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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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,5

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.

LINKS

Table of n, a(n) for n=2..87.

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.

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A008615, A321202, A321203.

Sequence in context: A099544 A036414 A234954 * A180649 A191238 A049310

Adjacent sequences:  A321198 A321199 A321200 * A321202 A321203 A321204

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Nov 05 2018

EXTENSIONS

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

STATUS

approved

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Last modified February 17 18:46 EST 2019. Contains 320222 sequences. (Running on oeis4.)