|
%I #14 Sep 08 2022 08:46:19
%S -3,-1,0,0,0,3,5,6,8,12,15,17,21,26,30,33,39,45,50,54,62,69,75,81,90,
%T 98,105,113,123,132,140,150,161,171,180,192,204,215,225,239,252,264,
%U 276,291,305,318,332,348,363,377,393,410,426,441,459,477,494,510,530
%N a(n) = negative of exponent of s(1) in factorization of s(n) where s(1)^2 * s(n+2) * s(n-2) = s(2)^2 * s(n+1) * s(n-1) - s(3) * s(1) * s(n)^2, s(n) = -s(-n) for all n in Z, with a(0) = -3, and s(1), s(2), s(3), s(4) are variables.
%C The sequence s(n) is a generalized Somos-4 sequence. Also, s(n) is a Laurent polynomial in the variables s(1), s(2), s(3), s(4) with only powers of s(1), s(2) appearing in the denominator.
%H G. C. Greubel, <a href="/A285631/b285631.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-2,1).
%F a(n) = a(n+19) - 6*n - 57 = a(-n) for all n in Z.
%F 0 = (a(n) - 2*a(n-1) + a(n-2) + 1) * (a(n) - 2*a(n+1) + a(n+2) + 1) for all n in Z.
%F G.f.: (-3 + 5*x - x^2 - x^3 + 3*x^5 - x^6 - x^7 + x^8 + 2*x^9 - x^10 - x^11 + 2*x^12 + x^13 - x^14 - x^15 + 3*x^16 - x^18 + 2*x^19 - x^20) / ((1 - x)^2 * (1 - x^19)).
%e G.f. = -3 - x + 3*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 12*x^9 + 15*x^10 + 17*x^11 + ...
%e Let s(1) = s1, s(2) = s2, s(3) = s3, s(4) = s4. Then s(5) = (s4*s2^3 -s3^3*s1) / s1^3 gives a(5) = -3, s(6) = s3*(s4*s2^5 -s4^2*s1^4 -s1*s2^2*s3^3) / (s2*s1^5) gives a(6) = -5, and s(7) = (-s4^3*s1^3*s2 +s4*s2^3*s3^3 -s1*s3^6) / s1^6 gives a(7) = 6.
%t a[ n_] := (3 n^2 + {35, 45, 30, 9, 39, 44, 24, 17, 42, 42, 17, 24, 44, 39, 9, 30, 45, 35, 0}[[Mod[n, 19, 1]]]) / 19 - 3;
%t LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-2,1},{-3,-1,0,0,0,3,5,6,8,12,15,17,21,26,30,33,39,45,50,54,62},60] (* _Harvey P. Dale_, Aug 05 2021 *)
%o (PARI) {a(n) = (3*n^2 + [0, 35, 45, 30, 9, 39, 44, 24, 17, 42, 42, 17, 24, 44, 39, 9, 30, 45, 35] [n%19+1]) / 19 - 3};
%o (PARI) {a(n) = my(m=abs(n)); polcoeff( (-3 + 5*x - x^2 - x^3 + 3*x^5 - x^6 - x^7 + x^8 + 2*x^9 - x^10 - x^11 + 2*x^12 + x^13 - x^14 - x^15 + 3*x^16 - x^18 + 2*x^19 - x^20) / ((1 - x)^2 * (1 - x^19)) + x * O(x^m), m)};
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-3 +5*x-x^2-x^3+3*x^5-x^6-x^7+x^8+2*x^9-x^10-x^11+2*x^12+x^13-x^14-x^15 + 3*x^16-x^18+2*x^19-x^20)/((1-x)^2*(1-x^19)))); // _G. C. Greubel_, Jul 29 2018
%Y Cf. A285640.
%K sign,easy
%O 0,1
%A _Michael Somos_, Apr 22 2017
|
