A165997 Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 1, 4, 9, 9, 1, 5, 14, 23, 23, 1, 6, 20, 43, 66, 1, 7, 27, 70, 136, 136, 1, 8, 35, 105, 241, 377, 377, 1, 9, 44, 149, 390, 767, 1144, 1, 10, 54, 203, 593, 1360, 2504, 2504, 1, 11, 65, 268, 861, 2221, 4725, 7229, 7229, 1, 12, 77, 345, 1206

Offset

0,6

Comments

There are f(n) = floor((2*n+3)/3) = A004396(n+1) terms in row n.

Links

Table of n, a(n) for n=0..65.

Example

Triangle begins:
       k=0   1   2    3     4     5     6      7      8
  n=0:   1
  n=1:   1
  n=2:   1,  1
  n=3:   1,  2,  2
  n=4:   1,  3,  5
  n=5:   1,  4,  9,   9
  n=6:   1,  5, 14,  23,   23
  n=7:   1,  6, 20,  43,   66
  n=8:   1,  7, 27,  70,  136,  136
  n=9:   1,  8, 35, 105,  241,  377,  377
  n=10:  1,  9, 44, 149,  390,  767, 1144
  n=11:  1, 10, 54, 203,  593, 1360, 2504,  2504
  n=12:  1, 11, 65, 268,  861, 2221, 4725,  7229,  7229
  n=13:  1, 12, 77, 345, 1206, 3427, 8152, 15381, 22610
  ...

Prog

(PARI) f(n) = floor((2*(n-1)+3)/3); s=14; M=matrix(s, s); for(n=1, s, M[n, 1]=1); for(n=2, s, for(k=2, f(n), M[n, k]=M[n, k-1]+M[n-1, k])); for(n=1, s, for(k=1, f(n), print1(M[n, k], ", ")))

Crossrefs

Cf. A004396 (row lengths).

Cf. A060941 (diagonal T(3*n, 2*n)).

Sequence in context: A289412 A266504 A111375 * A046752 A086350 A239830

Adjacent sequences: A165994 A165995 A165996 * A165998 A165999 A166000

Keyword

nonn,tabf

Author

Gerald McGarvey, Oct 03 2009

Status

approved