Search: id:A006003 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=0..1000 %H A006003 Index entries for sequences related to linear recurrences with constant coefficients %H A006003 Milan Janjic, Two Enumerative Functions %H A006003 J. D. Bell, A translation of Leonhard Euler's "De Quadratis Magicis", E795 %H A006003 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A006003 Index entries for sequences related to magic squares %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 Search completed in 0.002 seconds