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
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.
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
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.
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
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
H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001
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
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- T. J. Osler, Cardan polynomials and the reduction of radicals, Math. Mag., 74 (No. 1, 2001), 26-32.
- Index entries for linear recurrences with constant coefficients, signature (0,4,0,1).
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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
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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 *)
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{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 */
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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
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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
I made the old definition into a comment and gave the g.f. as an explicit definition. -
N. J. A. Sloane, Jan 05 2011
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