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A002419
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4-dimensional figurate numbers: a(n) = (6*n-2)*binomial(n+2,3)/4.
(Formerly M4699 N2008)
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20
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1, 10, 40, 110, 245, 476, 840, 1380, 2145, 3190, 4576, 6370, 8645, 11480, 14960, 19176, 24225, 30210, 37240, 45430, 54901, 65780, 78200, 92300, 108225, 126126, 146160, 168490, 193285, 220720, 250976, 284240, 320705, 360570, 404040, 451326, 502645, 558220
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OFFSET
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1,2
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COMMENTS
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a(n) = the sum of all the ways of adding the k-tuples of A016777(0) to A016777(n-1). For n=4, the terms are 1,4,7,10 giving (1)+(4)+(7)+(10)=22; (1+4)+(4+7)+(7+10)=33; (1+4+7)+(4+7+10)=33; (1+4+7+10)=22; adding 22+33+33+22=110. - J. M. Bergot, Jun 26 2017
Also the number of chordless cycles in the (n+2)-crown graph. - Eric W. Weisstein, Jan 02 2018
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = (3*n-1)*binomial(n+2, 3)/2.
G.f.: x*(1+5*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.
Sum_{n>=1} 1/a(n) = (-24+81*log(3) -9*Pi*sqrt(3))/14 = 1.143929... - R. J. Mathar, Mar 29 2011
a(n) = (3*n^4 + 8*n^3 + 3*n^2 - 2*n)/12. - Chai Wah Wu, Jan 24 2016
E.g.f.: x*(12 + 48*x + 26*x^2 + 3*x^3)*exp(x)/12. - G. C. Greubel, Jul 03 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(3*sqrt(3)*Pi - 32*log(2) + 8)/7. - Amiram Eldar, Feb 11 2022
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MATHEMATICA
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CoefficientList[Series[(1+5*x)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 10, 40, 110, 245}, 40] (* Harvey P. Dale, Nov 30 2014 *)
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PROG
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(Python)
A002419_list, m = [], [6, 1, 1, 1, 1]
for _ in range(10**2):
for i in range(4):
(Sage) [n*(n+1)*(n+2)*(3*n-1)/12 for n in (1..40)] # G. C. Greubel, Jul 03 2019
(GAP) List([1..40], n-> n*(n+1)*(n+2)*(3*n-1)/12) # G. C. Greubel, Jul 03 2019
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CROSSREFS
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Cf. A093563 ((6, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
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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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