login
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
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.
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
Row 1 of A245869.
Central spine of triangle in A245556. Cf. also A245557.
Sequence in context: A063147 A031014 A010899 * A354343 A106398 A179986
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