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 A090381 Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)). 7
 1, 6, 19, 36, 61, 90, 127, 168, 217, 270, 331, 396, 469, 546, 631, 720, 817, 918, 1027, 1140, 1261, 1386, 1519, 1656, 1801, 1950, 2107, 2268, 2437, 2610, 2791, 2976, 3169, 3366, 3571, 3780, 3997, 4218, 4447, 4680, 4921, 5166, 5419, 5676, 5941, 6210, 6487, 6768, 7057, 7350, 7651, 7956, 8269 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also degree of toric ideal associated with path with n+2 nodes [Eriksson]. 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 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 LINKS R. H. Hardin and N. J. A. Sloane, Table of n, a(n) for n = 0..1000 [First 210 terms from Hardin] N. Eriksson, Toric ideals of homogeneous phylogenetic models, arXiv:math/0401175 [math.CO], 2004. Nicolas Nagel, Example picture for angle (n+1)-sectors Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA G.f.: (1+4x+7x^2)/((1-x)^2*(1-x^2)). a(2t) = 12t^2+6t+1, a(2t+1) = 12t^2+18t+6 (t >= 0). 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. a(n) = 3*n*(n+1)+(1+(-1)^n)/2. - Wesley Ivan Hurt, May 06 2016 E.g.f.: 3*x*(2 + x)*exp(x) + cosh(x). - Ilya Gutkovskiy, May 06 2016 EXAMPLE Some triples for n=10 (from R. H. Hardin, Aug 04 2014): ..3....1....2....1....7....9....5....8....5....6....9....4...10....8....6....2 ..3....3....8....9....3....3....7....2....9....4....3...10....9....1....8....7 ..7....7...10....5....2....1....3....7....1....3....7....0....1....9....4....8 MAPLE f:=n-> if n mod 2 = 0 then t:=n/2; 12*t^2+6*t+1 else t:=(n-1)/2; 12*t^2+18*t+6; fi; [seq(f(n), n=0..100)]; MATHEMATICA CoefficientList[Series[(1 + 4 x + 7 x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 52}], x] (* Michael De Vlieger, May 07 2016 *) Table[3 n (n + 1) + (1 + (-1)^n)/2, {n, 0, 52}] (* or *) LinearRecurrence[{2, 0, -2, 1}, {1, 6, 19, 36}, 53] (* Michael De Vlieger, Sep 12 2016 *) PROG (PARI) x='x+O('x^99); Vec((1+4*x+7*x^2)/((1-x)^2*(1-x^2))) \\ Altug Alkan, May 12 2016 (Magma) [3*n*(n+1)+(1+(-1)^n)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 13 2016 CROSSREFS Cf. A090382, A090383, A090384, A090385. Row 1 of A245869. Central spine of triangle in A245556. Cf. also A245557. Sequence in context: A063147 A031014 A010899 * A354343 A106398 A179986 Adjacent sequences: A090378 A090379 A090380 * A090382 A090383 A090384 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Jan 30 2004 EXTENSIONS Edited by N. J. A. Sloane, Aug 04 2014 (merging the old A090381 and A245870). STATUS approved

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Last modified August 5 14:54 EDT 2024. Contains 374950 sequences. (Running on oeis4.)