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A001105 a(n) = 2*n^2. 121
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - Roberto E. Martinez II, 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.

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, Aug 05 2005

These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010

Integral areas of isosceles right triangles with rational legs (Legs are 2n and triangles are nondegenerate for n>0). - Rick L. Shepherd, Sep 29 2009]

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.

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

Sum of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 01 2013

Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective odd leg a (A180620); sequence gives values c-a, sorted with duplicates removed. - K. G. Stier, Nov 04 2013

Number of roots in the root systems of type B_n and C_n (for n>1). - Tom Edgar, Nov 05 2013

Area of a square with diagonal 2n. - Wesley Ivan Hurt, Jun 18 2014

This sequence appears also as the first and second member of the quartet [a(n), a(n), p(n), p(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p(n) = A046092(n). See an Oct 15 2014 comment on A147973 where also a reference is given. - Wolfdieter Lang, Oct 16 2014

a(n) are the only integers, m, where (A000005(m) + A000203(m)) = (number of divisors of m + sum of divisors of m) is an odd number. - Richard R. Forberg, Jan 09 2015

REFERENCES

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.

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Julio Antonio Gutierrez Samanez, More information

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013

V. Ladma, Magic Numbers

V. Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv preprint arXiv:1501.06098, 2015

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

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, Jun 12 2007

a(n) = A016742(n)/2. - Zerinvary Lajos, Jun 20 2008

a(n) = 2 * A000290(n). - Omar E. Pol, May 14 2008

a(n) = 4*n + a(n-1) - 2, n > 0. - Vincenzo Librandi

a(n) = A002378(n) + A002378(n+1). - Joerg M. Schuetze (joerg(AT)cyberheim.de), Mar 08 2010

a(n) = A176271(n,k) + A176271(n,n-k+1), 1<=k<=n. - Reinhard Zumkeller, Apr 13 2010

a(n) = A007607(A000290(n)). Reinhard Zumkeller, Feb 12 2011

For n > 0, a(n) = 1/coefficient of x^2 in the Maclaurin expansion of 1/(cos(x)+n-1). - Francesco Daddi, Aug 04 2011

a(n) = 3*a(n-1) - 3*a(n-1) + a(n-2). - Artur Jasinski, Nov 24 2011

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

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

a(n) = floor(1/(1-cos(1/n))), n > 0. - Clark Kimberling, Oct 08 2014

a(n) = A251599(3*n-1) for n > 0. - Reinhard Zumkeller, Dec 13 2014

a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+4)/3). - Wesley Ivan Hurt, Mar 12 2015

EXAMPLE

a(3) = 18; since 2(3) = 6 has 3 partitions with exactly two parts: (5,1), (4,2), (3,3).  Adding all the parts, we get: 1 + 2 + 3 + 3 + 4 + 5 = 18. - Wesley Ivan Hurt, Jun 01 2013

MAPLE

A001105:=n->2*n^2; seq(A001105(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

MATHEMATICA

2 Range[0, 50]^2 (* Harvey P. Dale, Jan 23 2011 *)

PROG

(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

CROSSREFS

Cf. A000290, A006331 (partial sums), A016742, A056106, A116471, A245508, A251599.

Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488.

Cf. A058331 and A247375. - Bruno Berselli, Sep 16 2014

Sequence in context: A067051 A074629 A209303 * A051787 A050804 A081324

Adjacent sequences:  A001102 A001103 A001104 * A001106 A001107 A001108

KEYWORD

nonn,easy

AUTHOR

Bernd.Walter(AT)frankfurt.netsurf.de

STATUS

approved

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Last modified August 24 08:01 EDT 2016. Contains 275769 sequences.