A227188 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.

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, 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, 0, 0, 0, 0, 0, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 2

Offset: 0

Author

Antti Karttunen, Jul 06 2013

Keywords

• nonn
• tabl
• base

Comments

A(n,k) is set to zero if there are fewer runs in n than k+1.

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.

Each row gives the partial sums of the terms on the corresponding row in A227186, up to the first zero.

Examples

			The top-left corner of the array:
row #  row starts as
   0    0, 0, 0, 0, 0, ...
   1    1, 0, 0, 0, 0, ...
   2    1, 2, 0, 0, 0, ...
   3    2, 0, 0, 0, 0, ...
   4    2, 3, 0, 0, 0, ...
   5    1, 2, 3, 0, 0, ...
   6    1, 3, 0, 0, 0, ...
   7    3, 0, 0, 0, 0, ...
   8    3, 4, 0, 0, 0, ...
   9    1, 3, 4, 0, 0, ...
  10    1, 2, 3, 4, 0, ...
  11    2, 3, 4, 0, 0, ...
  12    2, 4, 0, 0, 0, ...
  13    1, 2, 4, 0, 0, ...
  14    1, 4, 0, 0, 0, ...
  15    4, 0, 0, 0, 0, ...
  16    4, 5, 0, 0, 0, ...
etc.
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, ...
		

Links

• Antti Karttunen, The first 141 antidiagonals of the table, flattened

Crossrefs

Cf. A227192 (row sums). Number of nonzero terms on each row: A005811.

Cf. also A227186, A227189, A163575.

Programs

Maple

A227188 := proc(n,k)
    local bdgs,ru,i,b,a;
    bdgs := convert(n,base,2) ;
    if nops(bdgs) = 0 then
        return 0 ;
    end if;
    ru := 0 ;
    i := 1 ;
    b := op(i,bdgs) ;
    for i from 2 to nops(bdgs) do
        if op(i,bdgs) <> op(i-1,bdgs) then
            if ru = k then
                return i-1;
            end if;
            ru := ru+1 ;
        end if;
    end do:
    if ru =k then
        nops(bdgs) ;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Jul 23 2013

Mathematica

Table[PadRight[Rest@FoldList[Plus,0,Length/@Split[Reverse[IntegerDigits[j,2]]]],i+1-j][[i+1-j]],{i,0,12},{j,0,i}] (* Wouter Meeussen, Aug 31 2013 *)

Scheme

(define (A227188 n) (A227188bi (A002262 n) (A025581 n)))
(define (A227188bi n k) (cond ((< (A005811 n) (+ 1 k)) 0) ((zero? k) (A136480 n)) (else (+ (A136480 n) (A227188bi (A163575 n) (- k 1))))))

Formula

A(n,0) = A136480(n), n>0.