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%I #8 Mar 31 2012 23:00:47
%S 1,3,37,71,681,1291,12237,23183,219601,416019,3940597,7465175
%N Positions of the records of the negative integers in A179319; a(n) is the first position in A179319 equal to -n.
%C The g.f. of A059973 is (x+x^2-2*x^3)/(1-4*x^2-x^4).
%F Conjecture: the positions of the records of the negative integers in A179319 are given by:
%F * a(2n-1) = A059973(4n-1) - 1 for n>=1;
%F * a(2n) = A059973(4n) - 1 for n>=1.
%e Define WL(x) and WU(x) to be respectively the characteristic functions of the lower (A000201) and upper (A001950) Wythoff sequences:
%e * WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 +...+ x^[n*phi] +...
%e * WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 +...+ x^[n*(phi+1)] +...
%e then the g.f. of A179319 is the product:
%e * WL(-x)*WU(x) = 1 - x + x^2 - 2*x^3 + x^4 + x^6 + x^7 + x^10 - x^11 + x^12 + x^13 + x^14 + 2*x^15 +...+ A179319(n)*x^n +...
%e in which it is conjectured that the following holds:
%e * A179319(A059973(4n-1)-1) = -(2n-1) for n>=1;
%e * A179319(A059973(4n)-1) = -(2n) for n>=1.
%Y Cf. A183555, A179319, A059973, A183557, A000201, A001950.
%K nonn,more
%O 1,2
%A _Paul D. Hanna_, Jan 12 2011
%E Terms a(10) - a(12) computed by _D. S. McNeil_, Dec 28 2010.
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