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A348648 Triangle T(n, k), n >= 0, 0 <= k <= n, read by rows; the even terms of Pascal's triangle (A007318) are clustered as wXwXw subtriangles with w = 2^s-1 for some s > 0; if A007318(n, k) is even then T(n, k) gives the corresponding s otherwise T(n, k) = 0.

%I A348648 #18 Oct 27 2021 16:07:25
%S A348648 0,0,0,0,1,0,0,0,0,0,0,2,2,2,0,0,0,2,2,0,0,0,1,0,2,0,1,0,0,0,0,0,0,0,
%T A348648 0,0,0,3,3,3,3,3,3,3,0,0,0,3,3,3,3,3,3,0,0,0,1,0,3,3,3,3,3,0,1,0,0,0,
%U A348648 0,0,3,3,3,3,0,0,0,0,0,2,2,2,0,3,3,3,0,2,2,2,0
%N A348648 Triangle T(n, k), n >= 0, 0 <= k <= n, read by rows; the even terms of Pascal's triangle (A007318) are clustered as wXwXw subtriangles with w = 2^s-1 for some s > 0; if A007318(n, k) is even then T(n, k) gives the corresponding s otherwise T(n, k) = 0.
%C A348648 It is possible to build this triangle with the following procedure:
%C A348648 - T_0 is the 2X2X2 triangle with all 0's:
%C A348648           ^
%C A348648   T_0 =  /0\
%C A348648         /0 0\
%C A348648        +-----+
%C A348648 - for s > 0, T_s is obtained by arranging 3 copies of T_{s-1} and one wXwXw triangle (where w=2^s-1) will all s's as follows:
%C A348648                  ^
%C A348648                 / \
%C A348648                /   \
%C A348648               /     \
%C A348648              /T_{s-1}\
%C A348648   T_s =     +---------+
%C A348648            / \s ... s/ \
%C A348648           /   \.   ./   \
%C A348648          /     \. ./     \
%C A348648         /T_{s-1}\s/T_{s-1}\
%C A348648        +---------+---------+
%C A348648 - the triangle {T(n, k)} is the limit of T_s as s tends to infinity.
%C A348648 For any n >= 0, A001316(n) gives the number of 1's in row 4*n + 2 (and there are no 1's elsewhere).
%H A348648 Rémy Sigrist, <a href="/A348648/b348648.txt">Table of n, a(n) for n = 0..8255</a> (rows for n = 0..127, flattened)
%H A348648 Rémy Sigrist, <a href="/A348648/a348648.png">Colored representation of the first 2^9 rows</a> (black pixels correspond to 0's)
%H A348648 Rémy Sigrist, <a href="/A348648/a348648_1.png">Colored logarithmic scatterplot of A014428</a> (where the colors are given by the present sequence)
%H A348648 Rémy Sigrist, <a href="/A348648/a348648_2.png">Colored logarithmic scatterplot of A007318</a> (where the colors are given by the present sequence, and black pixels denotes 0's)
%H A348648 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A348648 T(n, k) = 0 iff A047999(n, k) = 1.
%F A348648 2^T(n, 0) OR 2^T(n, 1) OR ... OR 2^T(n, n) = 1 + A129760(n+1) (where OR denotes the bitwise OR operator).
%e A348648 Triangle T(n, k) begins (with dots instead of 0's):
%e A348648                     .
%e A348648                    . .
%e A348648                   . 1 .
%e A348648                  . . . .
%e A348648                 . 2 2 2 .
%e A348648                . . 2 2 . .
%e A348648               . 1 . 2 . 1 .
%e A348648              . . . . . . . .
%e A348648             . 3 3 3 3 3 3 3 .
%e A348648            . . 3 3 3 3 3 3 . .
%e A348648           . 1 . 3 3 3 3 3 . 1 .
%e A348648          . . . . 3 3 3 3 . . . .
%e A348648         . 2 2 2 . 3 3 3 . 2 2 2 .
%e A348648        . . 2 2 . . 3 3 . . 2 2 . .
%e A348648       . 1 . 2 . 1 . 3 . 1 . 2 . 1 .
%e A348648      . . . . . . . . . . . . . . . .
%o A348648 (PARI) T(n,k) = { if (n<=1, return (0)); my (s=#binary(n)-1); n-=2^s; if (k<=n, return (T(n,k)), k<2^s, return (s), return (T(n,k-2^s))) }
%Y A348648 Cf. A001316, A007318, A047999, A014428, A129760.
%K A348648 nonn,look,tabl
%O A348648 0,12
%A A348648 _Rémy Sigrist_, Oct 27 2021