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A006003 a(n) = n*(n^2 + 1)/2.
(Formerly M3849)
84
0, 1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335, 6095, 6924, 7825, 8801, 9855, 10990, 12209, 13515, 14911, 16400, 17985, 19669, 21455, 23346, 25345, 27455, 29679, 32020, 34481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and add the groups. In other words, "sum of the next n natural numbers". - Felice Russo

Number of rhombi in an n X n rhombus, if 'crossformed' rhombi are allowed. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

Also the sum of the integers between T(n-1)+1 and T(n), the n-th triangular number (A000217). Sum of n-th row of A000027 regarded as a triangular array.

Unlike the cubes which have a similar definition, it is possible for 2 elements of this sequence to sum to a third. E.g., a(36) + a(37) = 23346 + 25345 = 48691 = a(46). Might be called 2nd-order triangular numbers, thus defining 3rd-order triangular numbers (A027441) as n(n^3+1)/2, etc. - Jon Perry, Jan 14 2004

Also as a(n)=(1/6)*(3*n^3+3*n), n>0: structured trigonal diamond numbers (vertex structure 4) (cf. A000330 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

The sequence M(n) of magic constants for n X n magic squares (numbered 1 through n^2) from n=3 begins M(n) = 15, 34, 65, 111, 175, 260, ... - Lekraj Beedassy, Apr 16 2005 [comment corrected by Colin Hall, Sep 11 2009]

The sequence Q(n) of magic constants for the n-queens problem in chess begins 0, 0, 0, 0, 34, 65, 111, 175, 260, ... - Paul Muljadi, Aug 23 2005

Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005

Also partial differences of A063488(n) = (2*n-1)*(n^2-n+2)/2. a(n) = A063488(n) - A063488(n-1) for n>1. - Alexander Adamchuk, Jun 03 2006

In an n X n grid of numbers from 1 to n^2, select -- in any manner -- one number from each row and column. Sum the selected numbers. The sum is independent of the choices and is equal to the n-th term of this sequence. - F.-J. Papp (fjpapp(AT)umich.edu), Jun 06 2006

Sequence allows us to find X values of the equation:(X-Y)^3-(X+Y)=0. To find Y values: b(n)=(n^3-n)/2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 16 2006

For the equation: m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 and m is an odd number the X values are given by the sequence defined by: a(n)=(m*n^k+n)/2. The Y values are given by the sequence defined by: b(n)=(m*n^k-n)/2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 16 2006

If X is an n-set and Y a fixed 3-subset of X then a(n-3) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

(m*(2n)^k+n, m*(2n)^k-n) solves the Diophantine equation: 2m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 where m is a positive integer. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Oct 02 2007

Also c^(1/2) in a^(1/2) + b^(1/2) = c^(1/2) such that a^2 + b = c. - Cino Hilliard, Feb 09 2008

Number of units of a(n) belongs to a periodic sequence: 0, 1, 5, 5, 4, 5, 1, 5, 0, 9, 5, 1, 0, 5, 9, 5, 6, 5, 5, 9. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009

a(n) = n*A000217(n) - Sum_{i=0..n-1} A001477(i). - Bruno Berselli, Apr 25 2010

a(n) is the number of triples (w,x,y) having all terms in {0,...,n} such that at least one of these inequalities fails: x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012

Sum of n-th row of the triangle in A209297. - Reinhard Zumkeller, Jan 19 2013

The sequence starting with "1" is the third partial sum of (1, 2, 3, 3, 3, ...). - Gary W. Adamson, Sep 11 2015

a(n) is the largest eigenvalue of the matrix returned by the Matlab command magic(n) for n > 0. - Altug Alkan, Nov 10 2015

a(n) is the number of triples (x,y,z) having all terms in {1,...,n} such that all these triangle inequalities are satisfied: x+y > z, y+z > x, z+x > y. - Heinz Dabrock, Jun 03 2016

Shares its digital root with the stella octangula numbers (A007588). See A267017. - Peter M. Chema, Aug 28 2016

Can be proved to be the number of nonnegative solutions of a system of three linear Diophantine equations for n>=0 even: 2a_{11}+a_{12}+a_{13} = n, 2a_{22}+a_{12}+a_{23} = n and 2a_{33}+a_{13}+a_{23} = n. The number of solutions is f(n)=1/16*(n+2)*(n^2+4n+8) and a(n) = n*(n^2 + 1)/2 is obtained by remapping n -> 2*n-2. - Kamil Bradler, Oct 11 2016

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 15, pp 5, Ellipses, Paris 2008.

F.-J. Papp, Colloquium Talk, Department of Mathematics, University of Michigan-Dearborn, 2006 March 6

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

J. D. Bell, A translation of Leonhard Euler's "De Quadratis Magicis", E795, arXiv:math/0408230 [math.CO], 2004-2005.

Milan Janjic, Two Enumerative Functions

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

M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, Journal of Integer Sequences, 17 (2014), Article 14.3.5. - Felix Fröhlich, Oct 11 2016

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

A. J. Turner, J. F. Miller, Recurrent Cartesian Genetic Programming Applied to Famous Mathematical Sequences, preprint, Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation.

Eric Weisstein's World of Mathematics, Magic Constant.

Wikipedia, Floyd's triangle - Paul Muljadi, Jan 25 2010

Index entries for sequences related to magic squares

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = binomial(n, 3) + binomial(n-1, 3) + binomial(n-2, 3).

G.f.: x*(1+x+x^2)/(x-1)^4. - Floor van Lamoen, Feb 11 2002

Partial sums of A005448, centered triangular numbers: 3n(n-1)/2 + 1. - Jonathan Vos Post, Mar 16 2006

Binomial transform of [1, 4, 6, 3, 0, 0, 0, ...] = (1, 5, 15, 34, 65, ...). - Gary W. Adamson, Aug 10 2007

a(-n) = -a(n). - Michael Somos, Dec 24 2011

a(n) = Sum_{k = 1..n} A(k-1, k-1-n) where A(i, j) = i^2 + i*j + j^2 + i + j + 1. - Michael Somos, Jan 02 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=0, a(1)=1, a(2)=5, a(3)=15. - Harvey P. Dale, May 16 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3. - Ant King, Jun 13 2012

a(n) = A000217(n) + n*A000217(n-1). - Bruno Berselli, Jun 07 2013

a(n) = A057145(n+3,n). - Luciano Ancora, Apr 10 2015

E.g.f.: (1/2)*(2*x + 3*x^2 + x^3)*exp(x). - G. C. Greubel, Dec 18 2015; corrected by Ilya Gutkovskiy, Oct 12 2016

a(n) = T(n) + T(n-1) + T(n-2), where T means the tetrahedral-numbers, A000292. - Heinz Dabrock, Jun 03 2016

From Ilya Gutkovskiy, Oct 11 2016: (Start)

Convolution of A001477 and A008486.

Convolution of A000217 and A158799.

Sum_{n>=1} 1/a(n) = H(-i) + H(i) = 1.343731971048019675756781..., where H(k) is the harmonic number, i is the imaginary unit. (End)

MAPLE

with (combinat):seq((fibonacci(4, n)+n^3)/4, n=0..41); # Zerinvary Lajos, May 25 2008

MATHEMATICA

Table[ n(n^2 + 1)/2, {n, 0, 45}]

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 5, 15}, 50] (* Harvey P. Dale, May 16 2012 *)

CoefficientList[Series[x (1 + x + x^2)/(x - 1)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)

PROG

(PARI) {a(n) = n * (n^2 + 1) / 2}; /* Michael Somos, Dec 24 2011 */

(Haskell)

a006003 n = n * (n ^ 2 + 1) `div` 2

a006003_list = scanl (+) 0 a005448_list

-- Reinhard Zumkeller, Jun 20 2013

(MAGMA) [n*(n^2 + 1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015

(MAGMA) [Binomial(n, 3)+Binomial(n-1, 3)+Binomial(n-2, 3): n in [2..60]]; // Vincenzo Librandi, Sep 12 2015

(PARI) concat(0, Vec(x*(1+x+x^2)/(x-1)^4 + O(x^20))) \\ Felix Fröhlich, Oct 11 2016

CROSSREFS

Cf. A000330, A000537, A066886, A057587, A027480, A002817 (partial sums).

Cf. A000578 (cubes).

Cf. A007742, A005449, A005448, A118465, A226449, A034262, A080992, A267017.

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Antidiagonal sums of array in A000027. Row sums of the triangular view of A000027.

Cf. A063488 (sum of two consecutive terms).

Sequence in context: A147150 A238340 A162513 * A111385 A026101 A084288

Adjacent sequences:  A006000 A006001 A006002 * A006004 A006005 A006006

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

Better description from Albert Rich (Albert_Rich(AT)msn.com), March 1997

Corrected comment by Lekraj Beedassy on magic squares. n=2 does not exist, not strictly correct to set M(2)=0. - Colin Hall, Sep 11 2009

STATUS

approved

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Last modified December 7 04:57 EST 2016. Contains 278841 sequences.