A241894 The total number of squares and rectangles appearing in the Thue-Morse sequence (1, 0 version) logical matrices after n stages.

1, 2, 5, 18, 61, 242, 925, 3698, 14621, 58482, 233245, 932978, 3729181, 14916722, 59655965, 238623858, 954451741, 3817806962, 15271053085, 61084212338, 244336150301, 977344601202, 3909375608605

Offset: 0

Author

Kival Ngaokrajang, May 01 2014

Keywords

• nonn
• easy

Comments

a(n) is the total number of unit squares (A241891), 2 X 2

squares (A241892), 2 X 1 and 1 X 2 rectangles (A241893) that appear in the Thue-Morse sequence (another version starts with 1) logical matrices after n stages. See links for more details.

Links

• Kival Ngaokrajang, Illustration of initial terms
• Kival Ngaokrajang, illustration for n = 6
• Wikipedia, Thue-Morse sequence
• Index entries for linear recurrences with constant coefficients, signature (4,5,-20,-4,16).

Crossrefs

Cf. A010059, A241685.

Programs

Mathematica

LinearRecurrence[{4,5,-20,-4,16},{1,2,5,18,61},30] (* Harvey P. Dale, Aug 02 2016 *)

PARI

{a0=1; print1(a0,", "); for (n=2,50, b=(2^(n+1)+3+(-1)^n)/6; a=floor(b^2/2);if(Mod(n,2)==1, a=a+1); print1(a,", "))}

Formula

a(n) = A000982(A005578(n+1)).

G.f.: ( -1+2*x+8*x^2-8*x^3-8*x^4 ) / ( (x-1)*(4*x-1)*(1+2*x)*(2*x-1)*(1+x) ). - R. J. Mathar, May 04 2014

18*a(n) = 7+6*2^n +4^(n+1) +(-1)^n*( 3-2^(n+1) ). - R. J. Mathar, May 04 2014