A241592 Degree of Somos4-oid polynomial s_n(x).

0, 0, 0, 1, 1, 2, 3, 2, 5, 6, 7, 9, 10, 12, 14, 14, 18, 20, 22, 25, 27, 30, 33, 34, 39, 42, 45, 49, 52, 56, 60, 62, 68, 72, 76, 81, 85, 90, 95, 98, 105, 110, 115, 121, 126, 132, 138, 142, 150, 156, 162, 169, 175, 182, 189, 194, 203, 210, 217, 225, 232, 240

Offset: 1

Author

N. J. A. Sloane, May 16 2014

Keywords

• nonn
• easy

Examples

			G.f. = x^4 + x^5 + 2*x^6 + 3*x^7 + 2*x^8 + 5*x^9 + 6*x^10 + 7*x^11 + ...
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT]. See pp. 10,11.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. A002378, A006720, A035608, A156859.

Programs

Magma

I:=[0,0,0,1,1,2,3,2,5,6]; [n le 10 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-8)-2*Self(n-9)+Self(n-10): n in [1..70]]; // Vincenzo Librandi, Jan 20 2015

Mathematica

a[ n_] := If[ Divisible[ n, 8], -2 + n^2 / 16, Quotient[ 2 n^2 - 5 (-1)^n + 5, 32]]; (* Michael Somos, Jan 20 2015 *)
CoefficientList[Series[x^3 (1 - x + x^2 - 2 x^4 + 4 x^5 - 2 x^6) / ((1 - x)^2 (1 - x^8)), {x, 0, 70}], x] (* Vincenzo Librandi, Jan 20 2015 *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{0,0,0,1,1,2,3,2,5,6},70] (* Harvey P. Dale, Feb 28 2023 *)

PARI

{a(n) = if( n%8==0, -2 + n^2 / 16, (2*n^2 - 5*(-1)^n + 5) \ 32)}; /* Michael Somos, Jan 19 2015 */

PARI

concat(vector(3), Vec(x^4*(2*x^6-4*x^5+2*x^4-x^2+x-1)/((x-1)^3*(x+1)*(x^2+1)*(x^4+1)) + O(x^100))) \\ Colin Barker, Jul 17 2015

Formula

Eight interlaced quadratic progressions: deg(s_{8q+r}) = (4q + r)q + [-2,0,0,0,1,1,2,3]_r, 0 <= r <= 7.

G.f.: x^4 * (1 - x + x^2 - 2*x^4 + 4*x^5 - 2*x^6) / ((1 - x)^2 * (1 - x^8)). - Michael Somos, Jan 19 2015

a(n) = a(-n) for all n in Z. - Michael Somos, Jan 19 2015

a(4*n + 1) = A035608(n), a(4*n + 2) = A002378(n), a(4*n + 3) = A156859(n). - Michael Somos, Jan 19 2015