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A261363 Triangle read by rows: partial row sums of Sierpinski's triangle.

%I A261363 #12 May 13 2025 02:39:56
%S A261363 1,1,2,1,1,2,1,2,3,4,1,1,1,1,2,1,2,2,2,3,4,1,1,2,2,3,3,4,1,2,3,4,5,6,
%T A261363 7,8,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,3,4,1,1,2,2,2,2,2,2,3,3,4,1,2,
%U A261363 3,4,4,4,4,4,5,6,7,8,1,1,1,1,2,2,2,2,3,3,3,3,4
%N A261363 Triangle read by rows: partial row sums of Sierpinski's triangle.
%C A261363 T(n,n) = number of distinct terms in row n = number of odd terms in row n+1 = A001316(n).
%C A261363 Central terms, for n > 0: T(2*n,n) = A048896(n-1).
%H A261363 Reinhard Zumkeller, <a href="/A261363/b261363.txt">Rows n = 0..125 of triangle, flattened</a>
%e A261363 .   n |  Sierpinski: A047999(n,*)  |  Partial row sums: T(n,*)
%e A261363 . ----+----------------------------+----------------------------
%e A261363 .   0 |              1             |              1
%e A261363 .   1 |             1 1            |             1 2
%e A261363 .   2 |            1 0 1           |            1 1 2
%e A261363 .   3 |           1 1 1 1          |           1 2 3 4
%e A261363 .   4 |          1 0 0 0 1         |          1 1 1 1 2
%e A261363 .   5 |         1 1 0 0 1 1        |         1 2 2 2 3 4
%e A261363 .   6 |        1 0 1 0 1 0 1       |        1 1 2 2 3 3 4
%e A261363 .   7 |       1 1 1 1 1 1 1 1      |       1 2 3 4 5 6 7 8
%e A261363 .   8 |      1 0 0 0 0 0 0 0 1     |      1 1 1 1 1 1 1 1 2
%e A261363 .   9 |     1 1 0 0 0 0 0 0 1 1    |     1 2 2 2 2 2 2 2 3 4
%e A261363 .  10 |    1 0 1 0 0 0 0 0 1 0 1   |    1 1 2 2 2 2 2 2 3 3 4
%e A261363 .  11 |   1 1 1 1 0 0 0 0 1 1 1 1  |   1 2 3 4 4 4 4 4 5 6 7 8
%e A261363 .  12 |  1 0 0 0 1 0 0 0 1 0 0 0 1 |  1 1 1 1 2 2 2 2 3 3 3 3 4  .
%t A261363 row[n_] := Accumulate[Array[Boole[0 == BitAnd[n-#, #]] &, n + 1, 0]]; Array[row, 13, 0] // Flatten (* _Amiram Eldar_, May 13 2025 *)
%o A261363 (Haskell)
%o A261363 a261363 n k = a261363_tabl !! n !! k
%o A261363 a261363_row n = a261363_tabl !! n
%o A261363 a261363_tabl = map (scanl1 (+)) a047999_tabl
%Y A261363 Cf. A047999, A008949, A048896 (central terms), A001316 (right edge), A261366.
%K A261363 nonn,tabl,easy
%O A261363 0,3
%A A261363 _Reinhard Zumkeller_, Aug 16 2015