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A233775
Number of vertices in the n-th row of the Sierpinski gasket (cf. A047999).
9
1, 2, 3, 4, 5, 4, 6, 8, 9, 4, 6, 8, 10, 8, 12, 16, 17, 4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 33, 4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 65, 4, 6, 8, 10, 8, 12
OFFSET
0,2
COMMENTS
Partial sums give A233774.
The subsequence of odd terms is A083318. - Gary W. Adamson, Jan 13 2014
Equivalently, this is the coordination sequence for the Sierpinski gasket with respect to the apex. - N. J. A. Sloane, Sep 19 2020
FORMULA
a(0)=1, a(n) = (2^t(n) + 1) * 2^(c(n) - 1) where t(n) = A007814(n) is the number of trailing zeros in the binary representation of n and c(n) = A000120(n) is the total number of ones in the binary representation of n. - Johan Falk, Jun 24 2020
EXAMPLE
Illustration of initial terms:
--------------------------------------------------------
Diagram n a(n) A233774(n)
--------------------------------------------------------
* 0 1 1
/T\
*---* 1 2 3
/T\ /T\
*---*---* 2 3 6
/T\ /T\
*---* *---* 3 4 10
/T\ /T\ /T\ /T\
*---*---*---*---* 4 5 15
/T\ /T\
*---* *---* 5 4 19
--------------------------------------------------------
After five stages the number of "black" triangles in the structure is A006046(5) = 11 and the number of "black" triangles in row 5 is A001316(5-1) = 2. The number of vertices in row 5 is equal to 4, so a(5) = 4.
Written as an irregular triangle the sequence begins:
1;
2;
3;
4,5;
4,6,8,9;
4,6,8,10,8,12,16,17;
4,6,8,10,8,12,16,18,8,12,16,20,16,24,32,33;
...
MAPLE
A000120 := n -> add(i, i=convert(n, base, 2)):
A007814 := n -> padic[ordp](n, 2):
A233775 := n->(2^A007814(n)+1)*(2^(A000120(n)-1); # N. J. A. Sloane, Sep 19 2020
MATHEMATICA
A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];
Array[A233775, 100, 0] (* Paolo Xausa, Aug 05 2024 *)
PROG
(PARI) print1("1, "); for(k=1, 70, print1((2^valuation(k, 2)+1) *2^(hammingweight(k)-1), ", ")) \\ Hugo Pfoertner, Jul 27 2020
CROSSREFS
Right border gives A094373.
Sequence in context: A137912 A324196 A269597 * A118577 A377070 A135681
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 16 2013
STATUS
approved