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A179319 G.f.: WL(-x)*WU(x), where WL, WU are respectively the characteristic functions of the lower (A000201) and upper (A001950) Wythoff sequences.

%I A179319 #8 Mar 30 2012 18:37:22
%S A179319 1,-1,1,-2,1,0,1,1,0,0,1,-1,1,1,1,2,-1,1,1,0,1,-1,1,1,0,0,1,-1,1,-2,1,
%T A179319 0,1,-1,1,-2,1,-3,1,-1,1,0,1,-1,1,-2,1,0,1,1,0,0,1,-1,1,-2,1,0,1,-1,1,
%U A179319 -2,1,-3,1,-1,2,-2,1,-3,1,-4,1,-2,1,-1,2
%N A179319 G.f.: WL(-x)*WU(x), where WL, WU are respectively the characteristic functions of the lower (A000201) and upper (A001950) Wythoff sequences.
%C A179319 Mentioned in a posting by Paul D. Hanna to the Sequence Fans Mailing List, Dec 28 2010.
%F A179319  It appears that the records for positive integers occur at positions A059973(4n+1)-2 and A059973(4n+2)-2, while the records for negative integers occur at positions A059973(4n-1)-1 and A059973(4n)-1;
%F A179319 that is, the records seem to obey the following rule:
%F A179319 * a(A059973(4n+1)-2) = 2n-1 for n>1,
%F A179319 * a(A059973(4n+2)-2) = 2n for n>=1,
%F A179319 * a(A059973(4n-1)-1) = -(2n-1) for n>=1,
%F A179319 * a(A059973(4n)-1) = -(2n) for n>=1;
%F A179319 see A183555 and A183556.
%e A179319 WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 + x^12 +...+ x^[n*phi] + ...
%e A179319 WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 + x^18 +...+ x^[n*(phi+1)] + ...
%e A179319 G.f.: 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 - x^16 +...+ a(n)*x^n +...
%e A179319 Positions of records for positive coefficients (A183555) in WL(-x)*WU(x) begin:
%e A179319 1: 0
%e A179319 2: 15
%e A179319 3: 159
%e A179319 4: 303
%e A179319 5: 2887
%e A179319 6: 5471
%e A179319 7: 51839
%e A179319 8: 98207
%e A179319 9: 930247
%e A179319 10: 1762287
%e A179319 ...
%e A179319 Positions of records for negative coefficients (A183556) in WL(-x)*WU(x) begin:
%e A179319 -1: 1
%e A179319 -2: 3
%e A179319 -3: 37
%e A179319 -4: 71
%e A179319 -5: 681
%e A179319 -6: 1291
%e A179319 -7: 12237
%e A179319 -8: 23183
%e A179319 -9: 219601
%e A179319 -10: 416019
%e A179319 ...
%e A179319 Now compare the above positions to A059973:
%e A179319 [1,1, 2,4, 9,17, 38,72, 161,305, 682,1292, 2889,5473, 12238,23184, 51841,98209, 219602,416020, 930249,1762289, ...].
%o A179319  (PARI) {a(n)=local(phi=(1+sqrt(5))/2,WL=sum(m=0,ceil(n/phi),(-x)^floor(m*phi))+x*O(x^n),WU=sum(m=0,ceil(n/(phi+1)),x^floor(m*(phi+1)))+x*O(x^n));polcoeff(WL*WU,n)}
%Y A179319 Cf. A000201, A001950; A183555, A183556, A183557; A059973.
%K A179319 sign
%O A179319 0,4
%A A179319 _N. J. A. Sloane_, Jan 05 2011
%E A179319 Formula, examples, and program added by _Paul D. Hanna_, Jan 07 2011