A233775 Number of vertices in the n-th row of the Sierpinski gasket (cf. A047999).

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Eric Weisstein's World of Mathematics, Sierpinski Graph

Wikipedia, Sierpinski triangle

Index entries for coordination sequences

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;
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;

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

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.

Cf. A001316, A006046, A047999, A233774, A083318.

Sequence in context: A137912 A324196 A269597 * A118577 A135681 A135680

Adjacent sequences: A233772 A233773 A233774 * A233776 A233777 A233778

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 16 2013

Status

approved