A239333 Number of n X 1 0..3 arrays with no element equal to one plus the sum of elements to its left or three plus the sum of elements above it, modulo 4.

1, 2, 5, 12, 28, 66, 156, 368, 868, 2048, 4832, 11400, 26896, 63456, 149712, 353216, 833344, 1966112, 4638656, 10944000, 25820224, 60917760, 143723520, 339087488, 800010496, 1887468032, 4453111040, 10506243072, 24787422208, 58481066496, 137974619136, 325524082688

Offset

0,2

Comments

Number of n-length words on {0,1,2} in which 0 appears only in runs of length 2. - Milan Janjic, Feb 14 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..210 from R. H. Hardin)

D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.

Sela Fried, Proof of the conjectures stated in A239333 and A255115, 2026.

Index entries for linear recurrences with constant coefficients, signature (2,0,2).

Formula

a(n) = 2*a(n-1) + 2*a(n-3).

a(n) = A052912(n) + A052912(n-2) for n>=2. - R. J. Mathar, Jun 18 2015

G.f.: (1 + x^2) / (1 - 2*x - 2*x^3). - Colin Barker, Feb 18 2018

Example

Some solutions for n=5:
..0....2....2....2....0....0....0....0....2....2....2....2....2....0....2....2
..2....0....2....0....2....0....2....0....0....0....0....2....2....2....0....3
..2....2....2....3....2....2....3....0....2....2....3....0....0....0....3....2
..0....0....0....2....2....2....2....0....2....2....3....0....2....0....2....3
..2....2....2....0....2....0....0....0....3....0....2....2....3....2....3....2

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <2|0|2>>^n. <<1, 2, 5>>)[1, 1]:
seq(a(n), n=0..31);  # Alois P. Heinz, Feb 28 2026

Mathematica

RecurrenceTable[{a[0] == 1, a[1] == 2, a[2]== 5, a[n] == 2 a[n - 1] + 2 a[n - 3]}, a[n], {n, 0, 29}] (* Milan Janjic, Feb 14 2015 *)

Prog

(PARI) Vec(-(x^2+1)/(2*x^3+2*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

Crossrefs

Column k=1 of A239340.

Cf. A000930, A052912, A254657, A254600, A254664.

Sequence in context: A297496 A302020 A255115 * A166297 A024960 A291234

Adjacent sequences: A239330 A239331 A239332 * A239334 A239335 A239336

Keyword

nonn,easy

Author

R. H. Hardin, Mar 16 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 27 2026

Content from A255115 moved into this page by Alois P. Heinz, Feb 28 2026

Status

approved