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%I #21 Apr 02 2017 00:31:18
%S 0,0,1,0,0,1,0,0,2,2,0,0,0,0,2,0,0,0,0,3,1,0,0,0,0,0,2,1,0,0,0,0,0,3,
%T 3,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,0,3,1,0,0,
%U 0,0,0,0,0,0,0,4,2,2,0,0,0,0,0,0,0,0,0,0,3,3,2
%N Square array A(n,k) read by antidiagonals: the one-based bit-index where the (k+1)-st run in the binary expansion of n ends, as read from the least significant end.
%C A(n,k) is set to zero if there are fewer runs in n than k+1.
%C Equally, when A005811(n) > 1, A(n,k) gives the zero-based bit-index where the (k+2)-th run in the binary expansion of n starts, counted from the least significant end.
%C Each row gives the partial sums of the terms on the corresponding row in A227186, up to the first zero.
%H Antti Karttunen, <a href="/A227188/b227188.txt">The first 141 antidiagonals of the table, flattened</a>
%F A(n,0) = A136480(n), n>0.
%e The top-left corner of the array:
%e row # row starts as
%e 0 0, 0, 0, 0, 0, ...
%e 1 1, 0, 0, 0, 0, ...
%e 2 1, 2, 0, 0, 0, ...
%e 3 2, 0, 0, 0, 0, ...
%e 4 2, 3, 0, 0, 0, ...
%e 5 1, 2, 3, 0, 0, ...
%e 6 1, 3, 0, 0, 0, ...
%e 7 3, 0, 0, 0, 0, ...
%e 8 3, 4, 0, 0, 0, ...
%e 9 1, 3, 4, 0, 0, ...
%e 10 1, 2, 3, 4, 0, ...
%e 11 2, 3, 4, 0, 0, ...
%e 12 2, 4, 0, 0, 0, ...
%e 13 1, 2, 4, 0, 0, ...
%e 14 1, 4, 0, 0, 0, ...
%e 15 4, 0, 0, 0, 0, ...
%e 16 4, 5, 0, 0, 0, ...
%e etc.
%e For example, for n = 8, whose binary expansion is "1000", we get the run lengths 3 and 1 (scanning from the right), partial sums of which are 3 and 4, thus row 8 begins as A(8,0)=3, A(8,1)=4, A(8,2)=0, ...
%p A227188 := proc(n,k)
%p local bdgs,ru,i,b,a;
%p bdgs := convert(n,base,2) ;
%p if nops(bdgs) = 0 then
%p return 0 ;
%p end if;
%p ru := 0 ;
%p i := 1 ;
%p b := op(i,bdgs) ;
%p for i from 2 to nops(bdgs) do
%p if op(i,bdgs) <> op(i-1,bdgs) then
%p if ru = k then
%p return i-1;
%p end if;
%p ru := ru+1 ;
%p end if;
%p end do:
%p if ru =k then
%p nops(bdgs) ;
%p else
%p 0 ;
%p end if;
%p end proc: # _R. J. Mathar_, Jul 23 2013
%t Table[PadRight[Rest@FoldList[Plus,0,Length/@Split[Reverse[IntegerDigits[j,2]]]],i+1-j][[i+1-j]],{i,0,16},{j,0,i}]; _Wouter Meeussen_, Aug 31 2013
%o (Scheme)
%o (define (A227188 n) (A227188bi (A002262 n) (A025581 n)))
%o (define (A227188bi n k) (cond ((< (A005811 n) (+ 1 k)) 0) ((zero? k) (A136480 n)) (else (+ (A136480 n) (A227188bi (A163575 n) (- k 1))))))
%Y Cf. A227192 (row sums). Number of nonzero terms on each row: A005811.
%Y Cf. also A227186, A227189, A163575.
%K nonn,tabl,base
%O 0,9
%A _Antti Karttunen_, Jul 06 2013