A178568 Triangle read by rows, antidiagonals of an array (row r >= 1, column n >= 1) generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).

1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 1, 4, 4, 4, 3, 1, 5, 5, 9, 5, 2, 1, 6, 6, 16, 7, 6, 3, 1, 7, 7, 25, 9, 12, 7, 1, 1, 8, 8, 36, 11, 20, 13, 8, 4, 1, 9, 9, 49, 13, 30, 21, 27, 9, 3, 1, 10, 10, 64, 15, 42, 31, 64, 16, 10, 5, 1, 11, 11, 81, 17, 56, 43, 125, 25, 21, 11, 2

Offset: 1

Author

Gary W. Adamson, May 29 2010

Keywords

• nonn
• tabl

Comments

Companion to A178239 (the latter generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1)).

Row sums of the triangle = A169826: (1, 2, 5, 8, 16, 27, 45, 69, 109, ...).

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

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

Right border of the triangle = A002487: (1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, ...).

Terms in r-th row mod r (for r > 1): (1, 0, 1, 0, 1, 0, ...).

Examples

			First few rows of the array:
  1,   1,   2,   1,   3,   2,   3,   1,   4,   3,   5,   2,   5,   3, ..
  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14, ..
  1,   3,   4,   9,   7,  12,  13,  27,  16,  21,  19,  36,  25,  39, ..
  1,   4,   5,  16,   9,  20,  21,  64,  25,  36,  29,  80,  41,  84, ..
  1,   5,   6,  25,  11,  30,  31, 125,  36,  55,  41, 150,  61, 155, ..
  1,   6,   7,  36,  13,  42,  43, 216,  49,  78,  55, 252,  85, 258, ..
  1,   7,   8,  49,  15,  56,  57, 343,  64, 105,  71, 392, 113, 399, ..
  1,   8,   9,  64,  17,  72,  73, 512,  81, 136,  89, 576, 145, 584, ..
  ...
First few rows of the triangle:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  3,   1;
  1,  4,  4,   4,  3;
  1,  5,  5,   9,  5,   2;
  1,  6,  6,  16,  7,   6,  3;
  1,  7,  7,  25,  9,  12,  7,   1;
  1,  8,  8,  36, 11,  20, 13,   8,  4;
  1,  9,  9,  49, 13,  30, 21,  27,  9,  3;
  1, 10, 10,  64, 15,  42, 31,  64, 16, 10,  5;
  1, 11, 11,  81, 17,  56, 43, 125, 25, 21, 11,  2;
  1, 12, 12, 100, 19,  72, 57, 216, 36, 36, 19, 12,  5;
  1, 13, 13, 121, 21,  90, 73, 343, 49, 55, 29, 36, 13,  3;
  1, 14, 14, 144, 23, 110, 91, 512, 64, 78, 41, 80, 25, 14, 4;
  ...
		

Crossrefs

Array rows r=1 to r=10: A002487, A000027, A178590, A244643, A342610, A237711, A342611, A342614, A342615, A178569.

Cf. A178239.

Programs

PARI

A(r,n) = my(x=0,y=1); forstep(i=if(n,logint(n,2)),0,-1, if(bittest(n,i), x+=y;y*=r, y+=x;x*=r)); x;
T(r,n) = A(r-n+1,n); \\ Kevin Ryde, Mar 18 2021

Formula

a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).

Given (1, r, 1, 0, 0, 0, ...) in each column of an infinite lower triangular matrix M; shifted down twice from the previous column. r-th row of the array = lim_{n->inf} M^n.

For the r-th row, a(2^k+n) = r*a(n) + a(2^k-n). - Andrey Zabolotskiy, Oct 21 2021