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 A166556 #18 Apr 09 2025 07:24:55
%S A166556 1,2,1,3,1,1,4,2,2,1,5,2,2,1,1,6,3,2,1,2,1,7,3,3,1,3,1,1,8,4,4,2,4,2,
%T A166556 2,1,9,4,4,2,4,2,2,1,1,10,5,4,2,4,2,2,1,2,1,11,5,5,2,4,2,2,1,3,1,1,12,
%U A166556 6,6,3,4,2,2,1,4,2,2,1
%N A166556 Triangle read by rows, A000012 * A047999.
%H A166556 G. C. Greubel, <a href="/A166556/b166556.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A166556 Triangle read by rows, A000012 * A047999; where A000012 = an infinite lower triangular matrix with all 1's: [1; 1,1; 1,1,1;..]; and A047999 = Sierpinski's gasket.
%F A166556 The operation takes partial sums of Sierpinski's gasket terms, by columns.
%F A166556 From _G. C. Greubel_, Dec 02 2024: (Start)
%F A166556 T(n, k) = Sum_{j=k..n} (binomial(j,k) mod 2).
%F A166556 T(n, 0) = A000027(n+1).
%F A166556 T(n, 1) = A004526(n+1).
%F A166556 T(n, 2) = A004524(n+1).
%F A166556 T(2*n, n) = A080100(n).
%F A166556 Sum_{k=0..n} T(n, k) = A006046(n+1).
%F A166556 Sum_{k=0..n} (-1)^k*T(n, k) = A006046(floor(n/2)+1).
%F A166556 Sum_{k=0..floor(n/2)} T(n-k, k) = A007729(n). (End)
%e A166556 First few rows of the triangle =
%e A166556 1;
%e A166556 2, 1;
%e A166556 3, 1, 1;
%e A166556 4, 2, 2, 1;
%e A166556 5, 2, 2, 1, 1;
%e A166556 6, 3, 2, 1, 2, 1;
%e A166556 7, 3, 3, 1, 3, 1, 1;
%e A166556 8, 4, 4, 2, 4, 2, 2, 1;
%e A166556 9, 4, 4, 2, 4, 2, 2, 1, 1;
%e A166556 10, 5, 4, 2, 4, 2, 2, 1, 2, 1;
%e A166556 11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1;
%e A166556 12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1;
%e A166556 13, 6, 6, 3, 5, 2, 2, 1, 5, 2, 2, 1, 1;
%e A166556 ...
%p A166556 A166556 := proc(n,k)
%p A166556 local j;
%p A166556 add(A047999(j,k),j=k..n) ;
%p A166556 end proc: # _R. J. Mathar_, Jul 21 2016
%t A166556 A166556[n_, k_]:= Sum[Mod[Binomial[j,k], 2], {j,k,n}];
%t A166556 Table[A166556[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 02 2024 *)
%o A166556 (Magma)
%o A166556 A166556:= func< n,k | (&+[(Binomial(j,k) mod 2): j in [k..n]]) >;
%o A166556 [A166556(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 02 2024
%o A166556 (Python)
%o A166556 def A166556(n,k): return sum(binomial(j,k)%2 for j in range(k,n+1))
%o A166556 print(flatten([[A166556(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Dec 02 2024
%Y A166556 Cf. A000027, A004524, A004526, A047999, A080100.
%Y A166556 Sums include: A006046 (row), A007729 (diagonal).
%K A166556 nonn,easy,tabl
%O A166556 0,2
%A A166556 _Gary W. Adamson_, Oct 17 2009