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.

Showing 1-10 of 13 results. Next

A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 5, 2, 2, 6, 2, 2, 6, 1, 2, 2, 6, 2, 2, 2, 6, 2, 1, 2, 2, 6, 2, 2, 2, 2, 6, 2, 3, 2, 2, 6, 2, 4, 2, 2, 6, 2, 5, 2, 2, 6, 2, 6, 2, 2, 6, 2, 6, 1, 2, 2, 6, 2, 6, 2, 2, 2, 6, 2, 6, 2, 1, 2, 2, 6, 2, 6, 2, 2, 2, 2, 6, 2, 6, 2, 3, 2, 2, 6, 2, 6, 2, 4
Offset: 1

Views

Author

Paul Curtz, Jan 02 2014

Keywords

Comments

a(n) is not A173642, a compact Bohr-Stoner model (1924), modified by Charles Janet in 1930. The good distribution is A168208.
Only sequences N16(n) in A234398 are used:
N16(1)= 1 followed by 2's = A040000,
N16(2)= 1, 2, 3, 4, 5, followed by 6's = A101272,
N16(3)= 1 to 9, followed by 10's,
N16(4)= 1 to 13, followed by 14's, etc.
The distribution by rows are in the example.
The N16(n)'s are respectively on columns (hence triangle T)
1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, A002620(n+2)
3, 5, 8, 11, 15, 19, 24, 29, 35, A024206(n+2)
7, 10, 14, 18, 23, 28, 34, A014616(n+3)
13, 17, 22, 27, 33, A004116(n+4)
21, 26, 32,
31, etc.
See A163255.
Antidiagonals give the natural numbers A000027, like rows sums in the example.
A033638=1, 1, 2, 3, 5, 7,... is upon the triangle T.

Examples

			1,      H
2,       He
2, 1,    Li
2, 2,    Be
2, 2, 1,
2, 2, 2,
2, 2, 3,
2, 2, 4,
2, 2, 5,
2, 2, 6,
2, 2, 6, 1,
2, 2, 6, 2,
2, 2, 6, 2, 1,
2, 2, 6, 2, 2,
2, 2, 6, 2, 3,
2, 2, 6, 2, 4,
2, 2, 6, 2, 5,
2, 2, 6, 2, 6,
2, 2, 6, 2, 6, 1,
2, 2, 6, 2, 6, 2,
2, 2, 6, 2, 6, 2, 1,
2, 2, 6, 2, 6, 2, 2,
2, 2, 6, 2, 6, 2, 3, etc.

Crossrefs

Cf. A002061, A002522 (or A160457), A014206, A059100, diagonals of the triangle T. A004526.

A138469 Atomic numbers of p-block elements.

Original entry on oeis.org

5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 31, 32, 33, 34, 35, 36, 49, 50, 51, 52, 53, 54, 81, 82, 83, 84, 85, 86, 113, 114, 115, 116, 117, 118
Offset: 1

Views

Author

Paul Curtz, May 09 2008

Keywords

Comments

6 X 6 square of orbital p filling elements in Mendeleyev-Seaborg or Janet-Tarantola periodic table.
The sequence is considered full if we assume that all elements with 8 shells or more, i.e. of periods 8 and higher in Mendeleev's periodic table, have extremely unstable nuclei (extremely short half-life.)
The atomic numbers of elements in each block are
s-block: 1,2, 3,4, 11,12, 19,20, 37,38, 55,56, 87,88 for Mendeleev's periodic table (Cf. A160914)
s-block: 1,2, 3,4, 11,12, 19,20, 37,38, 55,56, 87,88, 119,120 for Janet's periodic table (Cf. A160914)
p-block: 5..10, 13..18, 31..36, 49..54, 81..86, 113..118 (Cf. A138469)
d-block: 21..30, 39..48, 71..80, 103..112 (Cf. A199934)
f-block: 57..70, 89..102 (Cf. A217923)

Links

Crossrefs

Cf. A130517, A131603, A167268.
Cf. A160914 Atomic numbers of s-block elements.
Cf. A138469 Atomic numbers of p-block elements.
Cf. A199934 Atomic numbers of d-block elements.
Cf. A217923 Atomic numbers of f-block elements.

Extensions

Edited by Daniel Forgues, May 15 2011
Ref. to A160914 and A199934 inserted by Jean-François Alcover, Oct 15 2012

A216607 The sequence used to represent partition binary diagram as an array.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3
Offset: 1

Views

Author

Mircea Merca, Sep 10 2012

Keywords

Comments

This sequence differs from A025672 first at index n=110.

Links

Crossrefs

Cf. A025672, A167268.

Programs

  • Maple
    seq(floor((1/4)*ceil(sqrt(4*n))^2)-n,n=1..50)
  • PARI
    A216607(n)=floor((1/4)*ceil(sqrt(4*n))^2)-n;

Formula

a(n) = floor((1/4)*ceiling(sqrt(4*n))^2) - n.
a(n^2) = a(n^2+n) = 0.
From Szymon Lukaszyk, Oct 27 2023: (Start)
a(n) = (-n) mod round(sqrt(n)).
a(n) = (A167268(n) - 2)/4. (End)

A160914 Extended s-block elements for Janet table.

Original entry on oeis.org

1, 2, 3, 4, 11, 12, 19, 20, 37, 38, 55, 56, 87, 88, 119, 120, 169, 170, 219, 220, 291, 292, 363, 364, 461, 462, 559, 560, 687, 688, 815, 816, 977, 978, 1139, 1140, 1339, 1340, 1539, 1540, 1781, 1782, 2023, 2024, 2311, 2312, 2599, 2600, 2937, 2938, 3275, 3276
Offset: 1

Views

Author

Paul Curtz, Oct 15 2011

Keywords

Comments

See A168342. Must be included in A167268. From right to left, first vertical is A168380 from 1 to 8. Second vertical is A168380-1. In (1) page 12, introducing elements 93 to 120, Janet says that there is a probable 8th row. For row 8, he proposes, like for row 7, 32 elements (89 to 120). Page 16 he presents 4 blocks: first has 2*8 elements, second: 6*6, third: 10*4, fourth: 14*2. Today, blocks are s,p,d,f for Mendeleyev-Moseley-Seaborg 118 elements periodic table. See (2), (3), A173592 and A138509. In 1927, only 88 on the first 92 elements were known; 41 (1937 discovered), 61 (1947), 85 (1940) and 87 (1939) were missing. Since 2010 (117 discovered) the first 118 elements are known. Janet predicted only 120 elements.

Examples

			The following is an s-block, 2*8=16 elements, i.e., a(n) written vertically, after p-block, 6*6, (A138469).
                          1   2
                          3   4
5   6   7   8   9   10   11  12
13  14  15  16  17  18   19  20
31  32  33  34  35  36   37  38
49  50  51  52  53  54   55  56
81  82  83  84  85  86   87  88
113 114 115 116 117 118  119 120

References

  • Charles JANET, La structure du Noyau de l'atome,considérée dans la Classification périodique, des éléments chimiques, 1927 (Novembre) N. 2 Beauvais, 67 pages, 3 leaflets.

Crossrefs

Cf. A099955.

A173592 Atomic numbers in the Mendeleyev-Moseley-Seaborg periodic table of elements read downwards columns, right to left.

Original entry on oeis.org

2, 10, 18, 36, 54, 86, 118, 1, 9, 17, 35, 53, 85, 117, 8, 16, 34, 52, 84, 116, 7, 15, 33, 51, 83, 115, 6, 14, 32, 50, 82, 114, 5, 13, 31, 49, 81, 113, 4, 12, 30, 48, 80, 112, 3, 11, 29, 47, 79, 111, 28, 46, 78, 110, 27, 45, 77, 109, 26, 44, 76, 108, 25, 43, 75, 107, 24, 42, 74
Offset: 1

Views

Author

Paul Curtz, Feb 22 2010

Keywords

Comments

A permutation of the natural numbers from 1 to 118.
The number of terms in the columns, also ordered right to left is: 7, 7, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2.
This is a consequence of finding 2*7=14, 6*6=36, 10*4=40, 14*2=28 elements with outer shells of s, p, d, and f-electrons.
Acronymic name: CMMSPT.

Examples

			The table contains 7 rows in 32 columns outlined as follows:
                                             1   2
                     3   4   5   6   7   8   9  10
                    11  12  13  14  15  16  17  18
      19  20....28  29  30  31  32  33  34  35  36
      37  38....46  47  48  49  50  51  52  53  54
55....69  70....78  79  80  81  82  83  84  85  86
87...101 102...110 111 112 113 114 115 116 117 118

Links

Crossrefs

Cf. A167268, A137583, A137325.

Programs

  • Mathematica
    elements = PadLeft[#, 32, 0] & /@ {{1, 2}, Range[3, 10], Range[11, 18], Range[19, 36], Range[37, 54], Range[55, 86], Range[87, 118]}; Transpose[elements] // Reverse // Flatten // Select[#, #!=0& ]& (* Jean-François Alcover, Oct 01 2012 *)

A199934 Extended d-block elements for Janet table.

Original entry on oeis.org

21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 347
Offset: 1

Views

Author

Paul Curtz, Nov 12 2011

Keywords

Comments

After the s-block (A160914) and the p-block (A138469), we present the d-block (4*10) elements for the Janet table, in A167268. Janet published his table in a 1928 book (see below), 24 of planche 7, in which he also introduced the s, p, d, f blocks (called nappes 1, 2, 3, 4).
a(n) also comes from one of the two classical Mendeleyev-Moseley-Seaborg periodic table(s) i.e. (see A138096):
1 2
3 4 5 to 10
11 12 13 to 18
19 20 21 to 30 31 to 36
37 38 39 to 48 49 to 54
55 56 57 to 70 71 to 80 81 to 86
87 88 89 to 102 103 to 112 113 to 118.
(The second is in A134982).

References

  • Charles Janet, Essais de classification hélicoidale des éléments chimiques, April 1928, N3, Beauvais, 2 + 104 pages, 4 leaflets (4 to 7).

Links

A249947 Number of available orbitals at increasing subshells in multi-electron atoms.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 5, 3, 1, 5, 3, 1, 7, 5, 3, 1, 7, 5, 3
Offset: 1

Views

Author

Max Alweiss, Nov 08 2014

Keywords

Comments

Follows electron configuration in a multi-electron atom, which is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. Each s subshell holds 1 orbital, each p holds 3 orbitals, each d holds 5 orbitals, and each f holds 7 orbitals.
Theoretically, the integers follow the pattern infinitely, but (conjecturally) the subshells stop after 7p, with 8s not existing as predicted.

Examples

			1s
2s 2p
3s 3p 3d
4s 4p 4d 4f
5s 5p 5d 5f
6s 6p 6d
7s 7p
(8s)
Read antidiagonals.

Links

Crossrefs

Cf. A167268.

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

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

A199426 Janet helicoidal classification of the periodic table.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 12, 13, 14, 15, 18, 17, 16, 19, 20, 21, 22, 23, 24, 25, 30, 29, 28, 27, 26, 31, 32, 33, 36, 35, 34, 37, 38, 39, 40, 41, 42, 43, 48, 47, 46, 45, 44, 49, 50, 51, 54, 53, 52, 55, 56, 57, 58, 59, 60, 61, 62, 63, 70, 69, 68, 67, 66, 65, 64
Offset: 1

Views

Author

Paul Curtz, Nov 06 2011

Keywords

Comments

A permutation of the natural numbers up to 120 (Janet table; in OEIS Wiki, Periodic table). Or more (extension).
Janet explicitly published his table in reference (1), leaflet 7. This was a consequence of his helicoidal classification of the periodic table created with four tangential increasing cylinders on which the numbers are written (2), leaflet 3, (for the first 3 cylinders):
(A) 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.
A boustrophedon path is used. 1 increases, 2 decreases.
a(n) is the vertical terms taken from bottom to top.
By 2 consecutive verticals the numbers of the terms are 2,2,6,2,6,2,10,6,2,... = A167268.

References

  • (1) Charles Janet, Essais de classification hélicoidale des éléments chimiques, avril 1928, N 3, Beauvais, 2+104 pages, 4 leaflets (3 to 7).
  • (2) Charles Janet, La classification hélicoidale des éléments chimiques, novembre 1928, N 4, Beauvais, 2+80 pages, 10 leaflets.

Formula

A167268/2 = 1,1,3,1,3,1,5,3,1,5,3,1,... = b(n). b(n) repeated is every term of A167268 shared in 2 equal parts: 1,1,1,1,3,3,1,1,5,5,3,3,1,1,... = c(n), distribution of verticals of (A).
a(n) is created by mixed increasing 1, 3, 5,6,7, 11, 13,14,15, via b(n) (or both via c(n))
and 2, 4, 10,9,8, 12, 18,17,16, (separately decreasing from right to left for 2, 4, 8,9,10, 11, 16,17,18).

A248928 Interleave (2*n+2)^2 with (2*n+3)^2, both listed n+1 times.

Original entry on oeis.org

4, 9, 16, 16, 25, 25, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 144, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 169, 196, 196, 196, 196, 196, 196, 196, 225, 225, 225, 225, 225, 225, 225
Offset: 0

Views

Author

Paul Curtz, Oct 17 2014

Keywords

Comments

Discovered via Janet's sequence A167268: the result of adding to A167268 the smallest increasing sequence (2, 7, 10, 14, 19, 23, 26, 30, 34, 39, 43, 47, ...) as to get a sequence of nondecreasing squares.
Even terms: 4, 16, 16, 36, 36, 36, ... = 4*A093995(n+1).
Odd terms: (A131507(n) + 2)^2.

Examples

			Seen as an irregular triangle:
4;
9;
16, 16;
25, 25;
36, 36, 36;
49, 49, 49;
64, 64, 64, 64;
81, 81, 81, 81;
...

Links

Crossrefs

Cf. A000267, A007395, A027434, A093995, A131507, A167268.

Programs

  • Mathematica
    Module[{nn=10,a,b},a=Table[PadRight[{},n+1,(2n+2)^2],{n,0,nn}];b= Table[ PadRight[ {},n+1,(2n+3)^2],{n,0,nn}];Riffle[a,b]]//Flatten (* Harvey P. Dale, Jun 10 2022 *)
  • PARI
    vector(60, n, (sqrtint(4*n-3)+1)^2) \\ after Charles R Greathouse IV, Michel Marcus, Oct 27 2014

Formula

a(n) = A027434(n+1)^2.
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.