A239040 Number of n X 1 0..3 arrays with no element equal to one plus the sum of elements to its left or one plus the sum of elements above it, modulo 4.

3, 7, 16, 38, 90, 212, 500, 1180, 2784, 6568, 15496, 36560, 86256, 203504, 480128, 1132768, 2672544, 6305344, 14876224, 35097536, 82805760, 195363968, 460923008, 1087457536, 2565643008, 6053132032, 14281179136, 33693644288, 79493552640, 187549463552, 442486215680, 1043959536640

Offset

1,1

Links

Paolo Xausa, Table of n, a(n) for n = 1..1000 (terms 1..210 from R. H. Hardin)

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

Formula

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

Empirical g.f.: x*(3 + x + 2*x^2) / (1 - 2*x - 2*x^3). - Colin Barker, Feb 24 2018

The above empirical formulas are correct by the matrix transfer method. - Andrew Howroyd, Jan 22 2026

From Bruce Nye, Jan 22 2026: (Start)

a(n) = A077999(n+2)/2.

Limit_{n->oo} a(n)/a(n-1) = A357101. (End)

Example

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

Mathematica

LinearRecurrence[{2, 0, 2}, {3, 7, 16}, 35] (* Paolo Xausa, Jan 28 2026 *)

Prog

(PARI) Vec(x*(3 + x + 2*x^2) / (1 - 2*x - 2*x^3) + O(x^30)) \\ Bruce Nye, Jan 22 2026

Crossrefs

Column 1 of A239047.

Cf. A077999, A357101.

Sequence in context: A192665 A052967 A297498 * A323225 A293065 A211278

Adjacent sequences: A239037 A239038 A239039 * A239041 A239042 A239043

Keyword

nonn,easy

Author

R. H. Hardin, Mar 09 2014

Status

approved