A249947 -id:A249947 - cp's OEIS frontend

A261046 Irregular triangle read by rows: the first column consists of the odd numbers repeated but without the first 1. Row n (n>=0) contains floor(n/2)=1 terms. Every row contains successive odd numbers.

1, 3, 3, 5, 5, 7, 5, 7, 9, 7, 9, 11, 7, 9, 11, 13, 9, 11, 13, 15, 9, 11, 13, 15, 17, 11, 13, 15, 17, 19, 11, 13, 15, 17, 19, 21, 13, 15, 17, 19, 21, 23, 13, 15, 17, 19, 21, 23, 25, 15, 17, 19, 21, 23, 25, 27, 13, 15, 17, 19, 21, 23, 25, 27

Offset: 0

Paul Curtz, Nov 19 2015

A131507(n), not in the same order.
a(n) multiplied by the triangle (extended A249947(n+1)) = (A167268(n+1))/2 is
   1,                 1,               1,
   3,                 1,               3,
   3, 5,              3, 1,            9, 5,
   5, 7,          *   3, 1,       =   15, 7,
   5, 7,  9,          5, 3, 1,        25, 21, 9
   7, 9, 11,          5, 3, 1,        35, 27, 11,
   etc.               etc.            etc.
The latter triangle is the odd numbers of A094728(n+1) which is
   1,
   4,  3,
   9,  8,  5,
  16, 15, 12,  7,
  25, 24, 21, 16, 9,
  etc.
Without the first column, the triangle is A120070(n+2). This gives a link between the frequencies of the spectral lines of the H-atom and the Janet periodic table of the elements.

			Triangle begins:
1,
3,
3,  5,
5,  7,
5,  7,  9,
7,  9, 11,
7,  9, 11, 13,
9, 11, 13, 15,
9, 11, 13, 15, 17,
....

Cf. A094728, A120070, A131507, A167268, A249947.

A264798 Irregular triangle read by rows: odd-valued terms of A094728(n+1).

Original entry on oeis.org

1, 3, 9, 5, 15, 7, 25, 21, 9, 35, 27, 11, 49, 45, 33, 13, 63, 55, 39, 15, 81, 77, 65, 45, 17, 99, 91, 75, 51, 19, 121, 117, 105, 85, 57, 21, 143, 135, 119, 95, 63, 23, 169, 165, 153, 133, 105, 69, 25, 195, 187, 171, 147, 115, 75, 27, 225, 221, 209, 189, 161, 125, 81, 29, 255, 247

Offset: 0

Paul Curtz, Nov 25 2015

A094728(n+1) comes from A120070(n+2). a(n) approximates frequencies of the spectral lines of the hydrogen atom.
Row sums: 1, 3, 14, 22, ... = A024598(n+1).
First column: A085046(n+1).
Row sums of A261046(n) = 1, 3, 8, 12, ... = A014255(n). See the formula.

			Irregular triangle begins:
1,
3,
9,  5,
15, 7,
25, 21,  9,
35, 27, 11,
49, 45, 33, 13,
63, 55, 39, 15,
...

Cf. A014255, A024598, A085046, A094728, A120070, A167268, A249947, A261046.

Table[n^2 - k^2, {n, 14}, {k, 0, n - 1}] /. n_ /; EvenQ@ n -> Nothing // Flatten (* Michael De Vlieger, Nov 25 2015 *)

for(n=1,20,for(k=0,n-1,s=n^2-k^2;if(s%2,print1(s,", ")))) \\ Derek Orr, Dec 24 2015

a(n) = A261046(n)*A167268(n+1)/2, where A167268 is Janet's sequence.

cp's OEIS Frontend

A261046 Irregular triangle read by rows: the first column consists of the odd numbers repeated but without the first 1. Row n (n>=0) contains floor(n/2)=1 terms. Every row contains successive odd numbers.

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A264798 Irregular triangle read by rows: odd-valued terms of A094728(n+1).

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