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A007585
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10-gonal (or decagonal) pyramidal numbers: n(n+1)(8n-5)/6.
(Formerly M4791)
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13
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0, 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375, 1826, 2366, 3003, 3745, 4600, 5576, 6681, 7923, 9310, 10850, 12551, 14421, 16468, 18700, 21125, 23751, 26586, 29638, 32915, 36425, 40176, 44176, 48433, 52955, 57750
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Binomial transform of [1, 10, 17, 8, 0, 0, 0,...] = (1, 11, 38, 90,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 18 2009]
a(n) = A000384(n)*n - sum[k=0..n-1]A000384(k) = n^2*(2*n-1)-sum[k=0..n-1)k*(2*k-1)=n*(n+1)*(8*n-5)/6, for d=4 in the general formula a(n)=n^2*(d*n-d+2)/2-sum[k=0..n-1]k*(d*k-d+2)/2=n*(n+1)*(2*d*n-2*d+3)/6. [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Apr 21 2010]
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REFERENCES
| A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| B. Berselli, a description of the recursive method n*[n*(d*n-d+2)/2] - sum[k=0..n-1] k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6: website Matem@ticamente. [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Apr 21 2010]
Harvey P. Dale, Table of n, a(n) for n = 0..1000
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FORMULA
| a(n)= (8*n-5)*binomial(n+1, 2)/3. G.f.: x*(1+7*x)/(1-x)^4.
a(n)=(8*n^3+3*n^2-5*n)/6 with n>=0 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 01 2010]
a(0)=0, a(1)=1, a(2)=11, a(3)=38, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4) [From Harvey P. Dale, Dec 20 2011]
G.f.: (7*x^2+x)/(x-1)^4 [From Harvey P. Dale, Dec 20 2011]
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EXAMPLE
| For n=0, a(0)=0; n=1, a(1)=6/6=1; n=2, a(2)=66/6=11; [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 01 2010]
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MATHEMATICA
| Table[n(n+1)(8n-5)/6, {n, 0, 80}] (* From Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 11, 38}, 40] (* From Harvey P. Dale, Dec 20 2011 *)
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CROSSREFS
| Cf. A001107.
Cf. A093565 ((8, 1) Pascal, column m=3). Partial sums of A001107.
Cf. A000384.
Sequence in context: A139276 A010002 A143109 * A024202 A133258 A103738
Adjacent sequences: A007582 A007583 A007584 * A007586 A007587 A007588
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy.
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