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A006003 n * (n^2 + 1) / 2.
(Formerly M3849)
73
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

Comment from Felice Russo: 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".

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 natural 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

The n-th row sums of Floyd's triangle are 1, 5, 15, 34, 65, 111, 175, 260, .... - Paul Muljadi, Jan 25 2010

a(n) = n*A000217(n) - sum(A001477(i), i=0..n-1). - 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

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

REFERENCES

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

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

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

Milan Janjic, Two Enumerative Functions

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

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

Eric Weisstein's World of Mathematics, Magic Constant.

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

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for sequences related to magic squares

FORMULA

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

G.f.: x*(1+x+x^2)/(x-1)^4. - Floor van Lamoen (fvlamoen(AT)hotmail.com), 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

EXAMPLE

G.f. = x + 5*x^2 + 15*x^3 + 34*x^4 + 65*x^5 + 111*x^6 + 175*x^7 + 260*x^8 + ...

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 *)

PROG

(PARI) { v=vector(100, i, i*(i^2+1)/2); x=vector(1275); c=0; for (i=1, 50, for (j=i, 50, x[c++ ]=v[j]-v[i])); for (k=1, 1275, for (l=1, 100, if (x[k]==v[l], print(x[k]); break))) } (Perry)

(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

CROSSREFS

Cf. A000330, A000537, A066886, A057587, A027480.

Cf. A000578 (cubes).

Cf. A007742, A005449.

(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.

Cf. A005448.

Cf. A063488. - Sum of two consecutive terms.

Cf. A118465.

Cf. A226449. - Bruno Berselli, Jun 09 2013

Cf. A034262.

Cf. A080992.

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

More terms from Robert G. Wilson v, Apr 15 2002

This is a second attempt at correction, first submission is hereby withdrawn. 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 August 19 20:32 EDT 2014. Contains 245794 sequences.