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 A118057 8*n^2-4*n-3. 5
 1, 21, 57, 109, 177, 261, 361, 477, 609, 757, 921, 1101, 1297, 1509, 1737, 1981, 2241, 2517, 2809, 3117, 3441, 3781, 4137, 4509, 4897, 5301, 5721, 6157, 6609, 7077, 7561, 8061, 8577, 9109, 9657, 10221, 10801, 11397, 12009, 12637, 13281, 13941, 14617 (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(1)=A001652(1)=3 and z(1)=A001653(1)=5; cf. A000290,A118058-A118061. Sequence found by reading the segment (1, 21) together with the line from 21, in the direction 21, 57,..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 04 2011 LINKS Harvey P. Dale, 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+18*x-3*x^2)/(1-x)^3. [Colin Barker, Jul 01 2012] a(n)+(a(n)+1)+...+(a(n)+8n+5)=(a(n)+8n+6)+...+a(n+1)-1; a(n+1)-1=a(n)+16n+3. a(n)+(a(n)+1)+...+(a(n)+8n+5)=(4n-1)(4n+1)(4n+3); e.g., 21+22+...+56=693=7*9*11. a(n) = 16*n+a(n-1)-12 (with a(1)=1). [From Vincenzo Librandi, Nov 13 2010] a(n) = A139098(n) - A004767(n). - Omar E. Pol, Sep 18 2012 EXAMPLE a(3)=8*3^2-4*3-3=57, a(4)=8*4^2-4*4-3=109 and 57+58+...+86=87+...+108. MATHEMATICA Table[8n^2-4n-3, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 21, 57}, 50] (* Harvey P. Dale, Sep 18 2012 *) CROSSREFS Sequence in context: A043382 A044123 A044504 * A020148 A037305 A223467 Adjacent sequences:  A118054 A118055 A118056 * A118058 A118059 A118060 KEYWORD nonn,easy AUTHOR Charlie Marion, Apr 26 2006 STATUS approved

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