A190866 Let E(n) be the average number of steps required for the addition of two binary numbers of length n; sequence gives a(n) = 2^(2n-1)*E(n).

1, 8, 46, 234, 1110, 5050, 22334, 96874, 414238, 1752634, 7355118, 30670346, 127243678, 525730394, 2164795918, 8888836906, 36411649918, 148852878458, 607462498670, 2475300829258, 10073160450270, 40945074731674, 166262166593486, 674512144772970, 2734211624758846, 11075312596363962, 44832399690121262, 181370501947392138, 733336266094313886, 2963615247763178714

Offset

1,2

Comments

The addition is carried out by a parallel adder as described by J. von Neumann. Pippenger describes E(n) as the expected number of times that a "carry-save" adder must be used to clear all carries.

Links

Table of n, a(n) for n=1..30.

Volker Claus, Die mittlere Additionsdauer eines Paralleladdierwerks, Acta Informat. 2 (1973), 283-291.

D. E. Knuth, The average time for carry propagation, Nederl. Akad. Wetensch. Indag. Math., 81 (2) (1978), 238-242.

Nicholas Pippenger, Analysis of carry propagation in addition: an elementary approach, J. Algorithms 42 (2002), 317-333.

Formula

The formulas used in the Maple program are due to Claus. Knuth gives an asymptotic expansion for E(n).

Example

Values of E(n), n>=1: 1/2, 1, 23/16, 117/64, 555/256, 2525/1024, 11167/4096, 48437/16384, ...

Maple

q := proc(n, i)
        option remember ;
        if i > n then
                0 ;
        elif i = 0 then
                1;
        elif i = 1 then
                if n >= 2 then
                        (procname(n-1, 1)+1)/2 ;
                else
                        1/2 ;
                end if;
        elif i >=2 then
                procname(n-1, i)+(1-procname(n-i+1, i))/2^i ;
        end if;
end proc:
E := proc(n)
        add( q(n, i), i=1..n) ;
end proc:
[seq(2^(2*n-1)*E(n), n=1..30)];

Crossrefs

See A190868 for a closely related sequence.

Sequence in context: A182542 A026843 A026874 * A026867 A026885 A052244

Adjacent sequences: A190863 A190864 A190865 * A190867 A190868 A190869

Keyword

nonn,frac

Author

R. J. Mathar and N. J. A. Sloane, May 22 2011

Status

approved