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

0,2

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.

Links

Table of n, a(n) for n=0..63.

Mathematica

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

Crossrefs

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

Sequence in context: A057956 A151734 A175027 * A308982 A182298 A182732

Adjacent sequences: A269834 A269835 A269836 * A269838 A269839 A269840

Keyword

nonn,tabf

Author

Paul Curtz, Mar 06 2016

Status

approved