A006503 a(n) = n*(n+1)*(n+8)/6. (Formerly M2835)

0, 3, 10, 22, 40, 65, 98, 140, 192, 255, 330, 418, 520, 637, 770, 920, 1088, 1275, 1482, 1710, 1960, 2233, 2530, 2852, 3200, 3575, 3978, 4410, 4872, 5365, 5890, 6448, 7040, 7667, 8330, 9030, 9768, 10545, 11362, 12220, 13120, 14063, 15050, 16082, 17160, 18285

Offset

0,2

Comments

If Y is a 3-subset of an n-set X then, for n>=4, a(n-4) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

The coefficient of x^3 in (1-x-x^2)^{-n} is the coefficient of x^3 in (1+x+2x^2+3x^3)^n. Using the multinomial theorem one then finds that a(n)=n(n+1)(n+8)/3!. - Sergio Falcon, May 22 2008

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Rowan Rowlands, Sheila Sundaram, and Lei Xue, Topology of Cut Complexes of Graphs, arXiv:2304.13675 [math.CO], 2023.

G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)

G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48.

Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) Article 10.7.8, section 3.

Vitaly M. Khamitov, Dmitriy Dmitrishin, Alexander Stokolos, and Daniel Gray, Convolved Numbers of k-sections of the Fibonacci Sequence: Properties, Consequences, arXiv:2603.08636 [math.CA], 2026. See p. 10.

Pieter Moree, Convoluted Convolved Fibonacci Numbers, J. Int. Seq. 7 (2004), Article 04.2.2. See also arXiv:math/0311205 [math.CO], 2003.

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.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = n*(n+1)*(n+8)/6.

G.f.: x*(3-2*x)/(1-x)^4.

a(n) = A000292(n) + A002378(n). - Reinhard Zumkeller, Sep 24 2008

a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4) with a(0)=0, a(1)=3, a(2)=10, a(3)=22. - Harvey P. Dale, Jan 27 2016

From Amiram Eldar, Oct 20 2025: (Start)

Sum_{n>=1} 1/a(n) = 4437/7840.

Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(2)/7 - 7253/7840. (End)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1, n >= 3. - Bruce Nye, Feb 14 2026

Maple

A006503:=-(-3+2*z)/(z-1)**4; # [Simon Plouffe in his 1992 dissertation.]

Mathematica

Clear["Global`*"] a[n_] := n(n + 1)(n + 8)/3! Do[Print[n, " ", a[n]], {n, 1, 25}] (* Sergio Falcon, May 22 2008 *)
Table[n(n+1)(n+8)/6, {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 3, 10, 22}, 50] (* Harvey P. Dale, Jan 27 2016 *)

Prog

(PARI) my(x='x+O('x^50)); concat([0], Vec(x*(3-2*x)/(1-x)^4)) \\ G. C. Greubel, May 11 2017
(PARI) a(n) = n*(n+1)*(n+8)/6; \\ Bruce Nye, Feb 14 2026

Crossrefs

a(n) = A095660(n+2, 3): fourth column of (1, 3)-Pascal triangle.

Cf. A000027, A000096, A006504.

Cf. A000292, A002378.

Row n=3 of A144064.

Partial sums of A055998.

Sequence in context: A346166 A122795 A140066 * A248851 A023554 A294414

Adjacent sequences: A006500 A006501 A006502 * A006504 A006505 A006506

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Better description from Jeffrey Shallit, Aug 1995

Status

approved