A137583 -id:A137583 - cp's OEIS frontend

A137904 Rows 1, 3, 5, 7 of Mendeleyev-Seaborg (extended to 32 columns) periodic table elements.

1, 2, 11, 12, 13, 14, 15, 16, 17, 18, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 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

Offset: 0

Paul Curtz, Apr 30 2008

60 terms. See A137575 (58 terms for rows 2, 4, 6). Differences 9, 19, 33 leads to difference 3 before a(1)=1. Hence -2, antihelium. Like A137583, 2, 2, 8 for A018227, 2, 10, 18, "magic numbers".

R. Arsenescu, Antihelium-3 production in lead-lead collisions at 158 A GeV/c. New Journal of Physics 5:1,2003. doi:10.1088/1367-2630/5/1/301.

Cf. A134984, A137913.

A058318 Number of energy levels in atoms of the n-th element of the periodic table.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Labos Elemer, Dec 12 2000

Run lengths, i.e., how many elements have 1,2,3,4,5,6,7 energy levels, are 2,8,8,18,18,32,26 (see A137583).

			For n=79, element 79 (Gold) has a(79)=6 energy levels (which may have 2,8,18,32,18,1 electrons).

Y. Bentor, Chemical Elements.Com
M. Winter, WebElements Periodic Table.

Cf. A007656, A018227, A058317, A137583, A173592.

seq(i$[2, 8, 8, 18, 18, 32, 32][i], i=1..7); # Michel Lagneau, Apr 03 2024

a(n) = m for s(m-1) < n <= s(m) for m=1..7, where s(m) = A173592(m) and s(0) = 0. - Michel Lagneau, Apr 03 2024

a(106)-a(118) from Michel Lagneau, Apr 03 2024

A166071 Sum of the atomic numbers in the n-th column of the Janet table of the first 120 elements.

Original entry on oeis.org

146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 234, 238, 242, 246, 250, 254, 258, 262, 266, 270, 292, 298, 304, 310, 316, 322, 332, 340

Offset: 1

Paul Curtz, Oct 06 2009

			a(1) = 57 (La) +89 (Ac).
a(14) = 70 (Yb) +102 (No).
a(15) = 21 (Sc) +39 (Y) +71 (Lu) +103 (Lr).
a(24) = 30 (Zn) +48 (Cd) +80 (Hg) +112.
a(25) = 5+13+31+49+81+113.
a(31) = 1 (H) +3 (Li) +11 (Na) +19 (K) +37 (Rb) +55 (Cs) +187 (Fr) +119.

Albert Tarantola, PSE of Elements (Janet form).
Anonymous, Janet periodic table, Web Elements Chemistry Nexus

Cf. A137325, A137583, A138102

sum_{n=1..32} a(n) = A000217(120).

a(32) corrected, tutorial description of the PSE and historical anectodes removed. - R. J. Mathar, Oct 07 2009

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

A137937 A137904(n) - A137575(n).

Original entry on oeis.org

0, 0, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 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

Offset: 0

Paul Curtz, Apr 30 2008

To have 0 upon vertical 2, 10, 18, 36, 54, 86 (noble gases) is not a problem (with the help of A137583. A (second) 0 upon 1, 3, 11, 19, 37, 55, 87 (alkali metals), seems paradoxal. Is 120 terms (elements) sequence 0, 0, 1, 2, .. 117, 118 credible? This leads to a row 0 (0 upon hydrogen, 0 upon helium). Two common points with Janet periodic table elements.

60 terms: another, with A137913, theoritical companion to A137904.

Cf. A134984, A137575, A137904.

A137913 prepended with 0, 0.

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.

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

A137904 Rows 1, 3, 5, 7 of Mendeleyev-Seaborg (extended to 32 columns) periodic table elements.

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A058318 Number of energy levels in atoms of the n-th element of the periodic table.

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A166071 Sum of the atomic numbers in the n-th column of the Janet table of the first 120 elements.

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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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A137937 A137904(n) - A137575(n).

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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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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.