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A192054 Let u, v be binary vectors of length n, let f(u,v) be length of longest carry propagation when we form the binary sum u+v; then a(n) = Sum_{u,v} f(u,v).

%I A192054 #16 Mar 06 2016 18:58:58
%S A192054 0,1,9,57,307,1517,7103,32117,141711,614429,2629495,11141893,46846671,
%T A192054 195760429,813970695,3370693013,13910890431,57246635581,235011903671,
%U A192054 962772769829,3937069121647,16074491903309,65538899349479,266887332403125,1085630844057375,4411756408116573,17912600251244567,72670852531322949,294610539143446735
%N A192054 Let u, v be binary vectors of length n, let f(u,v) be length of longest carry propagation when we form the binary sum u+v; then a(n) = Sum_{u,v} f(u,v).
%C A192054 There are 2^{2n} choices for (u,v).
%C A192054 A carry propagation is started if u_i = v_i = 1, and is extended if one bit of either u or v is 0 and the other is 1.
%C A192054 The longest carry propagation is n, for instance if u = 111...11, v = 000...01. See A050602 for further examples.
%H A192054 N. J. A. Sloane, <a href="/A192054/b192054.txt">Table of n, a(n) for n = 0..250</a>
%H A192054 Nicholas Pippenger, <a href="http://dx.doi.org/10.1006/jagm.2002.1216">Analysis of carry propagation in addition: an elementary approach</a>, J. Algorithms 42 (2002), 317-333.
%F A192054 Pippenger's formula is given in the Maple code.
%p A192054 C:=proc(n) local t0,j,k;
%p A192054 t0:=0;
%p A192054 for k from 1 to n+1 do
%p A192054   for j from 1 to floor(n/k) do
%p A192054 if (j*(k-1) <= n) and (j <= n-j*(k-1)) then
%p A192054   t0:=t0+binomial(n-j*(k-1),j)*(-1)^(j+1)/2^((k+1)*j);
%p A192054   fi;
%p A192054   od;
%p A192054 od:
%p A192054 RETURN(4^n*t0);
%p A192054 end;
%K A192054 nonn
%O A192054 0,3
%A A192054 _N. J. A. Sloane_, Jun 21 2011