A321201 Irregular triangle T with the nontrivial solutions of 2*e2 + 3*e3 = n, for n >= 2, with nonnegative e2 and e3, ordered as pairs with increasing e2 values.

1, 0, 0, 1, 2, 0, 1, 1, 0, 2, 3, 0, 2, 1, 1, 2, 4, 0, 0, 3, 3, 1, 2, 2, 5, 0, 1, 3, 4, 1, 0, 4, 3, 2, 6, 0, 2, 3, 5, 1, 1, 4, 4, 2, 7, 0, 0, 5, 3, 3, 6, 1, 2, 4, 5, 2, 8, 0, 1, 5, 4, 3, 7, 1, 0, 6, 3, 4, 6, 2, 9, 0, 2, 5, 5, 3, 8, 1, 1, 6, 4, 4, 7, 2, 10, 0

Offset: 2

Author

Wolfdieter Lang, Nov 05 2018

Keywords

• nonn
• tabf

Comments

The length of row n is 2*A(n), with A(n) = A008615(n+2) for n >= 2: 2*[1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, ...].

The trivial solution for n = 0 is [0, 0]. There is no solution for n = 1.

The row sums are given in A321202.

If a partition of n with parts 2 or 3 (with inclusive or) is written as 2^{e2} 3^{e3}, where e2 and e3 are nonnegative numbers, then in row n, all pairs [e2, e3] are given, for n >= 2, ordered with increasing values of e2.

The corresponding irregular triangle with the multinomial numbers n!/((n - (e2 + e3))!*e2!*e3!) is given in A321203. It gives the coefficients of x^n = x^{2*{e2} + 3*{e3}} of (1 + x^2 + x^3)^n, for n >= 2.

Examples

			The triangle T(n, k) begins (pairs are separated by commas):
  n\k  0  1   2  3   4  5   6  7 ...
  2:   1  0
  3:   0  1
  4:   2  0
  5:   1  1
  6:   0  2,  3  0
  7:   2  1
  8:   1  2,  4  0
  9:   0  3,  3  1
  10:  2  2,  5  0
  11:  1  3,  4  1
  12:  0  4,  3  2,  6  0
  13:  2  3,  5  1,
  14:  1  4,  4  2,  7  0
  15:  0  5,  3  3,  6  1
  16:  2  4,  5  2,  8  0
  17:  1  5,  4  3,  7  1
  18:  0  6,  3  4,  6  2,  9  0
  19:  2  5,  5  3,  8  1
  20:  1  6,  4  4,  7  2, 10  0
  ...
n=8: the two solutions of 2*e2 + 3*e3 = 8 are [e2, e3] = [1, 2] and = [4, 0], and 1 < 4, therefore row 8 is 1  2  4  0, with a comma after the first pair.
		

Crossrefs

Cf. A008615, A321202, A321203.

Programs

Mathematica

row[n_] := Reap[Do[If[2 e2 + 3 e3 == n, Sow[{e2, e3}]], {e2, 0, n/2}, {e3, 0, n/3}]][[2, 1]];
Table[row[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Nov 23 2018 *)

Formula

T(n, k) gives all pairs [e2, e3] solving 2*e2 + 3*e3 = n, ordered with increasing value of e2, for n >= 2. The trivial solution [0, 0] for n = 0 is not recorded. There is no solution for n = 1.

Extensions

Missing row 2 inserted by Jean-François Alcover, Nov 23 2018