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A051673
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Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.
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8
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0, 1, 12, 47, 120, 245, 436, 707, 1072, 1545, 2140, 2871, 3752, 4797, 6020, 7435, 9056, 10897, 12972, 15295, 17880, 20741, 23892, 27347, 31120, 35225, 39676, 44487, 49672, 55245, 61220, 67611, 74432, 81697, 89420, 97615, 106296, 115477, 125172
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OFFSET
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0,3
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COMMENTS
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Also as a(n) = (1/6)*(14*n^3 - 12*n^2 + 4*n), n>0: structured cubeoctahedral numbers (vertex structure 7); and structured pentagonal anti-diamond numbers (vertex structure 7) (cf. A004466 = alternate vertex) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = binomial transform of [1, 11, 24, 14, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009
This is prime for a(3) = 47. The subsequence of semiprimes begins: 707, 7435, 10897, 20741, 115477, 341797, 825091, 897097, no more through a(100). - Jonathan Vos Post, May 27 2010
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REFERENCES
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T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
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LINKS
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FORMULA
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a(n) = n*(n*(7*n-6) + 2)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=12, a(3)=47. - Harvey P. Dale, Jul 22 2011
a(n) = Sum_{k=1..n} A214661(n, k), for n > 0 (row sums). (End)
E.g.f.: (x/3)*(3 + 15*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 10 2024
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EXAMPLE
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a(51) = 51*(51*(7*51-6)+2)/3 = 304351 = 17 * 17903 is semiprime. - Jonathan Vos Post, May 27 2010
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MAPLE
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MATHEMATICA
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Table[n^3+4Sum[i^2, {i, 0, n-1}], {n, 0, 40}] (* or *) LinearRecurrence[ {4, -6, 4, -1}, {0, 1, 12, 47}, 40] (* Harvey P. Dale, Jul 22 2011 *)
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PROG
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(SageMath) [n*(7*n^2-6*n+2)/3 for n in range(51)] # G. C. Greubel, Mar 10 2024
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
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EXTENSIONS
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Corrected by T. D. Noe, Nov 01 2006, Nov 08 2006
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STATUS
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approved
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