A227344 Triangle read by rows, partitions into distinct parts by perimeter.
1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1, 0, 0, 7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 16, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 20, 0
Offset: 1
Examples
Triangle starts (dots for zeros): 01: 1 02: . 1 03: . . 2 04: . . . 2 05: . . . . 3 06: . . . 1 . 3 07: . . . . . . 5 08: . . . . . . . 6 09: . . . . . 1 . . 7 10: . . . . 1 . . . . 9 11: . . . . . . . . 1 . 11 12: . . . . . . . 1 . 1 . 13 13: . . . . . . . . 1 . 1 . 16 14: . . . . . . 1 . . . . 1 . 20 15: . . . . . 1 . . . 1 . 1 1 . 23 16: . . . . . . . . . . 2 . 1 1 . 28 17: . . . . . . . . . . . 2 . 1 2 . 33 18: . . . . . . . . 1 . . 1 2 . 1 2 . 39 19: . . . . . . . . . 1 . . 1 1 1 1 3 . 46 20: . . . . . . . 1 . . . . . 1 1 2 1 3 . 55 21: . . . . . . 1 . . . . . . 2 2 1 2 1 4 . 63 22: . . . . . . . . . . 1 . 1 . 2 1 1 2 2 4 . 75 23: . . . . . . . . . . . 1 . 1 . 2 1 3 2 2 5 . 87 24: . . . . . . . . . . . . 1 . 1 2 3 . 4 2 3 5 . 101 25: . . . . . . . . . 1 . . . 1 . 1 1 3 . 6 2 3 7 . 117 26: . . . . . . . . . . 1 . 1 . . . 2 1 3 . 7 2 4 8 . 136 27: . . . . . . . . 1 . . . . 1 . . . 5 2 2 1 8 3 4 9 . 156 28: . . . . . . . 1 . . . . . . 1 1 . . 4 2 3 2 8 4 5 11 . 180 29: . . . . . . . . . . . . . . 1 2 1 . . 4 3 3 3 9 5 5 13 . 207 30: . . . . . . . . . . . 1 . . 1 1 1 1 . 3 6 2 2 5 9 6 6 14 . 238
Links
- Joerg Arndt, Table of n, a(n) for n = 1..5050
Crossrefs
Cf. A227345 (partitions by boundary size).
Cf. A227426 (diagonal: number of partitions with maximal perimeter).
Cf. A227538 (smallest k with positive T(n,k)), A227614 (second lower diagonal). - Alois P. Heinz, Jul 17 2013
Programs
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Maple
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x^(i+1), 1), expand(`if`(i<1, 0, `if`(t>1, x^(i+1), 1)*b(n, i-1, iquo(t, 2))+ `if`(i>n, 0, `if`(t=2, x^(i+1), 1)*b(n-i, i-1, iquo(t, 2)+2))))) end: T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n$2, 0)): seq(T(n), n=1..20); # Alois P. Heinz, Jul 16 2013 -
Mathematica
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t>1, x^(i+1), 1], Expand[If[i<1, 0, If[t>1, x^(i+1), 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t == 2, x^(i+1), 1]*b[n-i, i-1, Quotient[t, 2]+2]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n, 0]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)
Comments