A181208 Number of n X 4 binary matrices with no two 1's adjacent diagonally or antidiagonally.

16, 64, 484, 2704, 17424, 104976, 652864, 4000000, 24681024, 151782400, 934891776, 5754132736, 35428274176, 218096472064, 1342706197504, 8266039005184, 50888705511424, 313286601609216, 1928696564957184, 11873676328960000

Offset

1,1

Comments

Column 4 of A181212.

Links

Robert Israel, Table of n, a(n) for n = 1..1265 (n = 1..325 from R. H. Hardin)

Robert Israel, Maple-assisted proof of formula

Index entries for linear recurrences with constant coefficients, signature (6,8,-48,24,32,-16).

Formula

Empirical: a(n) = 6*a(n-1) + 8*a(n-2) - 48*a(n-3) + 24*a(n-4) + 32*a(n-5) - 16*a(n-6).

Formula confirmed by Robert Israel, Dec 25 2017 (see link).

G.f.: 4*x*(4 - 8*x - 7*x^2 + 14*x^3 + 4*x^4 - 4*x^5) / ((1 - 8*x + 12*x^2 - 4*x^3)*(1 + 2*x - 4*x^2 - 4*x^3)). - Colin Barker, Mar 26 2018

Maple

f:= gfun:-rectoproc({a(n)=6*a(n-1)+8*a(n-2)-48*a(n-3)+24*a(n-4)+32*a(n-5)-16*a(n-6), a(1)=16, a(2)=64, a(3)=484, a(4)=2704, a(5)=17424, a(6)=104976}, a(n), remember):
map(f, [$1..20]); # Robert Israel, Dec 25 2017

Mathematica

RecurrenceTable[{a[n] == 6*a[n-1] + 8*a[n-2] - 48*a[n-3] + 24*a[n-4] + 32*a[n-5] - 16*a[n-6], a[1] == 16, a[2] == 64, a[3] == 484, a[4] == 2704, a[5] == 17424, a[6] == 104976}, a, {n, 1, 20}] (* Jean-François Alcover, Aug 29 2022, after Robert Israel *)
LinearRecurrence[{6, 8, -48, 24, 32, -16}, {16, 64, 484, 2704, 17424, 104976}, 30] (* Harvey P. Dale, Aug 29 2024 *)

Prog

(PARI) Vec(4*x*(4 - 8*x - 7*x^2 + 14*x^3 + 4*x^4 - 4*x^5) / ((1 - 8*x + 12*x^2 - 4*x^3)*(1 + 2*x - 4*x^2 - 4*x^3)) + O(x^30)) \\ Colin Barker, Mar 26 2018

Crossrefs

Cf. A181212.

Sequence in context: A352738 A061449 A168091 * A175209 A141840 A203281

Adjacent sequences: A181205 A181206 A181207 * A181209 A181210 A181211

Keyword

nonn,easy

Author

R. H. Hardin, Oct 10 2010

Status

approved