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A006003
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a(n) = n * (n^2 + 1) / 2.
(Formerly M3849)
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75
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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
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OFFSET
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0,3
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COMMENTS
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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 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
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REFERENCES
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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).
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LINKS
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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
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).
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 with linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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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
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EXAMPLE
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G.f. = x + 5*x^2 + 15*x^3 + 34*x^4 + 65*x^5 + 111*x^6 + 175*x^7 + 260*x^8 + ...
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MAPLE
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with (combinat):seq((fibonacci(4, n)+n^3)/4, n=0..41); # Zerinvary Lajos, May 25 2008
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MATHEMATICA
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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 *)
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PROG
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(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))) } \\ Jon 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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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EXTENSIONS
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Better description from Albert Rich (Albert_Rich(AT)msn.com), March 1997
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
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STATUS
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approved
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