OFFSET
1,5
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, ...).
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
EXAMPLE
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;
...
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 29 2010
STATUS
approved