A168281 -id:A168281 - cp's OEIS frontend

A168380 Row sums of A168281.

2, 4, 12, 20, 38, 56, 88, 120, 170, 220, 292, 364, 462, 560, 688, 816, 978, 1140, 1340, 1540, 1782, 2024, 2312, 2600, 2938, 3276, 3668, 4060, 4510, 4960, 5472, 5984, 6562, 7140, 7788, 8436, 9158, 9880, 10680, 11480, 12362, 13244, 14212, 15180, 16238, 17296, 18448, 19600, 20850, 22100

Offset: 1

Paul Curtz, Nov 24 2009

The atomic numbers of the augmented alkaline earth group in Charles Janet's spiral periodic table are 0 and the first eight terms of this sequence (see Stewart reference). - Alonso del Arte, May 13 2011

Maximum number of 123 patterns in an alternating permutation of length n+3. - Lara Pudwell, Jun 09 2019

			From _Lara Pudwell_, Jun 09 2019: (Start)
a(1)=2. The alternating permutation of length 1+3=4 with the maximum number of copies of 123 is 1324.  The two copies are 124 and 134.
a(2)=4.  The alternating permutation of length 2+3=5 with the maximum number of copies of 123 is 13254.  The four copies are 124, 125, 134, and 135.
a(3)=12. The alternating permutation of length 3+3=6 with the maximum number of copies of 123 is 132546.  The twelve copies are 124, 125, 126, 134, 135, 136, 146, 156, 246, 256, 346, and 356. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 4.
Lara Pudwell, Packing patterns in restricted permutations, 2019.
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.
Philip Stewart, Charles Janet: unrecognized genius of the Periodic System. Foundations of Chemistry (2010), p. 9.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

[(n+1)*(3+2*n^2+4*n-3*(-1)^n)/12: n in [1..50] ]; // Vincenzo Librandi, Aug 06 2011

LinearRecurrence[{2,1,-4,1,2,-1},{2, 4, 12, 20, 38, 56},50] (* G. C. Greubel, Jul 19 2016 *)
Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12, {n, 50}] (* Michael De Vlieger, Jul 20 2016 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,2,1,-4,1,2]^(n-1)*[2;4;12;20;38;56])[1,1] \\ Charles R Greathouse IV, Jul 21 2016

a(n) = 2*A005993(n-1).
a(n) = (n+1)*(3 + 2*n^2 + 4*n - 3*(-1)^n)/12.
a(n+1) - a(n) = A093907(n) = A137583(n+1).
a(2n+1) = A035597(n+1), a(2n) = A002492(n).
a(n) = A099956(n-1), 2 <= n <= 7.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: 2*x*(1 + x^2) / ( (1+x)^2*(x-1)^4 ).
a(n) = A000292(n) + A027656(n-1). - Paul Curtz, Oct 26 2012
E.g.f.: (1/12)*(3*(x - 1) + (3 + 15*x + 12*x^2 + 2*x^3)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 19 2016

A137508 Successive structures of alkaline earth metals (periodic table elements from 2nd column).

Original entry on oeis.org

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

Offset: 1

Paul Curtz, Apr 23 2008

nonn

Apparently a(n) = A168281(n+1). - Georg Fischer, Nov 11 2021

			27 terms: 2, 2 for beryllium, ... Every structure is palindromic (even and odd mixed). Also 2*A106314.

Cf. A005993, A099956, A168281. Same numbers as in 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

cp's OEIS Frontend

A168380 Row sums of A168281.

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A137508 Successive structures of alkaline earth metals (periodic table elements from 2nd column).

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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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