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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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a(3)=8*3^2-4*3-3=57, a(4)=8*4^2-4*4-3=109 and 57+58+...+86=87+...+108.
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MATHEMATICA
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Table[8n^2-4n-3, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 21, 57}, 50] (* Harvey P. Dale, Sep 18 2012 *)
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CROSSREFS
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Sequence in context: A043382 A044123 A044504 * A020148 A037305 A223467
Adjacent sequences: A118054 A118055 A118056 * A118058 A118059 A118060
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KEYWORD
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nonn,easy
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AUTHOR
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Charlie Marion, Apr 26 2006
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STATUS
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approved
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