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A135713
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a(n) = n*(n+1)*(4*n+1)/2.
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5
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0, 5, 27, 78, 170, 315, 525, 812, 1188, 1665, 2255, 2970, 3822, 4823, 5985, 7320, 8840, 10557, 12483, 14630, 17010, 19635, 22517, 25668, 29100, 32825, 36855, 41202, 45878, 50895, 56265, 62000, 68112, 74613, 81515, 88830, 96570, 104747, 113373, 122460, 132020
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OFFSET
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0,2
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COMMENTS
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This sequence is related to A045944 by a(n) = n*A045944(n)-Sum_{i=0..n-1} A045944(i); this is the case d=6 in the identity n^2*(d*n+d-2)/2 - sum(k*(d*k+d-2)/2, k=0..n-1) = n*(n+1)*(2*d*n+d-3)/6 . - Bruno Berselli, Nov 19 2010
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REFERENCES
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J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.
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LINKS
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B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
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FORMULA
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a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) for n>3. - Bruno Berselli, Nov 19 2010
E.g.f.: (1/2)*x*(10 + 17*x + 4*x^2)*exp(x). - G. C. Greubel, Oct 29 2016
Sum_{n>=1} 1/a(n) = 2*(5 - 2*Pi/3 - 4*log(2)) = 0.26603235073404654... - Ilya Gutkovskiy, Oct 29 2016
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {0, 5, 27, 78}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
Table[n*(n+1)*(4*n+1)/2, {n, 0, 25}] (* G. C. Greubel, Oct 29 2016 *)
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PROG
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(Magma) [n*(n+1)*(4*n+1)/2: n in [0..40]]; // Bruno Berselli, Aug 23 2011
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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