A144961 Eigentriangle whose left border is the Padovan sequence, and whose right border and row sums are a modified Fibonacci sequence.

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 3, 1, 1, 0, 1, 0, 0, 5, 1, 1, 1, 0, 2, 0, 0, 8, 2, 1, 1, 1, 0, 3, 0, 0, 13, 2, 2, 1, 1, 2, 0, 5, 0, 0, 21, 3, 2, 2, 1, 2, 3, 0, 8, 0, 0, 34, 4, 3, 2, 2, 2, 3, 5, 0, 13, 0, 0, 55, 5, 4, 3, 2, 4, 3, 5

Offset

0,15

Comments

Left border = A000931, the Padovan sequence: (1, 0, 0, 1, 0, 1, 1, 1, 2, 2, ...).

Right border = (1, 1, 1, 1, 2, 3, 5, 8, 13, 21, ...)

Row sums = (1, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).

Sum of n-th row terms = rightmost term in next row.

Links

Nathaniel Johnston, Rows n = 0..150, flattened

Formula

Triangle read by rows, termwise products of a Padovan "decrescendo" triangle: (1; 0,1; 0,0,1; 1,0,0,1,...) and the Fibonacci series preceded by two 1's: (1, 1, 1, 1, 2, 3, 5, 8, ...); (i.e., the INVERT transform of the Padovan sequence).

a(n,k) = Fibonacci(max(k-1,1))*A000931(n-k). - Nathaniel Johnston, May 05 2011

Example

First few rows of the triangle:
  1;
  0,  1;
  0,  0,  1;
  1,  0,  0,  1;
  0,  1,  0,  0,  2;
  1,  0,  1,  0,  0,  3;
  1,  1,  0,  1,  0,  0,  5;
  1,  1,  1,  0,  2,  0,  0,  8;
  2,  1,  1,  1,  0,  3,  0,  0, 13;
  2,  2,  1,  1,  2,  0,  5,  0,  0, 21;
  3,  2,  2,  1,  2,  3,  0,  8,  0,  0, 34;
  4,  3,  2,  2,  2,  3,  5,  0, 13,  0,  0, 55;
  ...

Maple

with(combinat): A000931 := proc(n) option remember: if(n=0)then return 1: elif(n<=2)then return 0: else return procname(n-2)+procname(n-3): fi: end: A144961 := proc(n, k) return fibonacci(max(k-1, 1))*A000931(n-k) end:for n from 0 to 12 do for k from 0 to n do printf("%d, ", A144961(n, k)); od: printf("\n"); od: # Nathaniel Johnston, May 05 2011

Crossrefs

Cf. A000045, A000931.

Sequence in context: A283000 A145765 A157424 * A144627 A135929 A346377

Adjacent sequences: A144958 A144959 A144960 * A144962 A144963 A144964

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Sep 27 2008

Extensions

Example triangle corrected by Nathaniel Johnston, May 05 2011

Status

approved