A179901 Triangle read by rows, antidiagonals of an array generated from (1, r, r, r, ...) convolved with (1, 0, r, r, r, ...).

1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 6, 8, 4, 1, 5, 8, 15, 12, 5, 1, 6, 10, 24, 24, 16, 6, 1, 7, 12, 35, 40, 33, 20, 7, 1, 8, 14, 48, 60, 56, 42, 24, 8, 1, 9, 16, 63, 84, 85, 72, 51, 28, 9, 1, 10, 18, 80, 112, 120, 110, 88, 60, 32, 10, 1, 11, 20, 99, 144, 161, 156, 135, 104, 69, 36, 11

Offset

1,5

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Formula

Triangle read by rows, antidiagonals of an array generated from (1, r, r, r, ...) convolved with (1, 0, r, r, r, ...), such that the r-th row of the array = (1, r, 2*r, ...) then for n > 3, a(n) = r^2 + a(n-1).

Example

First few rows of the array:
.
1,.1,..2,...3,...4,...5,...6,...7,....8,...
1,.2,..4,...8,..12,..16...20,..24,...28,... = A019442
1,.3,..6,..15,..24,..33,..42,..51,...60,... = A179805
1,.4,..8,..24,..40,..56,..72,..88,..104,...
.
Example: row 4 = (1, 4, 8, 24, ...) = (1, 4, 4, 4, ...) * (1, 0, 4, 4, 4, ...) = (1, r, 2*r, (2*r + r^2), ...).
.
First few rows of the triangle:
.
1,
1, 1;
1, 2, 2;
1, 3, 4, 3;
1, 4, 6, 8, 4;
1, 5, 8, 15, 12, 5;
1, 6, 10, 24, 24, 16, 6;
1, 7, 12, 35, 40, 33, 20, 7;
1, 8, 14, 48, 60, 56, 42, 24, 8;
1, 9, 16, 63, 84, 85, 72, 51, 28, 9;
1, 10, 18, 80, 112, 120, 110, 88, 60, 32, 10;
1, 11, 20, 99, 144, 161, 156, 135, 104, 69, 36, 11;
1, 12, 22, 120, 180, 208, 210, 192, 160, 120, 78, 40, 12;
1, 13, 24, 143, 220, 261, 272, 259, 228, 185, 136, 87, 44, 13;
...

Prog

(PARI) A(r, n)=if(n<=2, if(n==2, r, 1), r^2*(n-3) + 2*r)
{ for(n=1, 10, print(vector(n, k, A(n+1-k, k)))) } \\ Andrew Howroyd, Sep 21 2025

Crossrefs

Row sums are A179902.

Cf. A019442, A179805.

Sequence in context: A182630 A208805 A208514 * A209561 A283822 A210789

Adjacent sequences: A179898 A179899 A179900 * A179902 A179903 A179904

Keyword

nonn,tabl

Author

Gary W. Adamson, Jul 31 2010

Extensions

a(52) corrected by Andrew Howroyd, Sep 21 2025

Status

approved