

A292689


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


4



1, 3, 5, 15, 31, 45, 119, 231, 325, 975, 2015, 2925, 8191, 16383, 23405, 61431, 118759, 166725, 499151, 1030623, 1495405, 4186623, 8372735, 11960685, 31392247, 60686823, 85197125, 255591375, 528222175, 766774125, 2147229695, 4294721535, 6135503725, 16103829495, 31132078055
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OFFSET

1,2


COMMENTS

Term a(n) is the decimal value of A292688 = concatenation of the terms in row n of A153490 considered as a binary number.
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 nth iteration, the upperleft 3^n X 3^n squares will always remain the same. Therefore this sequence, which considers the antidiagonals of this infinite matrix, is welldefined.
The nth term a(n) has n binary digits.
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,...)


LINKS

Table of n, a(n) for n=1..35.
Eric Weisstein's World of Mathematics, Sierpinski Carpet.
Wikipedia, Sierpinski carpet.


FORMULA

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
This is true, moreover, a(k) = 2^k1 for these k (and k' = k+1), and the neighboring antidiagonals (k1 and k+2) have bitmaps of the form {101}*(101 repeated).  M. F. Hasler, Oct 25 2017


EXAMPLE

The Sierpinski carpet matrix A153490 reads
1 1 1 1 1 1 1 1 1...
1 0 1 1 0 1 1 0 1...
1 1 1 1 1 1 1 1 1...
1 1 1 0 0 0 1 1 1...
1 0 1 0 0 0 1 0 1...
1 1 1 0 0 0 1 1 1...
1 1 1 1 1 1 1 1 1...
1 0 1 1 0 1 1 0 1...
1 1 1 1 1 1 1 1 1...
(...)
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, ...)
Considered as binary numbers and converted to base 10, this yields 1, 3, 5, 15, 31, 45, 119, 231, 325, ...


PROG

(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!=2j%3!=2))); sum(k=1, n, A[k, nk+1]<<k)/2}


CROSSREFS

Cf. A153490, A292688.
Sequence in context: A126087 A148498 A259921 * A286521 A127978 A018470
Adjacent sequences: A292686 A292687 A292688 * A292690 A292691 A292692


KEYWORD

nonn,base


AUTHOR

M. F. Hasler, Oct 23 2017


STATUS

approved



