A018227 -id:A018227 - cp's OEIS frontend

A271994 The chalcogen sequence (a(n) = A018227(n)-2).

8, 16, 34, 52, 84, 116, 166, 216, 288, 360, 458, 556, 684, 812, 974, 1136, 1336, 1536, 1778, 2020, 2308, 2596, 2934, 3272, 3664, 4056, 4506, 4956, 5468, 5980, 6558, 7136, 7784, 8432, 9154, 9876, 10676, 11476, 12358, 13240, 14208, 15176, 16234, 17292, 18444

Offset: 2

Natan Arie Consigli, May 28 2016

Terms up to 116 are the atomic numbers of the elements of group 16 in the periodic table. Those elements are also known as chalcogens.

Colin Barker, Table of n, a(n) for n = 2..1000
Wikipedia, Chalcogen
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Table[(2 n^3 + 12 n^2 + 25 n + (-1)^n 3 (n + 2) - 30)/12, {n, 2, 43}] (* or *)
Drop[#, 2] &@ CoefficientList[Series[2 x^2 (4 - 3 x^2 + x^4)/((1 - x)^4 (1 + x)^2), {x, 0, 43}], x] (* Michael De Vlieger, May 29 2016 *)
LinearRecurrence[{2,1,-4,1,2,-1},{8,16,34,52,84,116},50] (* Harvey P. Dale, Sep 24 2022 *)

Vec(2*x^2*(4-3*x^2+x^4)/((1-x)^4*(1+x)^2) + O(x^50)) \\ Colin Barker, May 29 2016

From Colin Barker, May 29 2016: (Start)
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6) for n>7.
G.f.: 2*x^2*(4-3*x^2+x^4) / ((1-x)^4*(1+x)^2).
(End)
a(n) = (2*n^3 + 12*n^2 + 25*n + (-1)^n*3*(n + 2) - 30)/12. - Ilya Gutkovskiy, May 29 2016

A271995 The Pnictogen sequence: a(n) = A018227(n)-3.

Original entry on oeis.org

7, 15, 33, 51, 83, 115, 165, 215, 287, 359, 457, 555, 683, 811, 973, 1135, 1335, 1535, 1777, 2019, 2307, 2595, 2933, 3271, 3663, 4055, 4505, 4955, 5467, 5979, 6557, 7135, 7783, 8431, 9153, 9875, 10675, 11475, 12357, 13239, 14207, 15175, 16233, 17291, 18443

Offset: 2

Natan Arie Consigli, Jun 18 2016

Terms up to 115 are the atomic numbers of the elements of group 15 in the periodic table. Those elements are also known as pnictogens.

Colin Barker, Table of n, a(n) for n = 2..1000
Wikipedia, Pnictogen.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A018227.

LinearRecurrence[{2,1,-4,1,2,-1},{7,15,33,51,83,115},50] (* Harvey P. Dale, Oct 29 2023 *)

Vec(x^2*(7+x-4*x^2-2*x^3+x^4+x^5)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jun 19 2016, corrected Jun 26 2016

From Colin Barker, Jun 19 2016, corrected Jun 26 2016: (Start)
a(n) = (6*(-7+(-1)^n)+(25+3*(-1)^n)*n+12*n^2+2*n^3)/12.
a(n) = (n^3+6*n^2+14*n-18)/6 for n even.
a(n) = (n^3+6*n^2+11*n-24)/6 for n odd.
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6) for n>7.
G.f.: x^2*(7+x-4*x^2-2*x^3+x^4+x^5) / ((1-x)^4*(1+x)^2).
(End)

A271996 The crystallogen sequence (a(n) = A018227(n)-4).

Original entry on oeis.org

6, 14, 32, 50, 82, 114, 164, 214, 286, 358, 456, 554, 682, 810, 972, 1134, 1334, 1534, 1776, 2018, 2306, 2594, 2932, 3270, 3662, 4054, 4504, 4954, 5466, 5978, 6556, 7134, 7782, 8430, 9152, 9874, 10674, 11474, 12356, 13238, 14206, 15174, 16232, 17290, 18442

Offset: 2

Natan Arie Consigli, Jun 18 2016

Terms up to 114 are the atomic numbers of the elements of group 14 in the periodic table. Those elements are also called crystallogens.

Colin Barker, Table of n, a(n) for n = 2..1000
Wikipedia, Carbon group.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A173592, A018227.

Table[(6*(-9+(-1)^n)+(25+3*(-1)^n)*n+12*n^2+2*n^3)/12, {n, 2, 10}] (* or *) LinearRecurrence[{2, 1, -4, 1, 2, -1}, {6, 14, 32, 50, 82, 114}, 50] (* G. C. Greubel, Jun 23 2016 *)

Vec(2*x^2*(3+x-x^2-2*x^3+x^5)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jun 19 2016

From Colin Barker, Jun 19 2016: (Start)
a(n) = (6*(-9 + (-1)^n) + (25 + 3*(-1)^n)*n + 12*n^2 + 2*n^3)/12.
a(n) = (n^3 + 6*n^2 + 14*n - 24)/6 for n even.
a(n) = (n^3 + 6*n^2 + 11*n - 30)/6 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>7.
G.f.: 2*x^2*(3 + x - x^2 - 2*x^3 + x^5) / ((1-x)^4*(1+x)^2).
(End)

A271997 The icosagen sequence : a(n) = A018227(n)-5, for n >= 2.

Original entry on oeis.org

5, 13, 31, 49, 81, 113, 163, 213, 285, 357, 455, 553, 681, 809, 971, 1133, 1333, 1533, 1775, 2017, 2305, 2593, 2931, 3269, 3661, 4053, 4503, 4953, 5465, 5977, 6555, 7133, 7781, 8429, 9151, 9873, 10673, 11473, 12355, 13237, 14205, 15173, 16231, 17289, 18441

Offset: 2

Natan Arie Consigli, Jun 19 2016

Terms up to 113 are the atomic numbers of the elements of group 13 in the periodic table. Those elements are also called icosagens.

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Table[n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 6, {n,2,10}] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{5, 13, 31, 49, 81, 113},50] (* G. C. Greubel, Jun 23 2016 *)

From G. C. Greubel, Jun 23 2016: (Start)
a(n) = n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 6, for n >= 2.
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.: x^2 * (5 + 3*x - 6*x^3 - x^4 + 3*x^5)/( (1-x)^4 * (1+x)^2 ). (End)

A271998 Volatile sequence: a(n) = A018227(n)-6.

Original entry on oeis.org

30, 48, 80, 112, 162, 212, 284, 356, 454, 552, 680, 808, 970, 1132, 1332, 1532, 1774, 2016, 2304, 2592, 2930, 3268, 3660, 4052, 4502, 4952, 5464, 5976, 6554, 7132, 7780, 8428, 9150, 9872, 10672, 11472, 12354, 13236, 14204, 15172, 16230, 17288, 18440

Offset: 3

Natan Arie Consigli, Jun 27 2016

Terms up to 112 are the atomic numbers of the elements of group 12 in the periodic table. The group is also known as the volatile metals since almost all elements with that property fall in to group 12.

Colin Barker, Table of n, a(n) for n = 3..1000
Wikipedia, Group 12 element.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

LinearRecurrence[{2,1,-4,1,2,-1},{30,48,80,112,162,212},50] (* Harvey P. Dale, Mar 07 2022 *)

Vec(2*x^3*(15-6*x-23*x^2+12*x^3+10*x^4-6*x^5)/((1-x)^4*(1+x)^2) + O(x^60)) \\ Colin Barker, Oct 25 2016

From Colin Barker, Oct 25 2016: (Start)
G.f.: 2*x^3*(15 - 6*x - 23*x^2 + 12*x^3 + 10*x^4 - 6*x^5)/((1 - x)^4*(1 + x)^2).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>8.
a(n) = (n^3 + 9*n^2 + 26*n - 24)/6 for n even.
a(n) = (n^3 + 9*n^2 + 29*n - 15)/6 for n odd. (End)

A271999 Halogen sequence: a(n) = A018227(n)-1.

Original entry on oeis.org

1, 11, 17, 35, 53, 85, 117, 167, 217, 289, 361, 459, 557, 685, 813, 975, 1137, 1337, 1537, 1779, 2021, 2309, 2597, 2935, 3273, 3665, 4057, 4507, 4957, 5469, 5981, 6559, 7137, 7785, 8433, 9155, 9877, 10677, 11477, 12359, 13241, 14209, 15177, 16235, 17293

Offset: 1

Natan Arie Consigli, Jul 02 2016

Terms from 11 to 117 are the atomic numbers of the elements in group 17 in the periodic table. The elements in this group are also called halogens.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 11, 17, 35, 53, 85, 117, 167}, 50] (* Paolo Xausa, Oct 21 2024 *)

Vec(x*(1 + 9*x - 6*x^2 - 6*x^3 + 9*x^4 - x^5 - 4*x^6 + 2*x^7) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Colin Barker, Nov 14 2017

From Colin Barker, Nov 14 2017: (Start)
G.f.: x*(1 + 9*x - 6*x^2 - 6*x^3 + 9*x^4 - x^5 - 4*x^6 + 2*x^7) / ((1 - x)^4*(1 + x)^2).
a(n) = (1/12)*(2*n^3 + 12*n^2 + 28*n - 12) for n>2 and even.
a(n) = (1/12)*(2*n^3 + 12*n^2 + 22*n - 24) for n>2 and odd.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>8.
(End)

A272000 Coinage sequence: a(n) = A018227(n)-7.

Original entry on oeis.org

3, 11, 29, 47, 79, 111, 161, 211, 283, 355, 453, 551, 679, 807, 969, 1131, 1331, 1531, 1773, 2015, 2303, 2591, 2929, 3267, 3659, 4051, 4501, 4951, 5463, 5975, 6553, 7131, 7779, 8427, 9149, 9871, 10671, 11471, 12353, 13235, 14203, 15171, 16229, 17287, 18439

Offset: 1

Natan Arie Consigli, Jul 02 2016

Terms from 29 to 111 are the atomic numbers of the elements of group 11 in the periodic table. The group is also known as the coinage metals since copper (element 29), silver (element 47) and gold (element 79) are in group 11.

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Group 11 element.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Other groups: 1(A219527), 2(A168380), 3(A168388), 12(A271998), 13(A271997), 14(A271996), 15(A271995), 16(A271994), 17(A271999), 18(A018227).

LinearRecurrence[{2,1,-4,1,2,-1},{3,11,29,47,79,111},50] (* Harvey P. Dale, Nov 26 2018 *)

Vec(x*(3+5*x+4*x^2-10*x^3-3*x^4+5*x^5)/((1-x)^4*(1+x)^2) + O(x^60)) \\ Colin Barker, Oct 25 2016

From Colin Barker, Oct 25 2016: (Start)
G.f.: x*(3 + 5*x + 4*x^2 - 10*x^3 - 3*x^4 + 5*x^5)/((1 - x)^4*(1 + x)^2).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.
a(n) = (n^3 + 9*n^2 + 26*n - 30)/6 for n even.
a(n) = (n^3 + 9*n^2 + 29*n - 21)/6 for n odd. (End)

A137306 Related to A018226 and A018227: due to the stable element 118 (last term in A018227) this is submitted as a suggested new observed list with 459 as a probable next new atomic weight stable island.

Original entry on oeis.org

2, 8, 20, 28, 50, 83, 118, 126, 168, 194, 298

Offset: 1

Roger L. Bagula, Apr 20 2008

"These are crucial to the notion of an 'island of stability', which Glenn Seaborg espoused."

Y. Oganessian, J. Phys. G: Nucl. Part. Phys, 2007, iop.org
Rosenfeld, L. (1948). Nuclear Forces. Interscience Publishers, New York, xvii.
G. T. Seaborg - Contemporary Physics, 2004, informaworld.com

Cf. A018226, A018227.

(* nuclear radius visualization from Rosenfeld*) Clear[r, A, r0, p, n, m] A[p_, n_] := If[p == 0, 1, p + n] r0 = 1.3214405*10^(-13); r[p_, n_] := r0*( 1 + 0.8*(n/A[p, n])^2 - 0.3/A[p, n]^(1/3) + 0.010*p^2/A[p, n]^(4/3)) a0 = Table[Table[r[p, n], {n, p, Floor[2*p]}], {p, 0, 120}]; ListPlot[Flatten[a0]]

Roughly a(n) = Floor[1.53*a(n-1)]

Definition not clear to me. - N. J. A. Sloane, Apr 25 2008

A018226 Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay.

Original entry on oeis.org

2, 8, 20, 28, 50, 82, 126

Offset: 1

John Raithel (raithel(AT)rahul.net)

In the shell model for the nucleus, magic numbers are the numbers of either protons or neutrons at which a shell is filled.
First seven positive terms of A162626. - Omar E. Pol, Jul 07 2009
Steppenbeck: "The results of the experiment indicate that 54Ca's first excited state lies at a relatively high energy, which is characteristic of a large nuclear shell gap, thus indicating that N = 34 in 54Ca is a new magic number, as predicted theoretically by the University of Tokyo group in 2001. By conducting a more detailed comparison to nuclear theory the researchers were able to show that the N = 34 magic number is equally as significant as some other nuclear shell gaps."

Dictionary of Science (Simon and Schuster), see the entry for "Magic number".

S. Bjornholm, Clusters, condensed matter in embryonic form, Contemp. Phys. 31 1990 pp. 309-324 (p. 312).
Encyclopedia Britannica, magic number
J. Fridmann et al., 'Magic' nucleus 42-Si, Nature, 435 (2005), 922-924 and 897-898.
J. Glanz, Uut and Uup Add Their Atomic Mass to Periodic Table, New York Times, Feb 01, 2003, pages 1 and 26.
R. V. F. Janssens, Unexpected doubly magic nucleus, Nature, 459 (Jun 25 2009), 1069-1070. [_Added by N. J. A. Sloane, Jul 05 2009]
Radoslav Jovanovic, Magic Numbers and the Pascal Triangle
Lutvo Kuric, Digital nuclear shell model, International Letters of Chemistry, Physics and Astronomy, 13(2) (2014) 160-173; ISSN 2299-3843.
V. Ladma, Magic Numbers
NAPC Isotope Hydrology Section, Chapter 2, Atomic Systematics and Nuclear Structure [Broken link?]
R. Nave, Shell Model of Nucleus
R. Nave, Enhanced Abundance of Magic Number Nuclei
Rachele Nerattini, Johann S. Brauchart, and Michael K.-H. Kiessling, Magic numbers in Smale's 7th problem, arXiv:1307.2834v1 [math-ph], July 10, 2013.
Phys.org, Evidence for a new nuclear 'magic number', Oct 9, 2013.
D. Steppenbeck et al., Evidence for a new nuclear 'magic number' from the level structure of 54Ca, Nature, 2013 DOI: 10.1038/nature12522.
D. Warner, Not-so-magic numbers, Nature, 430 (Jul 29 2004), 517-519.
D. Weise, The Pythagorean Approach to Problems of Periodicity in Chemistry and Nuclear Physics
Wikipedia, Magic number (physics)

Cf. A018227 Number of electrons (which equals number of protons) such that they are arranged into complete shells within the atom.

Cf. A033547, A110856, A130598, A162626, A046939, A046940.

If 1 <= n <= 3 then a(n)=n*(n+1)*(n+2)/3, else if 4 <= n <= 7 then a(n)=n(n^2+5)/3. - Omar E. Pol, Jul 07 2009 [This needs to be clarified. - Joerg Arndt, May 03 2011]
From Daniel Forgues, May 03 2011: (Start)
If 1 <= n <= 3 then a(n) = 2 T_n, else
if 4 <= n <= 7 then a(n) = 2 (T_n - t_{n-1}),
where T_n is the n-th tetrahedral number, t_n the n-th triangular number.
G.f.: (2*x*(1 - 6*x^3 + 14*x^4 - 11*x^5 + 3*x^6))/(1 - x)^4, 1 <= n <= 7.
Using those formulas for n >= 0 gives A162626. (End)
a(n) = n*(n^2+5)/3 + (4*n-6)*A171386(n). - Omar E. Pol, Aug 14 2013

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

cp's OEIS Frontend

A271994 The chalcogen sequence (a(n) = A018227(n)-2).

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A271995 The Pnictogen sequence: a(n) = A018227(n)-3.

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A271996 The crystallogen sequence (a(n) = A018227(n)-4).

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A271997 The icosagen sequence : a(n) = A018227(n)-5, for n >= 2.

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A271998 Volatile sequence: a(n) = A018227(n)-6.

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A271999 Halogen sequence: a(n) = A018227(n)-1.

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A272000 Coinage sequence: a(n) = A018227(n)-7.

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A137306 Related to A018226 and A018227: due to the stable element 118 (last term in A018227) this is submitted as a suggested new observed list with 459 as a probable next new atomic weight stable island.

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