%I A006003 M3849
%S A006003 0,1,5,15,34,65,111,175,260,369,505,671,870,1105,1379,1695,2056,2465,
%T A006003 2925,3439,4010,4641,5335,6095,6924,7825,8801,9855,10990,12209,13515,
%U A006003 14911,16400,17985,19669,21455,23346,25345,27455,29679,32020,34481
%N A006003 n(n^2+1)/2.
%C A006003 Comment from Felice Russo (felice.russo(AT)katamail.com): 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".
%C A006003 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
%C A006003 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.
%C A006003 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 (perry(AT)globalnet.co.uk), 
               Jan 14 2004
%C A006003 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. 7, 2004.
%C A006003 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 (blekraj(AT)yahoo.com), Apr 16 2005. [Comment corrected 
               by Colin Hall (colin.hall3(AT)gmail.com), Sep 11 2009]
%C A006003 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.
%C A006003 Alternate terms of A057587. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), 
               Apr 10 2005
%C A006003 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 (alex(AT)kolmogorov.com), 
               Jun 03 2006
%C A006003 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
%C A006003 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
%C A006003 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
%C A006003 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 R. Janjic (agnus(AT)blic.net), 
               Jul 30 2007
%C A006003 (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
%C A006003 Also c^(1/2) in a^(1/2) + b^(1/2) = c^(1/2) such that a^2 + b = c. - 
               Cino Hilliard (hillcino368(AT)hotmail.com), Feb 09 2008
%C A006003 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. [From Mohamed Bouhamida 
               (bhmd95(AT)yahoo.fr), Sep 04 2009]
%D A006003 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 15, pp 5, Ellipses, 
               Paris 2008.
%D A006003 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, 
               Chem. Ber. 30 (1897), 1917-1926.
%D A006003 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (11).
%D A006003 F.-J. Papp, Colloquium Talk, Department of Mathematics, University of 
               Michigan-Dearborn, 2006 March 6
%D A006003 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006003 T. D. Noe, <a href="b006003.txt">Table of n, a(n) for n=0..1000</a>
%H A006003 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006003 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A006003 J. D. Bell, <a href="http://arXiv.org/abs/math.CO/0408230">A translation 
               of Leonhard Euler's "De Quadratis Magicis", E795</a>
%H A006003 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MagicConstant.html">Link to a section of The World of Mathematics.</
               a>
%H A006003 <a href="Sindx_Mag.html#magic">Index entries for sequences related to 
               magic squares</a>
%F A006003 binomial(n, 3)+binomial(n-1, 3)+binomial(n-2, 3).
%F A006003 G.f.: x*(1+x+x^2)/(x-1)^4. - Floor van Lamoen (fvlamoen(AT)hotmail.com), 
               Feb 11 2002.
%F A006003 Partial sums of A005448, centered triangular numbers: 3n(n-1)/2 + 1. 
               - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 16 2006
%F A006003 Binomial transform of [1, 4, 6, 3, 0, 0, 0,...] = (1, 5, 15, 34, 65,...). 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2007
%p A006003 with (combinat):seq((fibonacci(4,n)+n^3)/4, n=0..41); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 25 2008
%t A006003 Table[ n(n^2 + 1)/2, {n, 0, 45}]
%o A006003 (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)
%Y A006003 Cf. A000330, A000537, A066886, A057587, A027480.
%Y A006003 Cf. A000578 (cubes).
%Y A006003 Cf. A007742, A005449.
%Y A006003 (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.
%Y A006003 Antidiagonal sums of array in A000027.
%Y A006003 Cf. A005448.
%Y A006003 Cf. A063488 - Sum of two consecutive terms.
%Y A006003 Cf. A118465.
%Y A006003 Sequence in context: A055004 A147264 A147150 this_sequence A111385 A026101 
               A084288
%Y A006003 Adjacent sequences: A006000 A006001 A006002 this_sequence A006004 A006005 
               A006006
%K A006003 nonn,easy,nice
%O A006003 0,3
%A A006003 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A006003 Better description from Albert Rich (Albert_Rich(AT)msn.com) 3/97.
%E A006003 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 15 2002
%E A006003 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 (colin.hall3(AT)gmail.com), 
               Sep 11 2009