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

%I A233774 #34 Aug 08 2024 11:05:14
%S A233774 1,3,6,10,15,19,25,33,42,46,52,60,70,78,90,106,123,127,133,141,151,
%T A233774 159,171,187,205,213,225,241,261,277,301,333,366,370,376,384,394,402,
%U A233774 414,430,448,456,468,484,504,520,544,576,610,618,630,646,666,682
%N A233774 Total number of vertices in the first n rows of Sierpinski gasket, with a(0) = 1.
%H A233774 Paolo Xausa, <a href="/A233774/b233774.txt">Table of n, a(n) for n = 0..10000</a>
%H A233774 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 A233774 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiGasketGraph.html">Sierpiński Gasket Graph</a>.
%H A233774 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sierpinski_triangle">Sierpinski triangle</a>.
%F A233774 a(2^k) = A067771(k), k >= 0.
%e A233774 Illustration of initial terms:
%e A233774 -----------------------------------------------------
%e A233774            Diagram            n     A233775(n)   a(n)
%e A233774 -----------------------------------------------------
%e A233774               *               0         1         1
%e A233774              /T\
%e A233774             *---*             1         2         3
%e A233774            /T\ /T\
%e A233774           *---*---*           2         3         6
%e A233774          /T\     /T\
%e A233774         *---*   *---*         3         4        10
%e A233774        /T\ /T\ /T\ /T\
%e A233774       *---*---*---*---*       4         5        15
%e A233774      /T\             /T\
%e A233774     *---*           *---*     5         4        19
%e A233774 -----------------------------------------------------
%e A233774 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 A233774 A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];
%t A233774 Accumulate[Array[A233775, 100, 0]] (* _Paolo Xausa_, Aug 07 2024 *)
%Y A233774 Partial sums of A233775.
%Y A233774 Cf. A001316, A006046, A047999, A067771.
%K A233774 nonn
%O A233774 0,2
%A A233774 _Omar E. Pol_, Dec 16 2013