0,2

Alois P. Heinz, Table of n, a(n) for n = 0..1000

a(5) = 2^5 - 8 = 24: 00000, 00001, 00011, 00110, 00111, 01100, 01101, 01110, 01111, 10000, 10001, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111. These 8 words are not counted: 00010, 00100, 00101, 01000, 01001, 01010, 01011, 10010.

b:= proc(n, t, c) option remember; `if`(c<0, 0, `if`(n=0, 1,

b(n-1, [2, 4, 6, 4, 6, 4, 6][t], c-`if`(t=5, 1, 0))+

b(n-1, [3, 5, 7, 5, 7, 5, 7][t], c+`if`(t=6, 1, 0))))

end:

a:= n-> b(n, 1, 0):

seq(a(n), n=0..40);

# Alternative:

a:= proc(n) option remember; `if`(n<6, [1, 2, 4, 7, 13, 24][n+1],

((680+1441*n-444*n^2+35*n^3) *a(n-1)

-(4*(-112+625*n-179*n^2+14*n^3)) *a(n-2)

+(2*(1521-656*n+63*n^2)) *a(n-3)

+(2*(-9442+5295*n-947*n^2+56*n^3)) *a(n-4)

-(4*(-6721+3413*n-591*n^2+35*n^3)) *a(n-5)

+(4*(2*n-11))*(7*n^2-79*n+254) *a(n-6)

)/((n+1)*(7*n^2-93*n+340)))

end:

seq(a(n), n=0..40);

b[n_, t_, c_] := b[n, t, c] = If[c < 0, 0, If[n == 0, 1,

b[n - 1, {2, 4, 6, 4, 6, 4, 6}[[t]], c - If[t == 5, 1, 0]] +

b[n - 1, {3, 5, 7, 5, 7, 5, 7}[[t]], c + If[t == 6, 1, 0]]]];

a[n_] := b[n, 1, 0];

Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Sep 16 2023, after Alois P. Heinz *)

Cf. A118430, A164146, A255386, A260505, A260697, A303430.

Sequence in context: A059633 A088353 A192654 * A371789 A018184 A192675

Adjacent sequences: A260665 A260666 A260667 * A260669 A260670 A260671

Alois P. Heinz, Nov 14 2015

approved