A178239 Triangle read by rows, antidiagonals of an array generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 5, 2, 1, 1, 1, 6, 1, 7, 3, 3, 1, 1, 1, 7, 1, 9, 4, 7, 1, 1, 1, 1, 8, 1, 11, 5, 13, 1, 4, 1, 1, 1, 9, 1, 13, 6, 21, 1, 7, 3, 1, 1, 1, 10, 1, 15, 7, 31, 1, 10, 5, 5, 1

Offset: 0

Author

Gary W. Adamson, May 23 2010

Keywords

• nonn
• easy
• tabl

Comments

Partial sums of array terms in groups of 1, next 2, next 4, ... 8 = powers of (r+2).

Row sums = A178240: (1, 2, 3, 5, 7, 11, 16, 23, ...).

Row 1 of the array = A002487.

Row 2 = .............A116528.

Row 3 = .............A342633.

Row 4 = .............A342634.

...

Row 10 = ............A178243.

Polcoeff row r of the array as f(x) satisfies f(x)/f(x^2) = (1 + x + r*x^2).

Let q(x) = (1 + x + r*x^2). Then polcoeff row 4 = q(x) * q(x^2) * q(x^4) * q(x^8) * ...

Examples

			First few rows of the array =
      n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9 n=10 n=11 n=12 n=13 n=14 n=15
  r=0:  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  r=1:  1,   1,   2,   1,   3,   2,   3,   1,   4,   3,   5,   2,   5,   3,   4, ...
  r=2:  1,   1,   3,   1,   5,   3,   7,   1,   7,   5,  13,   3,  13,   7,  15, ...
  r=3:  1,   1,   4,   1,   7,   4,  13,   1,  10,   7,  25,   4,  25,  13,  40, ...
  r=4:  1,   1,   5,   1,   9,   5,  21,   1,  13,   9,  41,   5,  41,  21,  85, ...
  r=5:  1,   1,   6,   1,  11,   6,  31,   1,  16,  11,  61,   6,  61,  31, 156, ...
  ...
Example: In row 3: (1, 1, 4, 1, 7, 4, 13, ...) = A342633, r = 3.
A342633(7) = 13 = 3*4 + 1. In blocks of 1, 2, 4, 8, ... terms, partial sums are powers of (r+2) = 5: (1, 5, 25, ...).
First few rows of the triangle =
  1;
  1, 1;
  1, 1,  1;
  1, 1,  2, 1;
  1, 1,  3, 1,  1;
  1, 1,  4, 1,  3,  1;
  1, 1,  5, 1,  5,  2,  1;
  1, 1,  6, 1,  7,  3,  3, 1;
  1, 1,  7, 1,  9,  4,  7, 1,  1;
  1, 1,  8, 1, 11,  5, 13, 1,  4,  1;
  1, 1,  9, 1, 13,  6, 21, 1,  7,  3,  1;
  1, 1, 10, 1, 15,  7, 31, 1, 10,  5,  5, 1;
  1, 1, 11, 1, 17,  8, 43, 1, 13,  7, 13, 2,  1;
  1, 1, 12, 1, 19,  9, 57, 1, 16,  9, 21, 3,  5, 1;
  1, 1, 13, 1, 21, 11, 73, 1, 19, 11, 31, 4, 13, 2, 1;
  ...
		

Crossrefs

Cf. A178240, A359250 (column polynomials).

Array rows r=1..10: A002487, A116528, A342633, A342634, A342635, A342603, A342636, A342637, A342638, A178243.

Formula

Antidiagonals of an array generated from a(n) = a(2n); a(2n+1) = r*a(n) + a(n+1).

Given a triangle M with columns stepped down twice from the previous column, for columns > 0, with (1, 1, r, 0, 0, 0, ...) in each column, r-th row of the array = lim_{n->oo} M^n.