A002419 4-dimensional figurate numbers: a(n) = (6*n-2)*binomial(n+2,3)/4. (Formerly M4699 N2008)

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

Offset

1,2

Comments

a(n) is the n-th antidiagonal sum of the convolution array A213761. - Clark Kimberling, Jul 04 2012

Convolution of A000027 with A000567 (excluding 0). - Bruno Berselli, Dec 07 2012

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

References

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).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Eric Weisstein's World of Mathematics, Chordless Cycle.

Eric Weisstein's World of Mathematics, Crown Graph.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Index to sequences related to pyramidal numbers.

Formula

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

a(n) = A080852(6,n-1). - R. J. Mathar, Jul 28 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

Mathematica

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 *)
Table[n(n+1)(n+2)(3n-1)/12, {n, 40}] (* Eric W. Weisstein, Jan 02 2018 *)

Prog

(Magma) /* A000027 convolved with A000567 (excluding 0): */ A000567:=func<n |  n*(3*n-2)>; [&+[(n-i+1)*A000567(i): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Dec 07 2012
(PARI) a(n)=(3*n-1)*binomial(n+2, 3)/2 \\ Charles R Greathouse IV, Sep 24 2015
(Python)
A002419_list, m = [], [6, 1, 1, 1, 1]
for _ in range(10**2):
    A002419_list.append(m[-1])
    for i in range(4):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016
(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

Crossrefs

Cf. A093563 ((6, 1) Pascal, column m=4).

Cf. A000027, A000567, A002414 (first differences), A016777, A080852, A213761.

Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

Sequence in context: A071233 A063490 A251121 * A199826 A227056 A027981

Adjacent sequences: A002416 A002417 A002418 * A002420 A002421 A002422

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Status

approved