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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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

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

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.

A174028 Triangle T(n,k) = 2+4k read by rows.

Original entry on oeis.org

2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 14, 18, 2, 6, 10, 14, 18, 22, 2, 6, 10, 14, 18, 22, 26, 2, 6, 10, 14, 18, 22, 26, 30, 2, 6, 10, 14, 18, 22, 26, 30, 34, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38
Offset: 0

Views

Author

Paul Curtz, Mar 06 2010

Keywords

Examples

			2;
2, 6;
2, 6, 10;
2, 6, 10, 14;
2, 6, 10, 14, 18;
2, 6, 10, 14, 18, 22;
2, 6, 10, 14, 18, 22, 26;

Links

Crossrefs

Cf. A001105 (row sums), A167268.

Formula

T(n,k) = A016825(k).
T(n,k) = 2*A158405(n+1,k).

A199502 From Janet helicoidal classification of the periodic table.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 11, 12, 13, 18, 19, 20, 21, 30, 31, 36, 37, 38, 39, 48, 49, 54, 55, 56, 57, 70, 71, 80, 81, 86, 87, 88, 89, 102, 103, 112, 113, 118, 119, 120, 121, 138, 139, 152, 153, 162, 163, 168, 169, 170, 171, 188, 189, 202, 203, 212, 213, 218, 219, 220, 221
Offset: 1

Views

Author

Paul Curtz, Nov 07 2011

Keywords

Comments

In A199426, we saw how Janet discovered
25 26 43 44
24 27 42 45
7 8 15 16 23 28 33 34 41 46 51 52
6 9 14 17 22 29 32 35 40 47 50 53
1 2 3 4 5 10 11 12 13 18 19 20 21 30 31 36 37 38 39 48 49 54 55 56 57
a(n) is the last row.
a(n+1) - a(n) = 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 9, 1, 5, 1, 1, 1, 9, 1, 5, 1, 1, 1, 13, 1, 9, 1, 5, 1, 1, 1, 13, 1, 9, 1, 5, 1, 1, 1,... = d(n).
Take d(n) by pairs: sums are 2, 2, 6, 2, 6, 2, 2, 10, 6, 2 = A167268.
Take d(n) by 2, 2, 4, 4, 6, 6, 8, 8, terms (in A052928): sums are 2, 2, 8, 8, 18, 18, 32, 32,... = extended A137583= 2, before A093907.

References

  • Charles Janet, La classification hélicoidale des éléments chimiques, novembre 1928, Beauvais, 2+80 pages + 10 leaflets (see 3).

Links

Formula

A167268 = 2, 2, 6, 2, 6, 2, repeated = r(n) = 2, 2, 2, 2, 6, 6, 2, 2, 6, 6, 2, 2, 10, 10, 6, 6, 2, 2,...
a(n+2) - a(n) = r(n+1) = 2, 2, 2, 6, 6, 2, 2, n=1,2,3,...
a(2*n+1) - a(2*n) = 1 = A000012.
Previous Showing 11-13 of 13 results.

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