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 A118059 288*n^2 - 168*n - 119. 3
 1, 697, 1969, 3817, 6241, 9241, 12817, 16969, 21697, 27001, 32881, 39337, 46369, 53977, 62161, 70921, 80257, 90169, 100657, 111721, 123361, 135577, 148369, 161737, 175681, 190201, 205297, 220969, 237217, 254041, 271441, 289417, 307969 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+(x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(3)=A001652(3)=119 and z(3)=A001653(3)=169; cf. A000290, A118057-A118058, A118060-A118061. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+694*x-119*x^2)/(1-x)^3. [Colin Barker, Jul 01 2012] a(n)+(a(n)+1)+...+(a(n)+288n+203)=(a(n)+288n+204)+...+a(n+1)-1; a(n+1)-1=a(n)+576n+119. a(n)+(a(n)+1)+...+(a(n)+288n+203)=6(24n-7)(24n+5)(24n+17); e.g., 1969+1970+...+3036=2672670=6*65*77*89. EXAMPLE a(3)=288*3^2-168*3-119=337, a(4)=288*4^2-168*4-119=3817 and 1969+1970+...+3036=3037+...+3816 MATHEMATICA Table[288*n^2 - 168*n - 119, {n, 100}] (* Vincenzo Librandi, Jul 08 2012 *) PROG (MAGMA) [288*n^2 - 168*n - 119: n in [1..50]]; // Vincenzo Librandi, Jul 08 2012 CROSSREFS Sequence in context: A111105 A137559 A185377 * A028500 A133251 A116338 Adjacent sequences:  A118056 A118057 A118058 * A118060 A118061 A118062 KEYWORD nonn,easy,less AUTHOR Charlie Marion, Apr 26 2006 EXTENSIONS Corrected by T. D. Noe, Nov 13 2006 STATUS approved

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Last modified May 20 21:27 EDT 2013. Contains 225464 sequences.