 68, 72, 76, 81, 85, 90, 95, 99, 105, 110, 115, 121, 126, 132, 138, 143, 150, 156, 162, 169, 175, 182, 189, 195, 203, 210, 217, 225, 232

Offset: 1

Author

N. J. A. Sloane, Dec 23 2012

Keywords

• sign
• easy

Comments

Given the generalized Somos-4 sequence with variables s(1), s(2), s(3), s(4), u, v and recursion s(n) = (u*s(n-1)*s(n-3) + v*s(n-2)^2)/s(n-4), then s(n) is a Laurent polynomial with denominator b(n) := s(1)^a(n)*s(2)^a(n-1)*s(3)^a(n-2)*s(4)^a(n-3) for all n in Z. Moreover, s(n)*b(n) is an irreducible polynomial for all n in Z. - Michael Somos, Sep 16 2023

Examples

			G.f. = -x + x^5 + x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 5*x^10 + 6*x^11 + 7*x^12 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, arXiv:1207.6072 [nlin.SI], 2012, See Example 3.6.
• A. P. Fordy, Periodic Cluster Mutations and Related Integrable Maps, arXiv preprint arXiv:1403.8061 [math-ph], 2014.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. A006720, A236294.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*( x^6-x^5+x^4-x^2+2*x-1)/((1-x)^2*(1-x^8)))); // G. C. Greubel, Aug 10 2018

Maple

A118825x := proc(n)
    coeftayl((1-2*x+x^2)/(x^4+1),x=0,n) ;
end proc:
A056594 := proc(n)
    coeftayl(1/(x^2+1),x=0,n) ;
end proc:
A220838 := proc(n)
    -9/32-1/8*n+1/16*n^2+1/32*(-1)^n ;
    %+A118825x(n)/4 - A056594(n+3)/8  ;
end proc:
seq(A220838(n),n=0..80) ; # R. J. Mathar, Jan 30 2013

Mathematica

LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 1, -2, 1}, {-1, 0, 0, 0, 1, 1, 2, 3, 3, 5}, 62] (* Jean-François Alcover, Nov 26 2017 *)
a[ n_] := With[{m = n-1}, Floor[m^2/16] - Boole[Mod[m, 8] == 0]]; (* Michael Somos, Sep 16 2023 *)

PARI

{a(n) = if( n<1, n = 2-n); polcoeff( x * (x^6 - x^5 + x^4 - x^2 + 2*x - 1) / ( (1 - x)^2 * (1 - x^8) ) + x * O(x^n), n)} /* Michael Somos, Dec 27 2012 */

PARI

{a(n) = n--; n^2\16 - !(n%8)}; /* Michael Somos, Sep 16 2023 */

Formula

From Michael Somos, Dec 27 2012: (Start)

G.f.: x * (x^6 - x^5 + x^4 - x^2 + 2*x - 1) / ( (1 - x)^2 * (1 - x^8) ).

a(2-n) = a(n). (End)

Second difference has period 8. - Michael Somos, Dec 27 2012

a(n) = A236294(n-5) = max( a(n-1) + a(n-3), 2*a(n-2) ) - a(n-4) for all n in Z. - Michael Somos, Sep 16 2023