A137583 -id:A137583 - 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

A172002 A permutation of the natural numbers in groups of 2*k^2, k=1,2,....

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 7, 10, 6, 11, 5, 12, 16, 17, 15, 18, 14, 19, 13, 20, 29, 30, 28, 31, 27, 32, 26, 33, 25, 34, 24, 35, 23, 36, 22, 37, 21, 38, 47, 48, 46, 49, 45, 50, 44, 51, 43, 52, 42, 53, 41, 54, 40, 55, 39, 56, 72, 73, 71, 74, 70, 75, 69, 76, 68, 77, 67, 78, 66, 79, 65, 80

Offset: 1

Paul Curtz, Jan 22 2010

nonn

The idea is based on the Janet table of the elements (see A138509 and A171710). Arrange the atomic numbers as if the rows of the table were centered. There are two rows with 2 =2*1^2 elements, 2 rows with 8=2*2^2 elements, 2 rows with 18=2*3^2 elements, and this is extended infinitely by adding 2 rows with 2*k^2 elements (see A137583), incrementing k:
...........................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.
The sequence is obtained by reading the numbers in each of the rows (top-down), starting with the center left column, then the center right column, and then alternating from the left to the right, increasing the distance to the center until all 2*k^2 numbers of the block are exhausted.

Michael De Vlieger, Table of n, a(n) for n = 1..10680 (First 76 rows)
Anonymous, Janet periodic table, Web Elements Chemistry
Anonymous, Periodic Table: Formulations, Chemogenesis web book
Albert Tarantola, PSE of Elements (Janet form).

Table[(Riffle[Reverse@ #, Length@ # + #] &@ Range[Ceiling[n/2]^2]) + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 7}] // Flatten (* Michael De Vlieger, Jul 19 2016, after Vincenzo Librandi at A168380 *)

Edited by R. J. Mathar, Mar 02 2010

A138509 Janet (or left-step) periodic table from right to left. 120 terms. 8 rows, 32 columns without spaces.

Original entry on oeis.org

2, 1, 4, 3, 12, 11, 10, 9, 8, 7, 6, 5, 20, 19, 18, 17, 16, 15, 14, 13, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89

Offset: 1

Paul Curtz, May 10 2008

Cf. A137583, A168380 (first number in each row), A171710, A172002.

Table[Reverse@ Range[2 Ceiling[n/2]^2] + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 8}] // Flatten (* Michael De Vlieger, Jul 20 2016 *)

There are respectively A137583 = 2, 2, 8, 8, 18, 18, 32, 32 terms from right to left.

A138100 The atomic numbers read along the odd-indexed rows of the Janet table of the elements.

Original entry on oeis.org

1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 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

Offset: 1

Paul Curtz, May 03 2008

The union with A138101 gives the first 120 terms of A000027.

			Starts with 1 and 2 of the first row (H and He). Next come 5 to 12 of the 3rd row (B to Mg).

Albert Tarantola, PSE of Elements (Janet form).

Cf. A000027, A138101 (even-indexed rows), A138509 (Left-step Janet periodic table), A137583, A172002.

Table[Range[2 Ceiling[n/2]^2] + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 1, 7, 2}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)

Edited by R. J. Mathar, Oct 07 2009

A138101 The atomic numbers read along the even-indexed rows of the Janet table of the elements.

Original entry on oeis.org

3, 4, 13, 14, 15, 16, 17, 18, 19, 20, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 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, 119, 120

Offset: 1

Paul Curtz, May 03 2008

The union with A138100 gives the first 120 terms of A000027.

			Starts with 3 and 4 of the 2nd row (Li and Be). Next come 13 to 20 of the 4th row (Al to Ca).

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

Cf. A000027, A138100 (odd-indexed rows), A138509 (Left-step Janet periodic table), A137583, A172002.

Table[Range[2 Ceiling[n/2]^2] + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 2, 8, 2}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)

Edited by R. J. Mathar, Oct 07 2009

A168142 Count downwards from 2, then from 8, then from 18, then from ... 2*k^2, k>=1.

Original entry on oeis.org

2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

Offset: 1

Paul Curtz, Nov 19 2009

nonn

Janet's extended enumeration of the periodic table of the elements.

The table is read from the right to the left.

Charles Janet, La structure du Noyau de l'atome,consideree dans la Classification periodique des elements chimiques, Nov. 1927, N. 2, Beauvais, 67 pages, 3 leafleats, see page 15.
Charles Janet, Considerations sur la structure du noyau de l'atome, Dec 1929, N 5, Beauvais, 2+45 pp.,4 leaflets, see leaflets 2 and 3.

Cf. A138100, A138101, A093907, A137583, A172002.

Table[Reverse@ Range[2 n^2], {n, 5}] // Flatten (* Michael De Vlieger, Jul 22 2016 *)

Edited by R. J. Mathar, Feb 15 2010

A138096 Number of elements in the n-th column of the second extension of the Mendeleyev-Seaborg periodic table of elements.

Original entry on oeis.org

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

Offset: 1

Paul Curtz, May 03 2008

A variant of A134982, with one term "4" changing place because (relative to the first table) the two elements number 21 and 39 are moved over to the next column of d-metals, placed above element 71.

Cf. A137583, A093907, A134984, A135559.

A138102 The number 2k^2 repeated 2k^2 times, k=1 to 4.

Original entry on oeis.org

2, 2, 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, 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: 1

Paul Curtz, May 03 2008

Vaguely related to the number of elements in the Janet table A138102.
Essentially the same as A137575.
a(n)= (A138101=3,4,13,14,15,16,17,) - (A138100=1,2,5,6,7,8,9,) . [From Paul Curtz, Oct 11 2009]

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

Cf. A138101, A137325, A137583, A166071, A001105.

Edited by R. J. Mathar, Oct 07 2009

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

Paul Curtz, Feb 22 2010

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.

			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

SilcoTek, Periodic table of the Elements
Mark R. Leach, 2012 Compact Mendeleev-Moseley-Seaborg Periodic Table (CMMSPT), Database of Periodic Tables (from Chemogenesis).

Cf. A167268, A137583, A137325.

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 *)

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

Original entry on oeis.org

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

cp's OEIS Frontend

A168380 Row sums of A168281.

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A172002 A permutation of the natural numbers in groups of 2*k^2, k=1,2,....

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A138509 Janet (or left-step) periodic table from right to left. 120 terms. 8 rows, 32 columns without spaces.

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A138100 The atomic numbers read along the odd-indexed rows of the Janet table of the elements.

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A138101 The atomic numbers read along the even-indexed rows of the Janet table of the elements.

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A168142 Count downwards from 2, then from 8, then from 18, then from ... 2*k^2, k>=1.

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A138096 Number of elements in the n-th column of the second extension of the Mendeleyev-Seaborg periodic table of elements.

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A138102 The number 2*k^2 repeated 2*k^2 times, k=1 to 4.

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

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A138102 The number 2k^2 repeated 2k^2 times, k=1 to 4.