A093907 -id:A093907 - cp's OEIS frontend

A134982 Number of elements in the n-th column of Medeleiev's periodic system of elements.

7, 6, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 7

Offset: 1

Paul Curtz, Feb 05 2008

a(1)=7 represents Hydrogen and the alkali metals. Ends with a(32)=7 for 7 noble gases. These are the block counts (grouping) of the atomic numbers A134984. - R. J. Mathar, Feb 08 2008
The corresponding row numbers are A093907; the sum over all a(n) equals 118 in both cases. - R. J. Mathar, Feb 08 2008
Comment from Paul Curtz, May 03 2008 (Start):
1................................................................................................................2
3..4.........................................................................................5...6...7...8...9..10
11.12.......................................................................................13..14..15..16..17..18
19.20.21................................................22..23..24..25..26..27..28..29..30..31..32..33..34..35..36
37.38.39................................................40..41..42..43..44..45..46..47..48..49..50..51..52..53..54
55.56.57.58.59.60.61.62.63.64.65.66.67..68..69..70..71..72..73..74..75..76..77..78..79..80..81..82..83..84..85..86
87.88.89.90.91.92.93.94.95.96.97.98.99.100.101.102.103.104.105.106.107.108.109.110.111.112.113.114.115.116.117.118
(End)

Cf. A134984, A093907, A137583, A135559.

Edited by R. J. Mathar, Feb 08 2008

A333662 Number of electrons per shell in element Z=n expressed as a 32-bit unsigned integer.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 34, 98, 162, 226, 290, 354, 418, 482, 546, 2594, 4642, 4706, 4770, 4834, 2914, 4962, 5026, 5090, 5154, 3234, 5282, 7330, 9378, 11426, 13474, 15522, 17570, 148642, 279714, 281762, 283810

Offset: 1

Zachary Russ, Sep 02 2020

This is a memory-efficient way of encoding the number of electrons per shell of all known elements.

			  |-----|------------|------------|----------------------------------------|
  |     |            |            |               a(n) [bin]               |
  |  n  | a(n) [dec] | a(n) [hex] |           Electrons per Shell          |
  |     |            |            |  Q  |  P  |   O  |  N   |  M  |  L | K |
  |-----|------------|------------|-----|-----|------|------|-----|----|---|
  |   1 |          1 | 0x00000001 | 0000 00000 000000 000000 00000 0000 01 |
  |   2 |          2 | 0x00000002 | 0000 00000 000000 000000 00000 0000 10 |
  |   3 |          6 | 0x00000006 | 0000 00000 000000 000000 00000 0001 10 |
  |   4 |         10 | 0x0000000a | 0000 00000 000000 000000 00000 0010 10 |
  |   5 |         14 | 0x0000000e | 0000 00000 000000 000000 00000 0011 10 |
  |   6 |         18 | 0x00000012 | 0000 00000 000000 000000 00000 0100 10 |
  |  .  |     .      |      .     |                    .                   |
  |  .  |     .      |      .     |                    .                   |
  |  .  |     .      |      .     |                    .                   |
  | 113 |  960562338 | 0x394104a2 | 0011 10010 100000 100000 10010 1000 10 |
  | 114 | 1228997794 | 0x494104a2 | 0100 10010 100000 100000 10010 1000 10 |
  | 115 | 1497433250 | 0x594104a2 | 0101 10010 100000 100000 10010 1000 10 |
  | 116 | 1765868706 | 0x694104a2 | 0110 10010 100000 100000 10010 1000 10 |
  | 117 | 2034304162 | 0x794104a2 | 0111 10010 100000 100000 10010 1000 10 |
  | 118 | 2302739618 | 0x894104a2 | 1000 10010 100000 100000 10010 1000 10 |
  |-----|------------|------------|----------------------------------------|

Zachary Russ, Table of n, a(n) for n = 1..118
Wikipedia, Electron configurations of the elements (data page)

Cf. A333997, A168208, A093907.

A333997 Number of electrons per subshell in element Z=n expressed as a 56-bit unsigned integer.

Original entry on oeis.org

1, 2, 6, 10, 26, 42, 58, 74, 90, 106, 234, 362, 874, 1386, 1898, 2410, 2922, 3434, 68970, 134506, 138602, 142698, 146794, 89450, 154986, 159082, 163178, 167274, 109930, 175466, 437610, 699754, 961898, 1224042, 1486186, 1748330

Offset: 1

Zachary Russ, Sep 05 2020

This is a memory-efficient way of encoding the number of electrons per subshell of all known elements.

			  |-----|----------------------------------------------------------------------------|
  |     |                                a(n) [bin]                                  |
  |  n  |                          Electrons per Subshell                            |
  |     |  7p|7s| 6d | 6p|6s| 5f | 5d | 5p|5s| 4f | 4d | 4p|4s| 3d | 3p|3s| 2p|2s|1s |
  |-----|----|--|----|---|--|----|----|---|--|----|----|---|--|----|---|--|---|--|---|
  |   1 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 00 01 |
  |   2 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 00 10 |
  |   3 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 01 10 |
  |   4 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 10 10 |
  |   5 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 001 10 10 |
  |   6 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 010 10 10 |
  |  .  |                                     .                                      |
  |  .  |                                     .                                      |
  |  .  |                                     .                                      |
  | 113 | 001 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 114 | 010 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 115 | 011 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 116 | 100 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 117 | 101 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 118 | 110 10 1010 011 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  |-----|----------------------------------------------------------------------------|

Zachary Russ, Table of n, a(n) for n = 1..118
Wikipedia, Electron configurations of the elements (data page)

Cf. A333662, A168208, A093907.

A217927 Elements of the horizontal ADOMAH periodic table written from right to left, from bottom to top.

Original entry on oeis.org

2, 1, 4, 3, 10, 9, 8, 7, 6, 5, 12, 11, 18, 17, 16, 15, 14, 13, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 36, 35, 34, 33, 32, 31, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 38, 37, 54, 53, 52, 51, 50, 49, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89

Offset: 1

Paul Curtz, Oct 15 2012

We first write a variant of the ADOMAH periodic table:
                                                              1  2
                                           5  6  7  8  9 10   3  4
           21 22 23 24 25 26 27 28 29 30  13 14 15 16 17 18  11 12
57 to  70  39 40 41 42 43 44 45 46 47 48  31 32 33 34 35 36  19 20     (B)
89 to 102  71 72 73 74 75 76 77 78 79 80  49 50 51 52 53 54  37 38
          103104105106107108109110111112  81 82 83 84 85 86  55 56
                                         113114115116117118  87 88
                                                            119120
(See A219388).
It could be written vertically.
Differences of (2,4,12,20,38,56,88,120,... = A168380) = 2,8,8,18,18,... = A093907 from a 118 terms table.
The number of elements in the n-th period (2,8,18,32,32,18,8,2) is in A168281. Compare to A093907 = 2,8,8,18,18,32,32,50,...(extension of the Mendeleyev-Moseley-Seaborg table) and A137583 = 2,2,8,8,18,18,32,32. See the possible element 120 in A168208 (which must be clarified).
The horizontal ADOMAH periodic table (2006) is
                                                            119120
                                         113114115116117118  87 88
          103104105106107108109110111112  81 82 83 84 85 86  55 56
89 to 102  71 72 73 74 75 76 77 78 79 80  49 50 51 52 53 54  37 38     (A)
57 to  70  39 40 41 42 43 44 45 46 47 48  31 32 33 34 35 36  19 20
           21 22 23 24 25 26 27 28 29 30  13 14 15 16 17 18  11 12
                                           5  6  7  8  9 10   3  4
                                                              1  2
Generally it is written vertically.

Philip J. Stewart, "Charles Janet, Unrecognized genius of the periodic system", Foundations of Chemistry, January, 2009. ISSN 1386-4238.

V. Tsimmerman, ADOMAH Periodic Table

Cf. A137325.

Reference given by Jean-François Alcover, Oct 22 2012

Typos corrected in comments by Jean-François Alcover, Nov 16 2012

A219527 a(n) = (6n^2 + 7n - 9 + 2n^3)/12 - (-1)^n(n+1)/4.

Original entry on oeis.org

1, 3, 11, 19, 37, 55, 87, 119, 169, 219, 291, 363, 461, 559, 687, 815, 977, 1139, 1339, 1539, 1781, 2023, 2311, 2599, 2937, 3275, 3667, 4059, 4509, 4959, 5471, 5983, 6561, 7139, 7787, 8435, 9157, 9879, 10679, 11479, 12361, 13243

Offset: 1

Paul Curtz, Nov 21 2012

First column of the Mendeleyev-Moseley-Seaborg table (with alkali metals) or 31st column of the Janet table. See A138726.
(a(n+10) - a(n))/10 = 29, 36, 45, 54, ... = A061925(n+7) + 3.
b(n) = a(n+1) - 2*a(n) = 1, 5, -3, -1, -19, -23, -55, -69, -119, -147, -219, -265, -363, -431, ... contains -a(2*n).
b(2*n-1) - b(2*n-2) = 4, 2, -4, -14, -28, -46, -68, ... = A147973(n+3).

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A147973.

a[n_] := (6*n^2 + 7*n - 9 + 2*n^3)/12 - (-1)^n*(n + 1)/4; Table[ a[n], {n, 1, 42}] (* Jean-François Alcover, Apr 05 2013 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,3,11,19,37,55},50] (* Harvey P. Dale, Apr 01 2018 *)

a(n) = A168380(n+1) - 1.
a(n+2) - a(n+1) = A093907(n) = A137583(n+1).
a(n+3) - a(n+1) = 10,16,26,36,... = A137928(n+3).
G.f. x*(1 + x + 4*x^2 - 2*x^3 + x^5 - x^4) / ( (1+x)^2*(x-1)^4 ). - R. J. Mathar, Mar 27 2013

A346370 Upper bound for the number of solutions of the TRINTUM cube puzzle n X 1 X 1 (cuboid formed by 4n + 2 parts) different by the set of parts, which are distinguished by the amount of surface area they contribute to the assembled cuboid.

Original entry on oeis.org

3, 8, 10, 19, 22, 34, 38, 54, 59, 78, 84, 107, 114, 140, 148, 178, 187, 220, 230, 267, 278, 318, 330, 374, 387, 434, 448, 499, 514, 568, 584, 642, 659, 720, 738, 803, 822, 890, 910, 982, 1003, 1078, 1100, 1179, 1202, 1284, 1308, 1394, 1419, 1508, 1534, 1627

Offset: 1

Mikhail Kurkov, Jul 14 2021 [verification needed]

There are 6 parts in a basic set with a given surface area (with addition of an empty cell in the center): A-part (19), H-part (21), fish (21), rooster (27) x2 and tower (35). For purposes of this sequence, the H-part and the fish are equivalent, since they contribute the same surface area.
Equivalently, number of nonnegative integer solutions of the system of equations 19x + 21y + 27z + 35w = 96n + 54, x + y + z + w = 4n + 2. Here x = 3m + w, y = 2n - 4m, z = 2(n - w + 1) + m, so for nonnegative integers we have 0 <= m <= floor(n/2), 0 <= w <= n + floor(n/4) + 1 and number of solutions is Sum_{k=0..floor(n/2)} n + floor(k/2) + 2 = (n + 2)*(floor(n/2) + 1) + floor(floor(n/2)^2/4) for n > 0.
Conjecture: upper bound can be reduced to n+2 (based on attempts to construct cuboid using nonnegative integer solutions with m > 0, where it looks like impossible to place at least all roosters and towers somewhere).
The simplest way to construct cuboid of any size is to use cubes formed by 2 A-parts, 2 H-parts and 2 towers. We just remove an A-part from one cube and a tower from another to easily connect them together.

			a(1) = 3 because there are only 3 possible nonnegative integer solutions:
  19*0 + 21*2 + 27*4 + 35*0 = 150, 0 + 2 + 4 + 0 = 6;
  19*1 + 21*2 + 27*2 + 35*1 = 150, 1 + 2 + 2 + 1 = 6;
  19*2 + 21*2 + 27*0 + 35*2 = 150, 2 + 2 + 0 + 2 = 6.
a(2) = 8 because there are only 8 possible nonnegative integer solutions:
  19*0 + 21*4 + 27*6 + 35*0 = 246, 0 + 4 + 6 + 0 = 10;
  19*1 + 21*4 + 27*4 + 35*1 = 246, 1 + 4 + 4 + 1 = 10;
  19*2 + 21*4 + 27*2 + 35*2 = 246, 2 + 4 + 2 + 2 = 10;
  19*3 + 21*4 + 27*0 + 35*3 = 246, 3 + 4 + 0 + 3 = 10;
  19*3 + 21*0 + 27*7 + 35*0 = 246, 3 + 0 + 7 + 0 = 10;
  19*4 + 21*0 + 27*5 + 35*1 = 246, 4 + 0 + 5 + 1 = 10;
  19*5 + 21*0 + 27*3 + 35*2 = 246, 5 + 0 + 3 + 2 = 10;
  19*6 + 21*0 + 27*1 + 35*3 = 246, 6 + 0 + 1 + 3 = 10.

Mikhail Kurkov, Demonstration of the TRINTUM cube puzzle, YouTube video. [verification needed]
Mikhail Kurkov, Parts of the TRINTUM cube puzzle, image. [verification needed]
Mikhail Kurkov, Solutions for the cube, image. [verification needed]
Mikhail Kurkov, Some solutions for the 2 X 1 X 1 cuboid, image. [verification needed]
Mikhail Kurkov, Some solutions for the 3 X 1 X 1 cuboid, image. [verification needed]
Jon Maiga, Computer-generated formulas for A346370, Sequence Machine.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Cf. A102214.

LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {3, 8, 10, 19, 22, 34,
38}, 52] (* Robert P. P. McKone, Jul 16 2021 *)

a(n) = (n + 2)*(floor(n/2) + 1) + floor(floor(n/2)^2/4).
a(n) = A102214(floor(n/2)) + (1 + n mod 2)*(1 + floor(n/2)).
G.f.: x*(3 + 5*x - x^2 + 4*x^3 - 2*x^4 - 2*x^5 + 2*x^6)/((1 - x)^3*(1 + x)^2*(1 + x^2)). - Stefano Spezia, Jul 14 2021
Conjecture: a(n) = A008733(n-1) + A093907(n) for n > 0 (noticed by Sequence Machine). - Mikhail Kurkov, Oct 14 2021 [verification needed]

A138300 Differences of each column for atomic numbers of Mendeleyev-Seaborg 7*32 elements periodic table,first extension,A138096 table.86 terms.Horizontal lecture.

Original entry on oeis.org

2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32

Offset: 0

Paul Curtz, May 07 2008

			First column differences: 2, 8, 8, 18, 18, 32; second: 8, 8, 18, 18, 32.
Table: 1 2, 11 8's, 27 18's, 47 32's
2.............................................................................................8
8..8...........................................................................8..8..8..8..8..8
8..8..........................................................................18.18.18.18.18.18
18.18.18...........................................18.18.18.18.18.18.18.18.18.18.18.18.18.18.18
18.18.18...........................................32.32.32.32.32.32.32.32.32.32.32.32.32.32.32
32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32

Cf. A093907, A137508, A137575, A137583, A138102.

A173788 Nuclear charge numbers in Eric Scerri's periodic table of the elements read down columns.

Original entry on oeis.org

1, 9, 17, 35, 53, 85, 2, 10, 18, 36, 54, 86, 3, 11, 19, 37, 55, 87, 4, 12, 20, 38, 56, 88, 57, 89, 58, 90, 59, 91, 60, 92, 61, 93, 62, 94, 63, 95, 64, 96, 65, 97, 66, 98, 67, 99, 68, 100, 69, 101, 70, 102, 21, 39, 71, 103, 22, 40, 72, 104, 23, 41, 73, 105, 24, 42, 74, 106, 25

Offset: 1

Paul Curtz, Feb 24 2010

A permutation of the natural numbers from 1 to 116.
The table has 32 columns and 6 rows:
.H.He.Li.Be                                    .B..C..N..O
.F.Ne.Na.Mg                                    Al.Si..P..S
Cl.Ar..K.Ca     .Sc.Ti..V.Cr.Mn.Fe.Co.Ni.Cu.Zn.Ga.Ge.As.Se
Br.Kr.Rb.Sr     ..Y.Zr.Nb.Mo.Tc.Ru.Rh.Pd.Ag.Cd.In.Sn.Sb.Te
.I.Xe.Cs.Ba (La).Lu.Hf.Ta..W.Re.Os.Ir.Pt.Au.Hg.Tl.Pb.Bi.Po
At.Rn.Fr.Ra (Ac).Lr.Rf.Db.Sg.Bh.Hs.Mt.Ds.RgUubUutUuqUupUUh
The Lanthanides and Actinides are not expanded above.
The first 4 columns contain 6 elements each, the next 14 columns (not detailed above) are the Lanthanides and Actinides, then there are 10 columns with 4 elements each and the final 4 columns with 6 each.

			The first column is H (=1), F (=9), Cl (=17) down to At (=85).

Mark R. Leach, Eric Scerri's Triad Periodic Table

Cf. A093907

A173991 Bayley-Thomsen-Bohr periodic table(s) (1882-1895-1922) adapted by Scerri (1997).

Original entry on oeis.org

1, 2, 6, 7, 5, 8, 4, 9, 3, 10, 14, 15, 13, 16, 12, 17, 11, 18, 27, 28, 26, 29, 25, 30, 24, 31, 23, 32, 22, 33, 21, 34, 20, 35, 19, 36, 45, 46, 44, 47, 43, 48, 42, 49, 41, 50, 40, 51, 39, 52, 38, 53, 37, 54, 70, 71, 69, 72, 68, 73, 67, 74, 66, 75, 65, 76, 64, 77, 63, 78, 62, 79

Offset: 1

Paul Curtz, Mar 04 2010

This a compact (no spaces) symmetric table of (7 rows, 32 columns) 118 elements. Also from Mendeleyev-Moseley-Seaborg. A permutation of the numbers from 1 to 118. The writing is the same as A172002, from Janet table.

Writing begins from central (two) columns. Number of terms by columns: 2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,7,7,6,6,6,4,4,4,4,4,2,2,2,2,2,2,2; by rows: 2,8,8,18,18,32,32 (see A093907 and A137583).

a(n)= A172002(n+2) - 2.

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

Paul Curtz, Nov 07 2011

nonn

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.

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

Charles Janet, Three-Dimensional Spiral-Tube System

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.

cp's OEIS Frontend

A134982 Number of elements in the n-th column of Medeleiev's periodic system of elements.

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A333662 Number of electrons per shell in element Z=n expressed as a 32-bit unsigned integer.

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A333997 Number of electrons per subshell in element Z=n expressed as a 56-bit unsigned integer.

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A217927 Elements of the horizontal ADOMAH periodic table written from right to left, from bottom to top.

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A219527 a(n) = (6*n^2 + 7*n - 9 + 2*n^3)/12 - (-1)^n*(n+1)/4.

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A346370 Upper bound for the number of solutions of the TRINTUM cube puzzle n X 1 X 1 (cuboid formed by 4n + 2 parts) different by the set of parts, which are distinguished by the amount of surface area they contribute to the assembled cuboid.

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A138300 Differences of each column for atomic numbers of Mendeleyev-Seaborg 7*32 elements periodic table,first extension,A138096 table.86 terms.Horizontal lecture.

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A173788 Nuclear charge numbers in Eric Scerri's periodic table of the elements read down columns.

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A173991 Bayley-Thomsen-Bohr periodic table(s) (1882-1895-1922) adapted by Scerri (1997).

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A199502 From Janet helicoidal classification of the periodic table.

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A219527 a(n) = (6n^2 + 7n - 9 + 2n^3)/12 - (-1)^n(n+1)/4.