A245556 -id:A245556 - cp's OEIS frontend

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

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

N. J. A. Sloane, Jan 30 2004

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

			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

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).

Cf. A090382, A090383, A090384, A090385.
Row 1 of A245869.
Central spine of triangle in A245556. Cf. also A245557.

[3*n*(n+1)+(1+(-1)^n)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 13 2016

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)];

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 *)

x='x+O('x^99); Vec((1+4*x+7*x^2)/((1-x)^2*(1-x^2))) \\ Altug Alkan, May 12 2016

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

Edited by N. J. A. Sloane, Aug 04 2014 (merging the old A090381 and A245870).

A245557 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k and in which at least one of u,v,w is equal to n.

Original entry on oeis.org

1, 3, 6, 4, 3, 6, 15, 12, 7, 3, 6, 9, 24, 21, 18, 10, 3, 6, 9, 12, 33, 30, 27, 24, 13, 3, 6, 9, 12, 15, 42, 39, 36, 33, 30, 16, 3, 6, 9, 12, 15, 18, 51, 48, 45, 42, 39, 36, 19, 3, 6, 9, 12, 15, 18, 21, 60, 57, 54, 51, 48, 45, 42, 22

Offset: 0

N. J. A. Sloane, Aug 04 2014

The sum of (left-justified) rows 0 through n gives row n of A245556. For example, the sum of rows 0 thru 2 is 7, 12, 19, 12, 7, which is the n=2 row of A245556.

			Triangle begins:
[1]
[3, 6, 4]
[3, 6, 15, 12, 7]
[3, 6, 9, 24, 21, 18, 10]
[3, 6, 9, 12, 33, 30, 27, 24, 13]
[3, 6, 9, 12, 15, 42, 39, 36, 33, 30, 16]
[3, 6, 9, 12, 15, 18, 51, 48, 45, 42, 39, 36, 19]
[3, 6, 9, 12, 15, 18, 21, 60, 57, 54, 51, 48, 45, 42, 22]
...
Example. Suppose n = 2. We find:
triple count pair-sums 0  1  2  3  4
                       -------------
002      3     0,2     3     3
012      6     1,2,3      6  6  6
112      3     2,3           3  3
022      3     2,4           3     3
122      3     3,4              3  3
222      1     4                   1
                       -------------
Totals:                3  6 15 12  7, which is row 2 of the triangle.

Partial sums of the rows gives A245556.

Row sums are A082040.

Maple
```
See A245556.
```

T(n,k) = 3k (0 <= k <= n-1), T(n,k) = 12n-3k-3 (n <= k <= 2n-1), T(n,2n) = 3n+1.

cp's OEIS Frontend

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

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