%I #34 Aug 08 2024 11:05:14

%S 1,3,6,10,15,19,25,33,42,46,52,60,70,78,90,106,123,127,133,141,151,

%T 159,171,187,205,213,225,241,261,277,301,333,366,370,376,384,394,402,

%U 414,430,448,456,468,484,504,520,544,576,610,618,630,646,666,682

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

%H Paolo Xausa, <a href="/A233774/b233774.txt">Table of n, a(n) for n = 0..10000</a>

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiGasketGraph.html">Sierpiński Gasket Graph</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sierpinski_triangle">Sierpinski triangle</a>.

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

%e Illustration of initial terms:

%e -----------------------------------------------------

%e Diagram n A233775(n) a(n)

%e -----------------------------------------------------

%e * 0 1 1

%e /T\

%e *---* 1 2 3

%e /T\ /T\

%e *---*---* 2 3 6

%e /T\ /T\

%e *---* *---* 3 4 10

%e /T\ /T\ /T\ /T\

%e *---*---*---*---* 4 5 15

%e /T\ /T\

%e *---* *---* 5 4 19

%e -----------------------------------------------------

%e 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.

%t A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];

%t Accumulate[Array[A233775, 100, 0]] (* _Paolo Xausa_, Aug 07 2024 *)

%Y Partial sums of A233775.

%Y Cf. A001316, A006046, A047999, A067771.

%K nonn

%O 0,2

%A _Omar E. Pol_, Dec 16 2013