A269837 Irregular triangle read by rows: even terms of A094728(n+1) divided by 4.

1, 2, 4, 3, 6, 4, 9, 8, 5, 12, 10, 6, 16, 15, 12, 7, 20, 18, 14, 8, 25, 24, 21, 16, 9, 30, 28, 24, 18, 10, 36, 35, 32, 27, 20, 11, 42, 40, 36, 30, 22, 12, 49, 48, 45, 40, 33, 24, 13, 56, 54, 50, 44, 36, 26, 14, 64, 63, 60, 55, 48, 39, 28, 15

Offset: 0

Author

Paul Curtz, Mar 06 2016

Keywords

• nonn
• tabf

Comments

See A264798 and A261046 for the Hydrogen atom and the Janet periodic table.

a(n) odd terms are again A264798.

Decomposition by multiplication i.e. a(n) = b(n)*c(n) by irregular triangle:

1, 1 1,

2, 1 2,

4, 3, 2, 1, 2, 3,

6, 4, = 2, 1, * 3, 4,

9, 8, 5, 3, 2, 1, 3, 4, 5,

12, 10, 6, 3, 2, 1, 4, 5, 6,

16, 15, 12, 7, 4, 3, 2, 1, 4, 5, 6, 7,

etc. etc. etc.

b(n) is duplicated A004736(n) or mirror of A122197(n+1). c(n) = A138099(n+1).

Decomposition by subtraction, a(n) = d(n) - e(n):

1, 1 0,

2, 2, 0,

4, 3, 4, 3, 0, 0,

6, 4, = 6, 5, - 0, 1,

9, 8, 5, 9, 8, 7, 0, 0, 2,

12, 10, 6, 12, 11, 10, 0, 1, 4,

16, 15, 12, 7, 16, 15, 14, 13, 0, 0, 2, 6,

20, 18, 14, 8, 20, 19, 18, 17, 0, 1, 4, 9,

etc. etc. etc.

d(n) is the natural numbers A000027 inverted by lines. e(n) will be studied (see A239873).

Sum of a(n) by diagonals: 1, 5, 13, 27, 48, ... . The third differences have the period 2: repeat 2, 1. See A002717.

Crossrefs

Cf. A000027, A000034, A002717, A004736, A094728, A122197, A138099, A167430, A239873, A261046, A264798.

Programs

Mathematica

Table[(n + 1)^2 - k^2, {n, 15}, {k, 0, n - 1}]/4 /. Rational -> Nothing // Flatten (* _Michael De Vlieger, Mar 07 2016 *)