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A227849 a(n) = 2 * floor( 3/14 * n^2) if n even, a(n) = 2 * round( 3/14 * n^2) -1 if n odd.

%I A227849 #22 Sep 08 2022 08:46:05
%S A227849 0,-1,0,3,6,9,14,21,26,33,42,51,60,71,84,95,108,123,138,153,170,189,
%T A227849 206,225,246,267,288,311,336,359,384,411,438,465,494,525,554,585,618,
%U A227849 651,684,719,756,791,828,867,906,945,986,1029,1070,1113,1158,1203,1248
%N A227849 a(n) = 2 * floor( 3/14 * n^2) if n even, a(n) = 2 * round( 3/14 * n^2) -1 if n odd.
%C A227849 The degrees of the sequence of the G_2 polynomials defined by Bruce Westbury is conjectured to be a(n).
%H A227849 G. C. Greubel, <a href="/A227849/b227849.txt">Table of n, a(n) for n = 0..5000</a>
%H A227849 B. W. Westbury, <a href="https://arxiv.org/abs/math/0507112">Enumeration of non-positive planar trivalent graphs</a>, J. Algebraic Combin. 25 (2007), 357-373, arXiv:math/0507112 [math.CO], 2005.
%H A227849 B. W. Westbury, <a href="http://mathoverflow.net/questions/32020/">Finding recurrence relation for a sequence of polynomials</a>, MathOverflow, 15 July 2010.
%H A227849 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 0, 0, 1, -2, 1).
%F A227849 G.f.: (x^8 - 2*x^7 - 2*x^6 - 2*x^3 - 2*x^2 + x) / (x^9 - 2*x^8 + x^7 - x^2 + 2*x - 1).
%F A227849 G.f.: -x * (1 + x) * (1 - 3*x + x^2 - x^3 + x^4 - 3*x^5 + x^6) / ((1 - x)^2 * (1 - x^7)).
%F A227849 a(-n) = a(n). a(n+7) = a(n) + 3*(2*n + 7).
%e A227849 G.f. = -x + 3*x^3 + 6*x^4 + 9*x^5 + 14*x^6 + 21*x^7 + 26*x^8 + 33*x^9 + ...
%t A227849 CoefficientList[Series[(x^8-2*x^7-2*x^6-2*x^3-2*x^2+x)/(x^9-2*x^8+x^7 - x^2+2*x-1), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 08 2018 *)
%t A227849 LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,-1,0,3,6,9,14,21,26},60] (* _Harvey P. Dale_, Jul 26 2022 *)
%o A227849 (PARI) {a(n) = (n%2*7 + 3*n^2) \ 14 * 2 - n%2}
%o A227849 (PARI) {a(n) = (3*n^2 - [0, 10, 12, 6, 6, 12, 10][n%7 + 1]) / 7}
%o A227849 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^8-2*x^7-2*x^6-2*x^3-2*x^2+x)/(x^9-2*x^8+x^7-x^2+2*x-1))); // _G. C. Greubel_, Aug 08 2018
%Y A227849 Cf. A059710.
%K A227849 sign
%O A227849 0,4
%A A227849 _Michael Somos_, Oct 31 2013