A099956 -id:A099956 - 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

A099955 Atomic numbers in first column in the Mendeleyev periodic table of elements.

Original entry on oeis.org

1, 3, 11, 19, 37, 55, 87

Offset: 1

Parthasarathy Nambi, Nov 12 2004

Atomic numbers of hydrogen and then the alkali metals.

			The atomic number of sodium is 11.

Cf. A099956, alkaline earth metals; A101648, metalloids; A101647, nonmetals (except halogens and noble gases); A097478, halogens; A018227, noble gases; A101649, poor metals.

a(n) = A018227(n-1) + 1. - Lekraj Beedassy, Mar 31 2006

Edited by N. J. A. Sloane at the suggestion of Lekraj Beedassy, Jan 13 2008

A101649 Atomic numbers of the poor metals on the periodic table.

Original entry on oeis.org

13, 31, 49, 50, 81, 82, 83

Offset: 1

Robert Happelberg (roberthappelberg(AT)yahoo.com), Dec 10 2004

Elements 113, 114 and 115 are believed to be poor metals too. Element 116 could be either a poor metal or a metalloid. So I'm not putting the keyword full at this time. The poor metals are between the metalloids and the transition metals on the periodic table.

			The atomic number of aluminium is 13.

The New Encyclopaedia Britannica, 15th Edition, 1991, Volume 15 (Macropedia), Chemical Elements.

International Union of Pure and Applied Chemistry, Periodic Table of Elements
Wikipedia, Periodic table.
Wikipedia, Poor metals.

Cf. A099955, alkali metals; A099956, alkaline earth metals; A101648, metalloids; A101647, nonmetals (except halogens and noble gases); A097478, halogens; A018227, noble gases; A101649, poor metals.

A097478 Atomic numbers of halogens in the periodic table.

Original entry on oeis.org

9, 17, 35, 53, 85, 117

Offset: 1

Parthasarathy Nambi, Sep 19 2004

nonn

The halogens are all on the second rightmost column of the periodic table. (The rightmost column belongs to the noble gases.) - Alonso del Arte, Apr 26 2011

Halogens: 9, fluorine (F); 17;,chlorine (Cl); 35, bromine (Br); 53, iodine (I); 85, astatine (At); and 117, tennessine (Ts). - Daniel Forgues, Jul 13 2015, Luca Khan, Sep 09 2024

Monica Halka and Brian Nordstrom, Halogens and Noble Gases, Facts On File (2010), Part 1: The Halogens.

Bodner Research Web, The Chemistry of the Halogens
Royal Society of Chemistry, Periodic Table, 2015.
Wikipedia, Halogen

Cf. A099955, alkali metals; A099956, alkaline earth metals; A101648, metalloids; A101647, nonmetals (except halogens and noble gases); A018227, noble gases; A101649, poor metals.

a(1) = 9; a(n) = a(n-1) + 2 * (ceiling(n/2) + 1)^2, 2 <= n <= 6.

A101647 Atomic numbers of elements excluding metals, halogens, and noble gases in the periodic table.

Original entry on oeis.org

1, 6, 7, 8, 15, 16, 34

Offset: 1

Robert Happelberg (roberthappelberg(AT)yahoo.com), Dec 10 2004

"Nonmetals" as normally understood in elementary chemistry.
All nonmetals are sandwiched between the halogens and metalloids on the periodic table, with the exception of hydrogen, which is on the same group as the alkali metals. - Happelberg
There may appear to be some disagreement in regards to the definition of nonmetals. This sequence appears on the cover of the Halka and Nordstrom book on nonmetals, and that book has eight chapters, the first seven of which are each devoted to an element identified by this sequence.
The Baldwin book, on the other hand, flatly declares "All elements to the right of the stepped line [on the periodic table shown in that book] are nonmetals," meaning that the author also considers halogens and noble gases to be nonmetals.
But then there is the Halka and Nordstrom book on halogens and noble gases, which says "The halogens ... are nonmetals, but have such special properties that they are given their own classification. The same is true for the noble gases." - Alonso del Arte, Apr 26 2011

			Carbon is a nonmetal, and its atomic number is 6.

Carol Baldwin, Nonmetals. Raintree (2006) p. 13
Monica Halka and Brian Nordstrom, Halogens and Noble Gases, Facts On File (2010), p. xiii
Monica Halka and Brian Nordstrom, Nonmetals, Facts On File (2010)
The New Encyclopaedia Britannica, 15th Edition, 1991, Volume 15 (Macropedia), Chemical Elements.

Yinon Benton, Periodic Table: Non-Metals, from ChemicalElements.com.
Bodner Research Web, The Chemistry Of Nonmetals, from the Purdue University College of Science.
Wikipedia, Nonmetals.

Cf. A099955 (alkali metals), A099956 (alkaline earth metals), A101648 (metalloids), A097478 (halogens), A018227 (noble gases), A101649 (poor metals).

A101648 Atomic numbers of the metalloids or semimetals on the periodic table.

Original entry on oeis.org

5, 14, 32, 33, 51, 52, 84

Offset: 1

Robert Happelberg (roberthappelberg(AT)yahoo.com), Dec 10 2004

If ununhexium and ununseptium turn out to be metalloids, 116 and 117 should be added to this sequence, so I'm not putting the keyword full at this time. The metalloids are sandwiched between the nonmetals and the poor metals on the periodic table.

			The atomic number of boron is 5.

The New Encyclopaedia Britannica, 15th Edition, 1991, Volume 15 (Macropedia), Chemical Elements.

Wikipedia, Periodic table.
Wikipedia, Metalloids.

Cf. A099955, alkali metals; A099956, alkaline earth metals; A101647, nonmetals (except halogens and noble gases); A097478, halogens; A018227, noble gases; A101649, poor metals.

A168281 Triangle T(n,m) = 2*(min(n - m + 1, m))^2 read by rows.

Original entry on oeis.org

2, 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, 2, 8, 18, 32, 32, 18, 8, 2, 2, 8, 18, 32, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 72, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 98, 72

Offset: 1

Paul Curtz, Nov 22 2009

Row sums are A099956(n-1) = 2*A005993(n-1).
The flattened triangle is simply 2 followed by A137508.
If A106314 is interpreted as a triangle, T(n,m) = 2*A106314(n,m).

			The table starts in row n=1 with columns 1<=m<=n as:
  2;
  2,2;
  2,8,2;
  2,8,8,2;
  2,8,18,8,2;
  2,8,18,18,8,2;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10011 (First 141 rows).

Cf. A005993, A099956, A137508, A106314.

A168281 := proc(n,m) 2*(min(n+1-m,m))^2 ; end proc:
seq(seq(A168281(n,m),m=1..n),n=1..20) ;

Table[Map[2 Min[n + # - 1, #]^2 &, Drop[#, -Boole@ EvenQ@ n] ~Join~ Reverse@ # &@ Range@ Floor[n/2]], {n, 2, 14}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

Rephrased all comments in terms of a triangle by R. J. Mathar, Nov 24 2010
More terms from Michael De Vlieger, Jul 19 2016
Definition corrected by Georg Fischer, Nov 11 2021

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.

A244005 Atomic numbers of the pre-transition metals.

Original entry on oeis.org

3, 4, 11, 12, 13, 19, 20, 37, 38, 55, 56, 87, 88

Offset: 1

Arkadiusz Wesolowski, Jun 17 2014

Aluminium (Z = 13) is referred to as a pre-transition metal, along with the group 1 alkali metals (A099955) and group 2 alkaline earth metals (A099956).

P. A. Cox, Instant Notes; Inorganic Chemistry, 2nd edition, Taylor & Francis, 2004, pp. 188-189.

Cf. A099955, A099956.

cp's OEIS Frontend

A168380 Row sums of A168281.

Views

Author

Keywords

Comments

Examples

Links

Programs

Magma

Mathematica

PARI

Formula

A099955 Atomic numbers in first column in the Mendeleyev periodic table of elements.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A101649 Atomic numbers of the poor metals on the periodic table.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A097478 Atomic numbers of halogens in the periodic table.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A101647 Atomic numbers of elements excluding metals, halogens, and noble gases in the periodic table.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A101648 Atomic numbers of the metalloids or semimetals on the periodic table.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A168281 Triangle T(n,m) = 2*(min(n - m + 1, m))^2 read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A244005 Atomic numbers of the pre-transition metals.

Views

Author