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 1-4 of 4 results.

A183557 Positions of records in A179319 for both positive and negative integers; A183555 and A183556 merged together.

Original entry on oeis.org

0, 1, 3, 7, 15, 37, 71, 159, 303, 681, 1291, 2887, 5471, 12237, 23183, 51839, 98207, 219601, 416019, 930247, 1762287, 3940597, 7465175, 16692639, 31622991
Offset: 1

Views

Author

Paul D. Hanna, Jan 12 2011

Keywords

Crossrefs

Cf. A179319, A183555, A183556, A059973, A000201, A001950.

Extensions

Terms a(19) - a(25) were computed by D. S. McNeil, Dec 28 2010.

A059973 Expansion of x*(1 + x - 2*x^2) / ( 1 - 4*x^2 - x^4).

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 17, 38, 72, 161, 305, 682, 1292, 2889, 5473, 12238, 23184, 51841, 98209, 219602, 416020, 930249, 1762289, 3940598, 7465176, 16692641, 31622993, 70711162, 133957148, 299537289, 567451585, 1268860318, 2403763488, 5374978561
Offset: 0

Views

Author

H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001

Keywords

Comments

Based on fact that cube root of (2 +- 1 sqrt(5)) = sixth root of (9 +- 4 sqrt(5)) = ninth root of (38 +- 17 sqrt(5)) = ... = phi or 1/phi, where phi is the golden ratio.
Osler gives the first three of the above equalities with phi on page 27, stating they are simplified expressions from Ramanujan, but without hinting that the series continues.
Bisections: A001076 and A001077.

Examples

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 17*x^6 + 38*x^7 + 72*x^8 + 161*x^9 + ... - _Michael Somos_, Aug 11 2009

Links

Crossrefs

Cf. A000045, A001076, A001077.
Cf. A179319, A183555, A183556.

Programs

  • Magma
    I:=[0,1,1,2]; [n le 4 select I[n] else 4*Self(n-2)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Oct 10 2015
  • Mathematica
    CoefficientList[ Series[(x +x^2 -2x^3)/(1 -4x^2 -x^4), {x, 0, 33}], x]
LinearRecurrence[{0,4,0,1}, {0,1,1,2}, 50] (* Vincenzo Librandi, Oct 10 2015 *)
  • PARI
    {a(n) = if( n<0, n = -n; polcoeff( (-2*x + x^2 + x^3) / (1 + 4*x^2 - x^4) + x*O(x^n), n), polcoeff( (x + x^2 - 2*x^3) / ( 1 - 4*x^2 - x^4) + x*O(x^n), n))} /* Michael Somos, Aug 11 2009 */
  • PARI
    a(n) = if (n < 4, fibonacci(n), 4*a(n-2) + a(n-4));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015
  • Sage
    def a(n): return fibonacci(n) if (n<4) else 4*a(n-2) + a(n-4)
[a(n) for n in [0..40]] # G. C. Greubel, Jul 12 2021

Formula

From Michael Somos, Aug 11 2009: (Start)
a(2*n) = A001076(n).
a(2*n+1) = A001077(n). (End)
Recurrence: a(n) = 4*a(n-2) + a(n-4) for n >= 4; a(0)=0, a(1)=a(2)=1, a(3)=2. - Werner Schulte, Oct 03 2015
From Altug Alkan, Oct 06 2015: (Start)
a(2n) = Sum_{k=0..2n-1} a(k).
a(2n+1) = A001076(n-1) + Sum_{k=0..2n} a(k), n>0. (End)

Extensions

Edited by Randall L Rathbun, Jan 11 2002
More terms from Sascha Kurz, Jan 31 2003
I made the old definition into a comment and gave the g.f. as an explicit definition. - N. J. A. Sloane, Jan 05 2011
Moved g.f. from Michael Somos, into name to match terms. - Paul D. Hanna, Jan 12 2011

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

Views

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.
Showing 1-4 of 4 results.

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