A069778 q-factorial numbers 3!_q.

1, 6, 21, 52, 105, 186, 301, 456, 657, 910, 1221, 1596, 2041, 2562, 3165, 3856, 4641, 5526, 6517, 7620, 8841, 10186, 11661, 13272, 15025, 16926, 18981, 21196, 23577, 26130, 28861, 31776, 34881, 38182, 41685, 45396, 49321, 53466, 57837, 62440, 67281, 72366

Offset: 0

Author

Franklin T. Adams-Watters, Apr 07 2002

Keywords

• nonn
• easy
• changed

Comments

Number of proper n-colorings of the 4-cycle with one vertex color fixed (offset 2). - Michael Somos, Jul 19 2002

n such that x^3 + x^2 + x + n factors over the integers. - James R. Buddenhagen, Apr 19 2005

If Y is a 4-subset of an n-set X then, for n>=5, a(n-5) is the number of 5-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007

Equals row sums of the Connell (A001614) sequence read as a triangle. - Gary W. Adamson, Sep 01 2008

Binomial transform of 1, 5, 10, 6, 0, 0, 0 (0 continued). - Philippe Deléham, Mar 17 2014

Digital root is A251780. - Peter M. Chema, Jul 11 2016

Examples

			For 2-colorings only 1212 is proper so a(2-2)=1. The proper 3-colorings are: 1212,1313,1213,1312,1232,1323 so a(3-2)=6.
a(0) = 1*1 = 1;
a(1) = 1*1 + 5*1 = 6;
a(2) = 1*1 + 5*2 + 10*1 = 21;
a(3) = 1*1 + 5*3 + 10*3 + 6*1 = 52;
a(4) = 1*1 + 5*4 + 10*6 + 6*4 = 105; etc. - _Philippe Deléham_, Mar 17 2014
		

References

• T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Links

• Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
• M. Golafshan, M. Rigo, and M. Whiteland, Computing the k-binomial complexity of generalized Thue-Morse words, arXiv:2412.18425 [math.CO], 2024. See p. 29.
• Cemil Karaçam and Alper Vural, Enumerating 2D and 3D lattice paths with arbitrary steps, Mathematica Bohemica, pp. 1-13 (2025). See p. 10.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A069777, A069779, A218503, A056108 (first differences).

Cf. A001614. - Gary W. Adamson, Sep 01 2008

Cf. A226449. - Bruno Berselli, Jun 09 2013

Programs

Maple

A069778 := proc(n)
    (n+1)*(n^2+n+1) ;
end proc: # R. J. Mathar, Aug 24 2013

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 6, 21, 52}, 41] (* or *) Table[(n + 1) (n^2 + n + 1), {n, 0, 41}] (* Harvey P. Dale, Jul 11 2011 *)
Table[QFactorial[3, n], {n, 0, 41}] (* Arkadiusz Wesolowski, Oct 31 2012 *)

PARI

a(n)=(n+1)*(n^2+n+1)

Formula

a(n) = (n + 1)*(n^2 + n + 1).

a(n) = (n+1)^3-2*T(n) where T(n) =n*(n+1)/2= A000217(n) is the n-th triangular number. - Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 14 2006

a(n) = n^8 mod (n^3+n), with offset 1..a(1)=1. - Gary Detlefs, May 02 2010

a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4), n>3. - Harvey P. Dale, Jul 11 2011

G.f.: (1+2*x+3*x^2)/(1-x)^4. - Harvey P. Dale, Jul 11 2011

For n>0 a(n) = Sum_{k=A000217(n-1)...A000217(n+1)} k. - J. M. Bergot, Feb 11 2015

E.g.f.: (1 + 5*x + 5*x^2 + x^3)*exp(x). - Ilya Gutkovskiy, Jul 11 2016