cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane, Jan 05 2011

Keywords

Comments

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

Examples

			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, ...].

Crossrefs

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

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.

Extensions

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

A183555 Positions of the records of the positive integers in A179319; a(n) is the first position in A179319 equal to +n.

Original entry on oeis.org

0, 15, 159, 303, 2887, 5471, 51839, 98207, 930247, 1762287, 16692639, 31622991
Offset: 1

Views

Author

Paul D. Hanna, Jan 12 2011

Keywords

Comments

The g.f. of A059973 is (x+x^2-2*x^3)/(1-4*x^2-x^4).

Examples

			Define WL(x) and WU(x) to be respectively the characteristic functions of the lower (A000201) and upper (A001950) Wythoff sequences:
* WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 +...+ x^[n*phi] +...
* WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 +...+ x^[n*(phi+1)] +...
Then the g.f. of A179319 is the product:
* 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 +...
in which it is conjectured that the following holds:
* A179319(A059973(4n+1) - 2) = 2n-1 for n>=1;
* A179319(A059973(4n+2) - 2) = 2n for n>=1.

Crossrefs

Cf. A183556, A179319, A059973, A183557, A000201, A001950.

Formula

Conjecture: the positions of the records of the positive integers in A179319 are given by:
* a(2n-1) = A059973(4n+1) - 2 for n>1, with a(1) = 0;
* a(2n) = A059973(4n+2) - 2 for n>=1.

Extensions

Terms a(9) - a(12) computed by D. S. McNeil, Dec 28 2010.

A183556 Positions of the records of the negative integers in A179319; a(n) is the first position in A179319 equal to -n.

Original entry on oeis.org

1, 3, 37, 71, 681, 1291, 12237, 23183, 219601, 416019, 3940597, 7465175
Offset: 1

Views

Author

Paul D. Hanna, Jan 12 2011

Keywords

Comments

The g.f. of A059973 is (x+x^2-2*x^3)/(1-4*x^2-x^4).

Examples

			Define WL(x) and WU(x) to be respectively the characteristic functions of the lower (A000201) and upper (A001950) Wythoff sequences:
* WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 +...+ x^[n*phi] +...
* WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 +...+ x^[n*(phi+1)] +...
then the g.f. of A179319 is the product:
* 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 +...
in which it is conjectured that the following holds:
* A179319(A059973(4n-1)-1) = -(2n-1) for n>=1;
* A179319(A059973(4n)-1) = -(2n) for n>=1.

Crossrefs

Cf. A183555, A179319, A059973, A183557, A000201, A001950.

Formula

Conjecture: the positions of the records of the negative integers in A179319 are given by:
* a(2n-1) = A059973(4n-1) - 1 for n>=1;
* a(2n) = A059973(4n) - 1 for n>=1.

Extensions

Terms a(10) - a(12) computed by D. S. McNeil, Dec 28 2010.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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