Mathematical formulas of the periodic table

Formula-overview

A(n)	B(n)	C(n)		(n)	E(n)		T(n)	S(n)

2	0	2	(s)	0	0		1	0
2	2	6	(p)	1	2	[He]	4	2
8	8	10	(d)	2	10	[Ne]	12	10
8	18	14	(f)	3	18	[Ar]	21	28
18	32	18	(g)	4	36	[Kr]	39	60
18	50	22	(h)	5	54	[Xe]	58	110
32	72	26	(i)	6	86	[Rn]	90	182
32	98	30	(j)	7	118	[Uuo]	123	280
50	128	34	(k)	8	168	[Uho]	173	408
50	162	38	(l)	9	218	[Buo]	224	570
72	200	42		10	290		296	770
72	242	46		11	362		369	1012
98	288	50		12	460		467	1300
98	338	54		13	558		566	1638
128	392	58		14	686		694	2030
128	450	62		15	814		823	2480
162	512	66		16	976		985	2992
162	578	70		17	1138		1148	3570
200	648	74		18	1338		1348	4218
200	722	78		19	1538		1549	4940
242	800	82		20	1780		1791	5740
242	882	86		21	2022		2034	6622
288	968	90		22	2310		2322	7590
	Pauli				Noble-gas		Truss	Singularity
A(n)	B(n)	C(n)		(n)	E(n)		T(n)	S(n)

formel.zip [Excel-File with new formulas (53KB)]

Basic

Numbers are equal for all n> -3
A(n)= ((2n+3+(-1)^n)^2)/8

A(n)= Int((n+2)/2) ^ 2 * 2
   A(n)= E(n) - E(n-1)

B(n) = n ^ 2 * 2    (Pauli)
B(n) = B(n-1)+ C(n-1)

C(n)= 4 * n+ 2

C(n)= ( n +1 ) * 4 - 2

C(n)= 2 * ( 2*n+1 )

E(n) = E(n-1) + A(n)

T(n)=E(n)+(int(n+1)/2)+1
FNT = Int((n + 1) / 2) + 1

S(n)= S(n-1 ) + B(n)


S(n)= n* ( n +1 )*( 2* n+1)/3

Excel and References

Martin Gardner      Antimatter and chemistry

A(n)=QUADRATESUMME(ABRUNDEN((A(n)+2)/2;0))*2
  A(n)=SUMSQ(ROUNDDOWN((A(n)+2)/2,0))*2

B(n)=QUADRATESUMME(B(n))*2     B(n)=SUMSQ(B(n))*2
B(n)= S(n) - S(n-1)

C(n)= B( n +1 ) - B(n)

   William A. Stein

Outblocking of Elements

E(n)=QUADRATESUMME(ABRUNDEN((E(n)+2)/2;0))*2+E(n-1)
  E(n)=SUMSQ(ROUNDDOWN((E(n)+2)/2,0))*2+E(n-1)

Inert-gas algorithm (rare-gas)

T(n)=E(n)+(ABRUNDEN((n+1)/2;0))+1
T(n)=E(n)+(ROUNDDOWN((n+1)/2,0))+1

S(n)=QUADRATESUMME(S(n))*2+S(n-1)
S(n)=SUMSQ(S(n))*2+S(n-1)

Prof. Germain Kreweras    Singularity algorithm

ID Number: A006331 (Formerly M1963)

S(n) = Sequence: 0,2,10,28,60,110,182,280,408,570,770,1012,1300,1638,2030,
2 480,2992,3570,4218,4940,5740,6622,7590 , 8648, 9800, 11050,
12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820,
32412, 35150, 38038, 41080, 44280 ... (Numbers checked for correctness)

Name: n*(n+1)*(2n+1)/3.

Comments: Triangles in rhombic matchstick arrangement of side n.
Maximum accumulated number of electrons at energy level n -
Feb 28 2000

Mr. Scott A. Brown http://home.neo.rr.com/scottbrown/chall.html
webpage: http://home.neo.rr.com/scottbrown
e-mail: scottbrown@neo.rr.com

References Cahiers du Bureau Universitaire de Recherche Opérationnelle,
Institut de Statistique, Université de Paris, 6 (1965), circa p.82,
but I think volume, page or year must be wrong. Author is probably
Germain Kreweras.

Links: Basic atomic information

Formula: G.f.: x*(2+2*x)/(1-x)^4. a(n)=2*C(n+1,3)+2*C(n+2,3).
Program: (PARI) a(n)=n*(n+1)*(2*n+1)/3

Lit: Germain Kreweras

[1] "Les décisions collectives"; Mathématiques et Sciences Humaines;
No. 2; Spring, 1963; 25-35; #1500.

[2] "Représentation polyédrique des préordres complets finis et application
à l'agrégation des préférences"; La Décision; Colloque du CNRS;
Aix-en-Provence; 1969; 137-151; #1830.

[3]"Sur Quelques Problèmes Relatifs au Vote Pondéré"; Mathématiques et
Sciences Humaines; No. 84; Winter, 1983; 45-63; #258.

This produces A001105 doubled up.

References Martin Gardner, The Colossal Book of Mathematics, Classic Puzzles,
Paradoxes, and Problems, Chapter 2 entitled "The Calculus of Finite
Differences," W. W. Norton & Company, New York, 2001, pages 12-13.

Links: Index entries for sequences related to Chebyshev polynomials.
See also: a(n)= ((-1)^(n+1))*A053120(2*n,2).
Keywords: nonn
Offset: 0
Author: Bernd.Walter@frankfurt.netsurf.de

ID Number: A016825

C(n) = Sequence: 2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,
82,86,90,94,98,102,106,110,114,118,122,126,130,134,138,142,146,150,154,
158,162,166,170,174,178,182,186,190 (Numbers checked for correctness)

Name: 4n+2

Comments: Continued fraction for (e-1)/(e+1).
No solutions to a(n)=b^2-c^2 - Henry Bottomley
(se16@btinternet.com), Jan 13 2001
Apart from initial term(s), dimension of the space of weight 2n
cuspidal newforms for Gamma_0( 70 ).
References J. R. Goldman, The Queen of Mathematics, 1998, p. 70.

Links: William A. Stein was@math.berkeley.edu ,
Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein (was@math.berkeley.edu), The modular forms database
E. W. Weisstein, Link to a section of The World of Mathematics. (Currently unavailable)
Index entries for continued fractions for constants
Keywords: nonn,easy,nice
Offset: 0
Author: njas

ID Number: A000952 (Formerly M1574 and N0615)

Sequence: 2,6,10,14,18,26,30,38,42,46,50
Name: Orders n == 2 (mod 4) of conference matrices.
Comments: A conference matrix of order n is an n X n {-1,0,+1} matrix A such that A A' =
(n-1)I.
If n == 2 (mod 4) then a necessary condition is that n-1 is a sum of 2
squares. It is conjectured that this condition is also sufficient.
References V. Belevitch, Conference matrices and Hadamard matrices, Ann. Soc.
Scientifique Bruxelles, 82 (I) (1968), 13-32.
CRC Handbook of Combinatorial Designs, 1996, Chapter 52.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting
Codes, Elsevier-North Holland, 1978, p. 56.
Example: The essentially unique conference matrix of order 6:
0 +1 +1 +1 +1 +1
+1 0 +1 -1 -1 +1
+1 +1 0 +1 -1 -1
+1 -1 +1 0 +1 -1
+1 -1 -1 +1 0 +1
+1 +1 -1 -1 +1 0
Keywords: nonn,hard,nice
Offset: 1
Author(s): njas
Extension: 54 seems to be the smallest order for which it is not known whether a matrix
exists.
Show internal format for above sequence?

Arbeitsnotiz	My Notes
Ich habe mich sehr über die Formel S(n)=n(n+1)(2*n+1)/3 gefreut. Für den Atomkern/Protonen ist sie möglicherweise eine der wichtigsten Grenzfunktion für die Quarks, nicht nur die Tripplets werden als mögliche Unterfunktionen eines Produkts dargestellt, sondern die Werte sind eindeutig gleich mit der Singularität der Atomhülle. Die Formel wertet die Singularitäts- Theorie sehr stark auf. Dadurch ist ein starker Bezug zwischen Atomhülle und Atomkern (Quarks) zu erkennen.	I am quite pleased over the formula S(n)=n(n+1)(2*n+1)/3. It is possibly one of the most important limiting functions there are for the atomic nucleus/protons: not only are the triplets shown as possible subfunctions of a product, but the values are unambigously identical with the singularity of the atomic shell. The formula makes singularity theory much more important, revealing a strong connection between atomic shell and nucleus (quarks).
Neuester Theorie zu Folge lagern sich die Protonen im Atomkernzentrum an und die Neutronen bilden eine isolierende Schicht(en) darum. >>zu Einfach, da nur Ladungspezifisch betrachtet und die Art wie das Messergebnis erzeugt wurde, nicht mitberücksichtigt ist. Es stellt jedoch einer der möglichen Extremfälle von kompletter Ladungsverschiebung innerhalb des Atomkernes dar.<<	According to the most recent theory, protons are concentrated in the core of a nucleus and neutrons form an isolating layer(s) around them. My view: "Too simple, since this takes into consideration charge only, and the way in which the results of measurement were produced is not taken into account. It does, of course, represent one of the possible extreme cases of complete charge shift inside the atomic nucleus."
Für mich bleibt die Theorie der Protonen Neutronen Durchmischung weiterhin bestehen, da nur durch das Ladungswechseln [ Ladung des Protons(a) wird in ein beliebiges Neutron(b) verschoben, Proton(a) wird zum Neutron(a) und Neutron(b) wird Proton(b), usw..] Dadurch bleibt die jeweilige Ladung als Teilladung im Atomkern frei beweglich bzw. synchron zu der Elektronenladung in der Atomhülle.	In my view the theory of proton-neutron intermixture is still valid, since it is only thus that charge exchange can occur [the charge of proton(a) is transferred to any given neutron(b), and proton(a) becomes neutron(a) while neutron(b) becomes proton(b), etc.]. The respective charge thereby remains freely mobile as a partial charge in the nucleus or, as the case may be, synchronized with the electron charge in the atomic shell.
[ev. bedingt gerade dieses Verschieben der Protonenladung im Atomkern die Struktur der Atomhülle, über Zusammenhänge unbedingt nachdenken]. Die statische Masse des Atomkerns als Grundgerüst steht einer dynamischen Ladung der Protonen gegenüber?	[Possibly it is precisely this transfer of proton charge in the nucleus which determines the structure of the shell. We definitely have to think about connections of this sort.] Does the static mass of the nucleus, as a structural base, serve as a counterpoise to the dynamic charge of the protons?

APSIDIUM ©	Created:	2001-01-04		number.pdf
	Last Updated:	2003-04-26		number.pdf

Thanks for translation to Brian T. Regan

Thanks to Scott A. Brown and BOB