cp's OEIS Frontend

A090381 Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)).

%I A090381 #64 Sep 08 2022 08:45:12
%S A090381 1,6,19,36,61,90,127,168,217,270,331,396,469,546,631,720,817,918,1027,
%T A090381 1140,1261,1386,1519,1656,1801,1950,2107,2268,2437,2610,2791,2976,
%U A090381 3169,3366,3571,3780,3997,4218,4447,4680,4921,5166,5419,5676,5941,6210,6487,6768,7057,7350,7651,7956,8269
%N A090381 Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)).
%C A090381 Also degree of toric ideal associated with path with n+2 nodes [Eriksson].
%C A090381 Also number of triples (t_1, t_2, t_3) with all t_i in the range 0 <= t_i <= n such that at least one t_i + t_j = n (with i != j). - _R. H. Hardin_, Aug 04 2014
%C A090381 Conjecture: a(n) is the maximum number of areas that are defined by the 3n angle (n+1)-sectors in a triangle. - _Nicolas Nagel_, Sep 09 2016
%H A090381 R. H. Hardin and N. J. A. Sloane, <a href="/A090381/b090381.txt">Table of n, a(n) for n = 0..1000</a> [First 210 terms from Hardin]
%H A090381 N. Eriksson, <a href="http://arXiv.org/abs/math.CO/0401175">Toric ideals of homogeneous phylogenetic models</a>, arXiv:math/0401175 [math.CO], 2004.
%H A090381 Nicolas Nagel, <a href="/A090381/a090381.jpg">Example picture for angle (n+1)-sectors</a>
%H A090381 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A090381 G.f.: (1+4x+7x^2)/((1-x)^2*(1-x^2)).
%F A090381 a(2t) = 12t^2+6t+1, a(2t+1) = 12t^2+18t+6 (t >= 0).
%F A090381 The defining g.f. implies the recurrence a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), an empirical discovery of _R. H. Hardin_.
%F A090381 a(n) = 3*n*(n+1)+(1+(-1)^n)/2. - _Wesley Ivan Hurt_, May 06 2016
%F A090381 E.g.f.: 3*x*(2 + x)*exp(x) + cosh(x). - _Ilya Gutkovskiy_, May 06 2016
%e A090381 Some triples for n=10 (from _R. H. Hardin_, Aug 04 2014):
%e A090381 ..3....1....2....1....7....9....5....8....5....6....9....4...10....8....6....2
%e A090381 ..3....3....8....9....3....3....7....2....9....4....3...10....9....1....8....7
%e A090381 ..7....7...10....5....2....1....3....7....1....3....7....0....1....9....4....8
%p A090381 f:=n-> if n mod 2 = 0 then t:=n/2; 12*t^2+6*t+1 else
%p A090381 t:=(n-1)/2; 12*t^2+18*t+6; fi;
%p A090381 [seq(f(n), n=0..100)];
%t A090381 CoefficientList[Series[(1 + 4 x + 7 x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 52}], x] (* _Michael De Vlieger_, May 07 2016 *)
%t A090381 Table[3 n (n + 1) + (1 + (-1)^n)/2, {n, 0, 52}] (* or *)
%t A090381 LinearRecurrence[{2, 0, -2, 1}, {1, 6, 19, 36}, 53] (* _Michael De Vlieger_, Sep 12 2016 *)
%o A090381 (PARI) x='x+O('x^99); Vec((1+4*x+7*x^2)/((1-x)^2*(1-x^2))) \\ _Altug Alkan_, May 12 2016
%o A090381 (Magma) [3*n*(n+1)+(1+(-1)^n)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 13 2016
%Y A090381 Cf. A090382, A090383, A090384, A090385.
%Y A090381 Row 1 of A245869.
%Y A090381 Central spine of triangle in A245556. Cf. also A245557.
%K A090381 nonn,easy
%O A090381 0,2
%A A090381 _N. J. A. Sloane_, Jan 30 2004
%E A090381 Edited by _N. J. A. Sloane_, Aug 04 2014 (merging the old A090381 and A245870).