A014019 -id:A014019 - cp's OEIS frontend

A099443 A Chebyshev transform of Fib(n+1).

1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0

Offset: 0

Paul Barry, Oct 16 2004

The denominator is the 10th cyclotomic polynomial. It is also associated to the knots 4_1 and 5_1 by the Alexander and Jones polynomials respectively. The g.f. is the image of the g.f. of Fib(n+1) under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

With offset 1 this is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = y = z = 1.

			G.f. = 1 + x + x^2 + x^3 - x^5 - x^6 - x^7 - x^8 + x^10 + x^11 + x^12 + ...

Dror Bar-Natan, The Rolfsen Knot Table
C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).

Cf. A156174.

a[ n_] := With[ {m = n + 1}, Sign[Mod[m, 5]] (-1)^Quotient[m, 5]]; (* Michael Somos, Jun 17 2015 *)
LinearRecurrence[{1, -1, 1, -1}, {1, 1, 1, 1}, 50] (* G. C. Greubel, Aug 08 2017 *)

{a(n) = n++; sign(n%5) * (-1)^(n\5)}; /* Michael Somos, Sep 18 2006 */

G.f.: (1+x^2)/(1-x+x^2-x^3+x^4).
a(n) = sqrt(2*(10 - 4*sqrt(5))/25)*cos((6*Pi*n +Pi)/10) + sqrt(2*(4*sqrt(5) + 10)/25)*sin(Pi*(n+1)/5).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*Fib(n-2k+1).
a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*Fib(k+1)/2.
a(n) = Sum_{k=0..n} A014019(n-k)*binomial(1, k/2)(1+(-1)^k)/2.
With a leading zero, this is sum{k=0..floor(n/2), binomial(n-k-1, k)(-1)^kFib(n-2k)}, or the image of x/(1-x-x^2) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005
Euler transform of length 10 sequence [1, 0, 0, -1, -1, 0, 0, 0, 0, 1]. - Michael Somos, Sep 18 2006
G.f.: (1 - x^4) * (1 - x^5) / ((1 - x) * (1 - x^10)). - Michael Somos, Sep 18 2006
a(n) = -a(n-5) = -a(-2-n) for all n in Z. - Michael Somos, Sep 18 2006
Hankel transform is 1,0,0,1,0,0,0,... - Paul Barry, Jun 24 2008
0 = (a(n) - a(n+1)) * (a(n) - a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jul 07 2014
0 = a(n)*a(n-4) - a(n-1)*a(n-3) + a(n-2)*a(n-2) for all n in Z. - Michael Somos, Jul 07 2014
0 = a(n)*a(n+5) - a(n+1)*a(n+4) + a(n+2)*a(n+3) for all n in Z. - Michael Somos, Jul 07 2014
a(n) = (-1)^n * A156174(n). - Michael Somos, Oct 17 2018

A129920 Expansion of -1/(1 - x + 3x^2 - 2x^3 + x^4 - 2*x^5 + x^6).

Original entry on oeis.org

-1, -1, 2, 3, -4, -10, 5, 29, 2, -76, -45, 178, 212, -361, -750, 565, 2282, -306, -6206, -2428, 15176, 14353, -32719, -55104, 57933, 176234, -61524, -499047, -97429, 1271400, 921652, -2887641, -3948938, 5590078, 13380187, -7828378, -39536779, 108416, 104810904

Offset: 0

Roger L. Bagula, Jun 05 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-3,2,-1,2,-1).

Cf. A008620, A010892, A014019, A099443, A099479, A099480, A112712.

Cf. A125629, A129704, A129903.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) )); // G. C. Greubel, Sep 28 2024

CoefficientList[Series[-1/(1-x +3*x^2 -2*x^3 +x^4 -2*x^5 +x^6), {x,0,50}], x]

def A129920_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) ).list()
A129920_list(50) # G. C. Greubel, Sep 28 2024

a(n) = a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6), n >= 6. - Franck Maminirina Ramaharo, Jan 08 2019

Edited by Franck Maminirina Ramaharo, Jan 08 2019

A125629 Expansion of -1/(1 - x + x^2 - x^3 + x^4 + x^6).

Original entry on oeis.org

1, -1, 0, 0, 0, 1, 2, 2, 1, 0, -1, -3, -5, -5, -3, 0, 4, 9, 13, 13, 8, -1, -13, -26, -35, -34, -20, 6, 40, 74, 95, 89, 48, -26, -120, -209, -258, -232, -111, 98, 355, 587, 699, 601, 245, -342, -1040, -1641, -1887, -1545, -504, 1137, 3023, 4568, 5073, 3936, 912

Offset: 0

Roger L. Bagula, Jun 07 2007

The Knot Atlas, L6a3
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,0,-1).

Cf. A008620, A010892, A014019, A099443, A099479, A099480, A112712, A129920, A129704, A129903.

CoefficientList[Series[-1/(1 - x + x^2 - x^3 + x^4 + x^6), {x, 0, 50}], x]

G.f.: 1/(x^(17/2)*f(x)), where f(x) = -1/x^(5/2) - 1/x^(9/2) + 1/x^(11/2) + -1/x^(13/2) + 1/x^(15/2) - 1/x^(17/2) is the Jones polynomial for the link with Dowker-Thistlethwaite notation L6a3.

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) - a(n-6), n >= 6. - Franck Maminirina Ramaharo, Jan 08 2019

Edited by Franck Maminirina Ramaharo, Jan 08 2019

A134119 a(n) = floor(n^2/10) - floor((n-1)^2/10).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17

Offset: 0

Alexander R. Povolotsky, Jan 12 2008

Note that for n >=1 there is a pattern that keeps steadily alternating between 4 terms and 6 terms for the each two consecutive groups. The terms value remains the same within each 4-term or 6-term group, while during the switch from the 4-group to the 6-group and then back from the 6-group to the 4-group, etc., the term value is getting bumped by 1.

Assuming this obeys the recurrence a(n) = a(n-10) + 2, this has generating function G(x) = x^4*(1+x^4)/[(-1+x)^2*(x+1)*(x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1)] = (1 - 3x^2 - 3x^3)/[10(x^4 + x^3 + x^2 + x + 1)]+1/[10(x+1)] + 1/[5(-1+x)^2] +(-1 + 2x - 3x^2 - x^3)/[10(x^4 - x^3 + x^2 - x + 1)] + 3/[10(-1+x)]. The first term can be rewritten as a linear superposition of A104384(n), A104384(n+2), A103483(n+3); the second, ~1/(x+1), with the alternating A033999, the third component ~1/(x-1)^2 with a(n)=n+1, the next ~1/(x^4 - x^3 + x^2 - x + 1) = A014019 and the last is proportional to 1/(1-x) = A000012. So a(n) is a sum of these sequences. - R. J. Mathar, Jan 16 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A000012, A014019, A033999, A056865.

Table[Floor[n^2/10] - Floor[(n - 1)^2/10], {n, 0, 50}] (* G. C. Greubel, Feb 22 2017 *)

PARI
```
a(n)= floor(n^2/10) - floor((n-1)^2/10)
```

Empirical g.f.: x^4*(x^4+1) / (x^11 - x^10 - x + 1). - Colin Barker, Aug 08 2013

The above conjectured g.f. is correct. - Sela Fried, Dec 08 2024

More terms from N. J. A. Sloane, Jan 22 2008

A281861 Riordan transform of the Fibonacci sequence with the Riordan matrix A053121.

Original entry on oeis.org

0, 1, 1, 4, 6, 18, 32, 85, 165, 411, 839, 2013, 4237, 9933, 21317, 49236, 107014, 244750, 536500, 1218888, 2687362, 6077644, 13453606, 30329434, 67326816, 151439158, 336842092, 756452890, 1684953360, 3779590010, 8427441240

Offset: 0

Wolfdieter Lang, Feb 18 2017

The Riordan matrix A053121 of the Bell type R = (c(x^2), x*c(x^2)), with the o.g.f. c of A000108 (Catalan), is the inverse of the Riordan matrix A049310 (Chebyshev S).
The Riordan transform of a sequence {a_n}, n >= 0 to a sequence {b_n}, n >= 0 is b = R a
  in matrix notation, with the lower triangular Riordan matrix (N x N, with arbitrary large N).

Cf. A000045, A000108, A049310, A053121, A014019.

G.f.: c(x^2)*A(x*c(x^2)) with the g.f. c of A000108 (Catalan) and A of A000045 (Fibonacci).

a(n) = Sum_{m=0..n} R(n, m)*F(n), with R(n, m) = A053121(n, m) and F(m) = A000045(m), n >= 0.

A099494 A Chebyshev transform of Fibonacci(n)+(-1)^n.

Original entry on oeis.org

1, 0, 1, 1, -1, 0, 0, -2, 0, 1, -1, 1, 2, -1, 0, 1, -2, -1, 1, -1, 0, 2, 0, 0, 1, -1, -1, 0, -1, 0, 1, 0, 1, 1, -1, 0, 0, -2, 0, 1, -1, 1, 2, -1, 0, 1, -2, -1, 1, -1, 0, 2, 0, 0, 1, -1, -1, 0, -1, 0, 1, 0, 1, 1, -1, 0, 0, -2, 0, 1, -1, 1, 2, -1, 0, 1, -2, -1, 1, -1, 0, 2, 0, 0, 1, -1, -1

Offset: 0

Paul Barry, Oct 19 2004

A Chebyshev transform of A008346, which has g.f. 1/(1-2x^2-x^3). The image of G(x) under the Chebyshev transform is (1/(1+x^2))*G(x/(1+x^2)).

Periodic with period length 30. - Ray Chandler, Sep 08 2015

Index entries for linear recurrences with constant coefficients, signature (0,-1,1,-1,0,-1).

Cf. A000484, A008346, A014019

G.f.: (1+x^2)^2/(1+x^2-x^3+x^4+x^6).
a(n) = -a(n-2)+a(n-3)-a(n-4)-a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(F(n-2*k)+(-1)^(n-2*k)).
a(n) = A014019(n-1) + A000484(n).

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A099443 A Chebyshev transform of Fib(n+1).

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A129920 Expansion of -1/(1 - x + 3*x^2 - 2*x^3 + x^4 - 2*x^5 + x^6).

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A125629 Expansion of -1/(1 - x + x^2 - x^3 + x^4 + x^6).

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A134119 a(n) = floor(n^2/10) - floor((n-1)^2/10).

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A281861 Riordan transform of the Fibonacci sequence with the Riordan matrix A053121.

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A099494 A Chebyshev transform of Fibonacci(n)+(-1)^n.

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A129920 Expansion of -1/(1 - x + 3x^2 - 2x^3 + x^4 - 2*x^5 + x^6).