A269658 Number of length-5 0..n arrays with no adjacent pair x,x+1 repeated.

1, 26, 225, 988, 3065, 7686, 16681, 32600, 58833, 99730, 160721, 248436, 370825, 537278, 758745, 1047856, 1419041, 1888650, 2475073, 3198860, 4082841, 5152246, 6434825, 7960968, 9763825, 11879426, 14346801, 17208100, 20508713, 24297390

Offset: 0

Author

R. H. Hardin, Mar 02 2016

Keywords

• nonn

Comments

The repeated pair is of the form (x,x+1) with 0 <= x <= n-1. Together with its repetition it occupies all but one position of the length-5 array. There are three choices for this position (beginning, middle, end; cf. example for n=1) and n+1 choices for the element in this position. This makes n*3*(n+1) forbidden arrays out of the (n+1)^5 possible ones. - M. F. Hasler, Feb 29 2020

Examples

			From _M. F. Hasler_, Feb 29 2020: (Start)
For n=0, there is only one array of length 5 with coefficients in 0..0, (0,0,0,0,0), and it satisfies the requirement, so a(0) = 1.
For n=1, the six arrays of length 5 with coefficients in 0..1 which do not satisfy the requirement are {(0,1,0,1,x), (0,1,x,0,1), (x,0,1,0,1); 0 <= x <= 1}, so a(1) = 2^5 - 6 = 26.
(End)
Some solutions for n=3:
  2  3  1  0  0  3  3  0  0  0  0  3  2  3  2  2
  2  0  3  3  2  3  3  1  2  3  0  3  3  1  0  3
  1  0  2  1  1  1  3  2  3  2  3  2  0  1  2  0
  0  2  0  1  0  2  2  2  1  3  3  2  2  1  3  3
  2  3  0  0  0  1  1  3  1  0  1  0  1  1  1  1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 0..210 (a(0) = 1 inserted by _M. F. Hasler_, Feb 29 2020).

Crossrefs

Row 5 of A269656.

Programs

PARI

apply( {A269658(n)=(n+1)^5-3*n*(n+1)}, [0..44]) \\ M. F. Hasler, Feb 29 2020

Formula

Empirical: a(n) = n^5 + 5*n^4 + 10*n^3 + 7*n^2 + 2*n + 1.

Conjectures from Colin Barker, Jan 25 2019: (Start)

G.f.: (1 + 20*x + 84*x^2 + 8*x^3 + 7*x^4) / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n >= 6.

(End)

a(n) = (n+1)^5 - 3*n*(n+1) = A000584(n+1) - A028896(n), cf. comment, which confirms the above conjectured formulas. - M. F. Hasler, Feb 29 2020

Extensions

Extended to a(0) = 1 by M. F. Hasler, Feb 29 2020