A241891 Total number of unit squares appearing in the Thue-Morse sequence of logical matrices (1, 0 version) after n stages.

1, 2, 4, 8, 20, 72, 244, 968, 3700, 14792, 58484, 233928, 932980, 3731912, 14916724, 59666888, 238623860, 954495432, 3817806964, 15271227848, 61084212340, 244336849352, 977344601204, 3909378404808, 15637502434420

Offset

0,2

Comments

a(n) is the total number of isolated "1s" (no adjacent 1s in the horizontal or vertical directions) which appear as unit squares in the Thue-Morse sequence (another version starts with 1) of logical matrices after n stages. See links for more details.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Kival Ngaokrajang, Illustration of initial terms

Wikipedia, Thue-Morse sequence

Index entries for linear recurrences with constant coefficients, signature (4,5,-20,-4,16).

Formula

a(n) = A033691(A001045(n)) for n > 2, a(0) = 1, a(1) = 2, a(2) = 4.

a(n) = 4*a(n-1)+5*a(n-2)-20*a(n-3)-4*a(n-4)+16*a(n-5). - Colin Barker, Jan 17 2015

G.f.: -(24*x^5+12*x^4+2*x^3-9*x^2-2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Jan 17 2015

Prog

(PARI) {a0=1; a1=2; print1(a0, ", ", a1, ", "); for (n=0, 50, b=ceil(2*(2^n-1)/3); a=1-(-1)^b+4*b+2*b^2; if(Mod(n, 2)==0, a=a+4); print1(a, ", "))}
(PARI) Vec(-(24*x^5+12*x^4+2*x^3-9*x^2-2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jan 17 2015

Crossrefs

Cf. A010059, A241682.

Sequence in context: A344490 A006407 A100447 * A112278 A309868 A059731

Adjacent sequences: A241888 A241889 A241890 * A241892 A241893 A241894

Keyword

nonn,easy

Author

Kival Ngaokrajang, May 01 2014

Status

approved