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

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, 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, -2, 1, -3, 1, -1, 2, -2, 1, -3, 1, -4, 1, -2, 1, -1, 2

Offset

0,4

Comments

Mentioned in a posting by Paul D. Hanna to the Sequence Fans Mailing List, Dec 28 2010.

Links

Table of n, a(n) for n=0..76.

Formula

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;

that is, the records seem to obey the following rule:

* a(A059973(4n+1)-2) = 2n-1 for n>1,

* a(A059973(4n+2)-2) = 2n for n>=1,

* a(A059973(4n-1)-1) = -(2n-1) for n>=1,

* a(A059973(4n)-1) = -(2n) for n>=1;

see A183555 and A183556.

Example

WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 + x^12 +...+ x^[n*phi] + ...
WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 + x^18 +...+ x^[n*(phi+1)] + ...
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 +...
Positions of records for positive coefficients (A183555) in WL(-x)*WU(x) begin:
1: 0
2: 15
3: 159
4: 303
5: 2887
6: 5471
7: 51839
8: 98207
9: 930247
10: 1762287
...
Positions of records for negative coefficients (A183556) in WL(-x)*WU(x) begin:
-1: 1
-2: 3
-3: 37
-4: 71
-5: 681
-6: 1291
-7: 12237
-8: 23183
-9: 219601
-10: 416019
...
Now compare the above positions to A059973:
[1,1, 2,4, 9,17, 38,72, 161,305, 682,1292, 2889,5473, 12238,23184, 51841,98209, 219602,416020, 930249,1762289, ...].

Prog

 (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)}

Crossrefs

Cf. A000201, A001950; A183555, A183556, A183557; A059973.

Sequence in context: A357525 A016102 A083905 * A321916 A257265 A045706

Adjacent sequences: A179316 A179317 A179318 * A179320 A179321 A179322

Keyword

sign

Author

N. J. A. Sloane, Jan 05 2011

Extensions

Formula, examples, and program added by Paul D. Hanna, Jan 07 2011

Status

approved