cp's OEIS Frontend

A292689 Decimal values of the antidiagonals of the Sierpinski carpet considered as binary numbers.

%I A292689 #25 Feb 16 2025 08:33:51
%S A292689 1,3,5,15,31,45,119,231,325,975,2015,2925,8191,16383,23405,61431,
%T A292689 118759,166725,499151,1030623,1495405,4186623,8372735,11960685,
%U A292689 31392247,60686823,85197125,255591375,528222175,766774125,2147229695,4294721535,6135503725,16103829495,31132078055
%N A292689 Decimal values of the antidiagonals of the Sierpinski carpet considered as binary numbers.
%C A292689 Term a(n) is the decimal value of A292688 = concatenation of the terms in row n of A153490 considered as a binary number.
%C A292689 The Sierpinski carpet is the fractal obtained by starting with a unit square and at subsequent iterations, subdividing each square into 3 X 3 smaller squares and removing the middle square. After the n-th iteration, the upper-left 3^n X 3^n squares will always remain the same. Therefore this sequence, which considers the antidiagonals of this infinite matrix, is well-defined.
%C A292689 The n-th term a(n) has n binary digits.
%C A292689 The Hamming weights of the terms (also row sums of A153490) are (1, 2, 2, 4, 5, 4, 6, 6, 4, 8, 10, 8, 13, 14, 10, 14, 13, 8, 14, 16, 12, 18, 18, 12, 16, ...).
%H A292689 Paolo Xausa, <a href="/A292689/b292689.txt">Table of n, a(n) for n = 1..729</a>
%H A292689 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiCarpet.html">Sierpinski Carpet</a>.
%H A292689 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sierpinski_carpet">Sierpinski carpet</a>.
%F A292689 a(k+1) = 2*a(k)+1 for all k in A003462 = (1, 4, 13, 40, 121, 364, ...). (Conjectured.) - _R. J. Cano_, Oct 25 2017
%F A292689 This is true, moreover, a(k) = 2^k-1 for these k (and k' = k+1), and the neighboring antidiagonals (k-1 and k+2) have bitmaps of the form {101}*(101 repeated). - _M. F. Hasler_, Oct 25 2017
%e A292689 The Sierpinski carpet matrix A153490 reads
%e A292689    1 1 1 1 1 1 1 1 1 ...
%e A292689    1 0 1 1 0 1 1 0 1 ...
%e A292689    1 1 1 1 1 1 1 1 1 ...
%e A292689    1 1 1 0 0 0 1 1 1 ...
%e A292689    1 0 1 0 0 0 1 0 1 ...
%e A292689    1 1 1 0 0 0 1 1 1 ...
%e A292689    1 1 1 1 1 1 1 1 1 ...
%e A292689    1 0 1 1 0 1 1 0 1 ...
%e A292689    1 1 1 1 1 1 1 1 1 ...
%e A292689    ...
%e A292689 The concatenation of the terms in the antidiagonals yields A292688 = (1, 11, 101, 1111, 11111, 101101, 1110111, 11100111, 101000101, 1111001111, 11111011111, 101101101101, 1111111111111, 11111111111111, 101101101101101, ...).
%e A292689 Considered as binary numbers and converted to base 10, this yields 1, 3, 5, 15, 31, 45, 119, 231, 325, ... .
%t A292689 A292689[i_]:=With[{a=Nest[ArrayFlatten[{{#,#,#},{#,0,#},{#,#,#}}]&,{{1}},i]},Array[FromDigits[Diagonal[a,#],2]&,3^i,1-3^i]];A292689[4] (* Generates 3^4 terms *) (* _Paolo Xausa_, May 13 2023 *)
%o A292689 (PARI) A292689(n,A=Mat(1))={while(#A<n,A=matrix(3*#A,3*#A,i,j,if(A[(i+2)\3,(j+2)\3],i%3!=2||j%3!=2)));sum(k=1,n,A[k,n-k+1]<<k)/2}
%Y A292689 Cf. A153490, A292688.
%K A292689 nonn,base
%O A292689 1,2
%A A292689 _M. F. Hasler_, Oct 23 2017