cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Paul Curtz, Nov 19 2015

Keywords

Comments

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.

Examples

			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,
....
		

Crossrefs

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

Views

Author

Paul Curtz, Nov 25 2015

Keywords

Comments

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.

Examples

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

Crossrefs

Programs

  • Mathematica
    Table[n^2 - k^2, {n, 14}, {k, 0, n - 1}] /. n_ /; EvenQ@ n -> Nothing // Flatten (* Michael De Vlieger, Nov 25 2015 *)
  • PARI
    for(n=1,20,for(k=0,n-1,s=n^2-k^2;if(s%2,print1(s,", ")))) \\ Derek Orr, Dec 24 2015

Formula

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