A174067 Triangle, row sums = A000041 starting (1, 2, 3, 5, 7, ...); derived from finite differences of p(x) = A(x)*A(x^2) = B(x)*B(x^3) = C(x)*C(x^4) = ...

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 1, 1, 0, 1, 5, 2, 2, 1, 0, 1, 7, 2, 3, 1, 1, 0, 1, 9, 4, 3, 3, 1, 1, 0, 1, 12, 5, 5, 3, 3, 0, 1, 0, 1, 15, 8, 6, 5, 3, 2, 1, 1, 0, 1, 19, 10, 9, 6, 5, 2, 2, 1, 1, 0, 1, 25, 13, 12, 10, 5, 5, 2, 2, 1, 1, 0, 1, 31, 17, 16, 12, 9, 5, 4, 2, 2, 1, 1, 0, 1

Offset

1,4

Comments

Row sums = A000041 starting with offset 1: (1, 2, 3, 5, 7, 11, ...).

Links

Table of n, a(n) for n=1..91.

Formula

Given an array of rows satisfying p(x) = A(x)*A(x^2) = row 1 = A174065; row = 2 A174068 satisfying p(x) = B(x)*B(x^3); row 3 satisfies p(x) = C(x)*C(x^4), ... and so on; take finite differences from the top, becoming rows of triangle A174067.

Example

First few rows of the array:
  1, 1, 1, 2, 3, 4,  5,  7,  9, 12, 15, 19, ... = A174065
  1, 1, 2, 2, 4, 5,  7,  9, 13, 17, 23, 29, ... = A174068
  1, 1, 2, 3, 4, 6,  9, 12, 16, 22, 29, 38, ... satisfies p(x) = C(x)*C(x^4)
  1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, ... analogous for k=5
  1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 49, ..................k=6
  1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, ..................k=7
  1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, ..................k=8
  1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, ..................k=9
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, ..................k=10
  ...
Finally, take finite differences from the top, deleting the first 1, to obtain triangle A174067:
   1;
   1,  1;
   2,  0,  1;
   3,  1,  0,  1;
   4,  1,  1,  0,  1;
   5,  2,  2,  1,  0,  1;
   7,  2,  3,  1,  1,  0,  1;
   9,  4,  3,  3,  1,  1,  0,  1;
  12,  5,  5,  3,  3,  0,  1,  0,  1;
  15,  8,  6,  5,  3,  2,  1,  1,  0,  1;
  19, 10,  9,  6,  5,  2,  2,  1,  1,  0,  1;
  25, 13, 12, 10,  5,  5,  2,  2,  1,  1,  0,  1;
  31, 17, 16, 12,  9,  5,  4,  2,  2,  1,  1,  0,  1;
  38, 24, 20, 18, 11,  8,  5,  4,  2,  2,  1,  1,  0,  1;
  ...

Crossrefs

Cf. A000041, A174065, A174066, A174068.

Sequence in context: A355536 A100260 A165317 * A124943 A169803 A099557

Adjacent sequences: A174064 A174065 A174066 * A174068 A174069 A174070

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 06 2010

Status

approved