A233774 Total number of vertices in the first n rows of Sierpinski gasket, with a(0) = 1.

1, 3, 6, 10, 15, 19, 25, 33, 42, 46, 52, 60, 70, 78, 90, 106, 123, 127, 133, 141, 151, 159, 171, 187, 205, 213, 225, 241, 261, 277, 301, 333, 366, 370, 376, 384, 394, 402, 414, 430, 448, 456, 468, 484, 504, 520, 544, 576, 610, 618, 630, 646, 666, 682

Offset: 0

Author

Omar E. Pol, Dec 16 2013

Keywords

• nonn

Examples

			Illustration of initial terms:
-----------------------------------------------------
           Diagram            n     A233775(n)   a(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. The total number of vertices is 19, so a(5) = 19.
		

Links

• Paolo Xausa, 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, Sierpiński Gasket Graph.
• Wikipedia, Sierpinski triangle.

Crossrefs

Partial sums of A233775.

Cf. A001316, A006046, A047999, A067771.

Programs

Mathematica

A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];
Accumulate[Array[A233775, 100, 0]] (* Paolo Xausa, Aug 07 2024 *)

Formula

a(2^k) = A067771(k), k >= 0.