This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A233775 #71 Aug 06 2024 22:02:11
%S A233775 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,
%T A233775 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,
%U A233775 24,32,36,16,24,32,40,32,48,64,65,4,6,8,10,8,12
%N A233775 Number of vertices in the n-th row of the Sierpinski gasket (cf. A047999).
%C A233775 Partial sums give A233774.
%C A233775 The subsequence of odd terms is A083318. - _Gary W. Adamson_, Jan 13 2014
%C A233775 Equivalently, this is the coordination sequence for the Sierpinski gasket with respect to the apex. - _N. J. A. Sloane_, Sep 19 2020
%H A233775 N. J. A. Sloane, <a href="/A233775/b233775.txt">Table of n, a(n) for n = 0..10000</a>
%H A233775 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 A233775 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiGasketGraph.html">Sierpiński Gasket Graph</a>.
%H A233775 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sierpinski_triangle">Sierpinski triangle</a>.
%H A233775 <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a>
%F A233775 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
%e A233775 Illustration of initial terms:
%e A233775 --------------------------------------------------------
%e A233775 Diagram n a(n) A233774(n)
%e A233775 --------------------------------------------------------
%e A233775 * 0 1 1
%e A233775 /T\
%e A233775 *---* 1 2 3
%e A233775 /T\ /T\
%e A233775 *---*---* 2 3 6
%e A233775 /T\ /T\
%e A233775 *---* *---* 3 4 10
%e A233775 /T\ /T\ /T\ /T\
%e A233775 *---*---*---*---* 4 5 15
%e A233775 /T\ /T\
%e A233775 *---* *---* 5 4 19
%e A233775 --------------------------------------------------------
%e A233775 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.
%e A233775 Written as an irregular triangle the sequence begins:
%e A233775 1;
%e A233775 2;
%e A233775 3;
%e A233775 4,5;
%e A233775 4,6,8,9;
%e A233775 4,6,8,10,8,12,16,17;
%e A233775 4,6,8,10,8,12,16,18,8,12,16,20,16,24,32,33;
%e A233775 ...
%p A233775 A000120 := n -> add(i, i=convert(n, base, 2)):
%p A233775 A007814 := n -> padic[ordp](n, 2):
%p A233775 A233775 := n->(2^A007814(n)+1)*(2^(A000120(n)-1); # _N. J. A. Sloane_, Sep 19 2020
%t A233775 A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];
%t A233775 Array[A233775, 100, 0] (* _Paolo Xausa_, Aug 05 2024 *)
%o A233775 (PARI) print1("1, "); for(k=1,70, print1((2^valuation(k,2)+1) *2^(hammingweight(k)-1),", ")) \\ _Hugo Pfoertner_, Jul 27 2020
%Y A233775 Right border gives A094373.
%Y A233775 Cf. A001316, A006046, A047999, A233774, A083318.
%K A233775 nonn,tabf
%O A233775 0,2
%A A233775 _Omar E. Pol_, Dec 16 2013