A173304 -id:A173304 - cp's OEIS frontend

A173302 Irregular triangle by rows, generated from an array with A000041 starting again at columns (2^n - 1), n=1,2,3,...

1, 1, 1, 2, 1, 3, 2, 1, 5, 3, 1, 7, 5, 2, 11, 7, 3, 15, 11, 5, 1, 22, 15, 7, 1, 30, 22, 11, 2, 42, 30, 15, 3, 56, 42, 22, 5, 77, 56, 30, 7, 101, 77, 42, 11, 135, 101, 56, 15, 176, 135, 77, 22, 1, 231, 176, 101, 30, 1, 297, 231, 135, 42, 2

Offset: 0

Gary W. Adamson, Feb 15 2010

Sum of finite difference terms of the array create triangle A173304.

Row sums = A173303.

			The generating array =
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176,...
...1, 1, 2, 3, 5,..7,.11, 15, 22, 30, 42, 56, .77, 101, 135,...
.........1, 1, 2,..3...5,..7, 11, 15, 22, 30, .42, .56, .77,...
.......................1,..1,..2,..3,..5,..7,..11,..15,..22,...
..........................................................1,...
...then columns become the rows of triangle A173302:
1;
1, 1;
2, 1;
3, 2, 1;
5, 3, 1;
7, 5, 2;
11, 7, 3;
15, 11, 5, 1;
22, 15, 7, 1;
30, 22, 11, 2;
42, 30, 15, 3;
56, 42, 22, 5;
77, 56, 30, 7;
101, 77, 4, 11;
135, 101, 56, 15;
176, 135, 77, 22, 1;
...

Cf. A000041, A173301, A173303, A173304.

Create an array with A000041 in row 0, then A000041 starts anew under successive columns (1, 3, 7, 15, 31,...). (Cf. A173301).

Columns of the array = rows of A173302.

A173303 Row sums of triangle A173302.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 21, 32, 45, 65, 90, 125, 170, 231, 307, 411, 539, 707, 917, 1188, 1522, 1950, 2475, 3137, 3949, 4962, 6195, 7725, 9579, 11856, 14610, 17971, 22012, 26919, 32798, 39892, 48367, 58541, 70647, 85123, 102291, 122724, 146891, 175545, 209322

Offset: 0

Gary W. Adamson, Feb 15 2010

nonn

Each term is a sum of partition numbers.

			a(7) = 32 = (1 + 5 + 11 + 15), = p(0) + p(4) + p(6) + p(7).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row sums of A173302.

Cf. A000041, A173304.

a(n) = sum(k=0, logint(n+1, 2), numbpart(n + 1 - 2^k)) \\ Andrew Howroyd, Aug 10 2018

a(n) = Sum_{k >=0, 2^k <= n + 1} A000041(n + 1 - 2^k). - Andrew Howroyd, Aug 10 2018

Terms a(21) and beyond from Andrew Howroyd, Aug 10 2018

cp's OEIS Frontend

A173302 Irregular triangle by rows, generated from an array with A000041 starting again at columns (2^n - 1), n=1,2,3,...

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A173303 Row sums of triangle A173302.

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