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0, 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882, 968, 1058, 1152, 1250, 1352, 1458, 1568, 1682, 1800, 1922, 2048, 2178, 2312, 2450, 2592, 2738, 2888, 3042, 3200, 3362, 3528, 3698, 3872, 4050, 4232, 4418
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OFFSET
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0,2
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COMMENTS
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Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002
"If each period in the periodic system ends in a rare gas ..., the number of elements in a period can be found from the ordinal number n of the period by the formula: L = ((2n+3+(-1)^n)^2)/8..." - Nature Jun 09 1951; Nature 411 (Jun 07 2001), p. 648. This produces the present sequence doubled up.
These numbers also occur as the limiting periods in the Harmonic Periodic Table of Gutierrez Samanez. See also the Klehr link.
Let z(1)=I (I^2=-1), z(k+1) = 1/(z(k)+2I); then a(n)=(-1)*Imag(z(n+1))/real(z(n+1)) - Benoit Cloitre, Aug 06 2002
Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - Jeremy Gardiner, Dec 19 2004
Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2,(15+21)/2,... - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2005
Integral areas of isosceles right triangles with rational legs (Legs are 2n and triangles are nondegenerate for n>0). [From Rick L. Shepherd, Sep 29 2009]
a(n) = A176271(n,k) + A176271(n,n-k+1), 1<=k<=n. [From Reinhard Zumkeller, Apr 13 2010]
Pattern similar to triangular numbers on the Ulam spiral [From Giovanna Roda (giovanna.roda(AT)gmail.com), Oct 20 2010]
a(n) = A007607(A000290(n)). [Reinhard Zumkeller, Feb 12 2011]
Even squares divided by 2. - Omar E. Pol, Aug 18 2011
Number of stars when distributed as in the U.S.A. flag: n rows with n+1 stars and, between each pair of these, one row with n stars (i.e., n-1 of these), i.e., n*(n+1)+(n-1)*n =2*n^2 = A001105(n). - César Eliud Lozada, Sep 17 2012.
a(n) = A070216(n,n) for n > 0. - Reinhard Zumkeller, Nov 11 2012
a(n) = A014132(2*n-1,n) for n > 0. - Reinhard Zumkeller, Dec 12 2012
Apparently the number of Dyck paths with semilength n+3 and an odd number of peaks and the central peak has height n-3. [David Scambler, Apr 29 2013]
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REFERENCES
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A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
Martin Gardner, The Colossal Book of Mathematics, Classic Puzzles, Paradoxes and Problems, Chapter 2 entitled "The Calculus of Finite Differences," W. W. Norton and Company, New York, 2001, pages 12-13.
Julio Antonio Gutierrez Samanez, "Sistema Periodico Armonico y leyes Geneticas de los Elementos Quimicos" (Harmonic Periodic System and Genetic Laws of Chemical Elements), Cusco, Peru 2004. ISBN: 9972-33-063-X.
L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44.
A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Julio Antonio Gutierrez Samanez, More information
Wolfram Klehr, Title? [Broken link?]
V. Ladma, Magic Numbers
Giovanna Roda Experimenting with the Ulam spiral [From Giovanna Roda (giovanna.roda(AT)gmail.com), Oct 20 2010]
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = ((-1)^(n+1))*A053120(2*n, 2).
G.f. 2*x*(1+x)/(1-x)^3.
a(n) = A100345(n, n).
sum_{n>=1} 1/a(n) = Pi^2/12 [Jolley eq. 319] - Gary W. Adamson, Dec 21 2006
a(n) = A049452(n)-A033991(n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007
a(n) = A016742(n)/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008
a(n) = 2*A000290(n). - Omar E. Pol, May 14 2008
a(n) = 4*n+a(n-1)-2, n>0. [From Vincenzo Librandi]
a(n) = A002378(n)+A002378(n+1). [From Joerg M. Schuetze (joerg(AT)cyberheim.de), Mar 08 2010]
For n>0, a(n)=1/coefficient of x^2 in the Maclaurin expansion of 1/(cos(x)+n-1). [From Francesco Daddi, Aug 04 2011]
a(n) = 3*a(n-1)-3*a(n-1)+a(n-2). [From Artur Jasinski, Nov 24 2011]
a(n) = A000217(n) + A000326(n). - Omar E. Pol, Jan 11 2013
(a(n) - A000217(k))^2 = A000217(2n-1-k)*A000217(2n+k) + n^2, for all k. - Charlie Marion, May 04 2013
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MATHEMATICA
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2 Range[0, 50]^2 (* Harvey P. Dale, Jan 23 2011 *)
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PROG
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(MAGMA) [2*n^2: n in [0..50] ]; // Vincenzo Librandi, Apr 30 2011
(PARI) a(n)=2*n^2; \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
a001105 = a005843 . a000290 -- Reinhard Zumkeller, Dec 12 2012
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CROSSREFS
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Cf. A000290, A016742, A116471, A006331 (partial sums)
Sequence in context: A067051 A074629 A209303 * A051787 A050804 A081324
Adjacent sequences: A001102 A001103 A001104 * A001106 A001107 A001108
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KEYWORD
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nonn,easy
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AUTHOR
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Bernd.Walter(AT)frankfurt.netsurf.de
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STATUS
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approved
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