A219080 Number of 3 X n binary arrays with every 0 a horizontal, diagonal or antidiagonal neighbor to some 1.

1, 41, 337, 2321, 17537, 134809, 1023441, 7759553, 58921537, 447447561, 3397273553, 25794057969, 195847550081, 1487017265657, 11290490704977, 85725445999521, 650888811968321, 4942012485147241, 37523285458920273, 284903562335955153, 2163191178033816193, 16424491254114254681

Offset

1,2

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Index entries for linear recurrences with constant coefficients, signature (6,5,46,61,-10,-1,-10).

Formula

Empirical: a(n) = 6*a(n-1) + 5*a(n-2) + 46*a(n-3) + 61*a(n-4) - 10*a(n-5) - a(n-6) - 10*a(n-7).

Empirical g.f.: x*(1 + 35*x + 86*x^2 + 48*x^3 - 21*x^4 - 11*x^5 - 10*x^6) / (1 - 6*x - 5*x^2 - 46*x^3 - 61*x^4 + 10*x^5 + x^6 + 10*x^7). - Colin Barker, Jul 25 2018

The above empirical formulas confirmed using the transfer matrix method. - Andrew Howroyd, Jul 17 2026

Example

Some solutions for n=3:
..1..1..0....0..1..1....0..1..1....1..1..1....1..1..1....0..1..0....0..1..0
..0..1..0....0..1..1....1..1..0....1..1..1....0..0..0....0..0..1....1..1..0
..0..0..1....0..1..0....1..0..0....0..0..1....0..1..1....0..1..1....1..1..1

Mathematica

LinearRecurrence[{6, 5, 46, 61, -10, -1, -10}, {1, 41, 337, 2321, 17537, 134809, 1023441}, 25] (* Paolo Xausa, Jul 18 2026 *)

Prog

(PARI) Vec(x*(1 + 35*x + 86*x^2 + 48*x^3 - 21*x^4 - 11*x^5 - 10*x^6) / (1 - 6*x - 5*x^2 - 46*x^3 - 61*x^4 + 10*x^5 + x^6 + 10*x^7) + O(x^26)) \\ Andrew Howroyd, Jul 17 2026

Crossrefs

Row 3 of A219078.

Sequence in context: A300775 A232857 A232793 * A184060 A300645 A268779

Adjacent sequences: A219077 A219078 A219079 * A219081 A219082 A219083

Keyword

nonn,easy,changed

Author

R. H. Hardin, Nov 11 2012

Extensions

New name from Andrew Howroyd, Jul 15 2026

Status

approved