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%I #29 Feb 28 2023 09:11:16
%S 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,
%T 49,52,56,60,62,68,72,76,81,85,90,95,98,105,110,115,121,126,132,138,
%U 142,150,156,162,169,175,182,189,194,203,210,217,225,232,240
%N Degree of Somos4-oid polynomial s_n(x).
%H Colin Barker, <a href="/A241592/b241592.txt">Table of n, a(n) for n = 1..1000</a>
%H R. W. Gosper and Richard C. Schroeppel, <a href="http://arxiv.org/abs/math/0703470">Somos Sequence Near-Addition Formulas and Modular Theta Functions</a>, arXiv:math/0703470 [math.NT]. See pp. 10,11.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,1,-2,1).
%F Eight interlaced quadratic progressions: deg(s_{8q+r}) = (4q + r)q + [-2,0,0,0,1,1,2,3]_r, 0 <= r <= 7.
%F 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
%F a(n) = a(-n) for all n in Z. - _Michael Somos_, Jan 19 2015
%F a(4*n + 1) = A035608(n), a(4*n + 2) = A002378(n), a(4*n + 3) = A156859(n). - _Michael Somos_, Jan 19 2015
%e G.f. = x^4 + x^5 + 2*x^6 + 3*x^7 + 2*x^8 + 5*x^9 + 6*x^10 + 7*x^11 + ...
%t a[ n_] := If[ Divisible[ n, 8], -2 + n^2 / 16, Quotient[ 2 n^2 - 5 (-1)^n + 5, 32]]; (* _Michael Somos_, Jan 20 2015 *)
%t 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 *)
%t 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 *)
%o (PARI) {a(n) = if( n%8==0, -2 + n^2 / 16, (2*n^2 - 5*(-1)^n + 5) \ 32)}; /* _Michael Somos_, Jan 19 2015 */
%o (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
%o (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
%Y Cf. A002378, A006720, A035608, A156859.
%K nonn,easy
%O 1,6
%A _N. J. A. Sloane_, May 16 2014
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