A174066 Irregular triangle, row sums = A000041, left border = A174065.

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 3, 1, 2, 7, 4, 2, 2, 9, 5, 3, 2, 3, 12, 7, 4, 4, 3, 15, 9, 5, 6, 3, 4, 19, 12, 7, 8, 6, 4, 25, 15, 9, 10, 9, 4, 5, 31, 19, 12, 14, 12, 8, 5, 38, 25, 15, 18, 15, 12, 5, 7, 48, 31, 19, 24, 21, 16, 10, 7, 60, 38, 25, 30, 27, 20, 15, 7, 9, 73, 48, 31, 38, 36

Offset

1,5

Comments

Left border = A174065: (1, 1, 1, 2, 3, 4, 5, 7, 9, 12,...) * its aerated variant (1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7,...) = A000041, the partition sequence: (1, 1, 2, 3, 5, 7, 11, 15, 22, 30,...).

Links

Table of n, a(n) for n=1..86.

Formula

The triangle is the result of three rules after beginning (1, 1, 1, 1,...):

Columns >1 are shifted down twice from previous columns; column terms = left border * (left border placed as a heading row); and row sums = A000041, the partition numbers. The rules force the next missing term in the triangle to be the leftmost term in column 1. This is found by taking p(n) for row n, then subtracting the sum of row n terms (minus leftmost term).

Example

Heading and first few rows of the triangle =
.1,...1,...1,...2,...3,...4,...5,...7,...9,... = A174065.
.1;........................................... = .. 1 (A000041)
.1;........................................... = .. 1
.1,...1;...................................... = .. 2
.2,...1;...................................... = .. 3
.3,...1,...1;................................. = .. 5
.4,...2,...1;................................. = .. 7
.5,...3,...1,...2;............................ = ..11
.7,...4,...2,...2;............................ = ..15
.9....5,...3,...2,...3;....................... = ..22
.12,..7,...4,...4,...3;....................... = ..30
.15,..9,...5,...6,...3;...4;.................. = ..42
.19,.12,...7,...8,...6,...4;.................. = ..56
.25,.15,...9,..10,...9,...4,...5;............. = ..77
.31,.19,..12,..14,..12,...8,...5;............. = .101
.38,.25,..15,..18,..15,..12,...5,...7;........ = .135
.48,.31,..19,..24,..21,..16,..10,...7;........ = .176
.60,.38,..25,..30,..27,..20,..15,...7,...9;... = .231
.73,.48,..31,..38,..36,..28,..20,..14,...9;... = .297
...
Example: leftmost term in 8th row has to be 7 = (15 - (4 + 2 + 2)); so we
place a 7 as next term in the heading, then multiply * leftmost column.
Finally, shift the columns down twice.

Crossrefs

Cf. A000041, A174065, A174067.

Sequence in context: A135841 A210992 A220484 * A089178 A187489 A355145

Adjacent sequences: A174063 A174064 A174065 * A174067 A174068 A174069

Keyword

nonn,tabf

Author

Gary W. Adamson, Mar 06 2010

Status

approved