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%I #8 Sep 16 2017 22:09:43
%S 1,0,1,0,2,0,1,0,1,1,2,0,2,0,1,0,1,0,1,1,3,1,2,0,1,1,2,0,2,0,1,0,1,0,
%T 1,0,2,0,1,1,2,2,3,1,3,1,2,0,1,0,1,1,3,1,2,0,1,1,2,0,2,0,1,0,1,0,1,0,
%U 2,0,1,0,1,1,2,0,2,0,1,1,2,1,2,2,4,2,3,1,2,2,3,1
%N Number of singletons in the integer partition having viabin number n.
%C The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20.
%F a(n) = A290260(2n).
%e a(37) = 2; indeed, the binary representation of 37 is 100101, leading to the integer partition [3',2',1,1] (the singletons are marked).
%p a := proc (n) local b: b := proc (n) if n = 1 then 0 elif `mod`(n, 2) = 0 and `mod`((1/2)*n, 2) = 1 then 1+b((1/2)*n) elif `mod`(n, 2) = 1 then b((1/2)*n-1/2) elif `mod`(n-4, 8) = 0 then b((1/2)*n)-1 else b((1/2)*n) end if end proc: b(2*n) end proc: seq(a(n), n = 1 .. 150);
%Y Bisection of A290260.
%K nonn
%O 1,5
%A _Emeric Deutsch_, Sep 16 2017
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