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.

A100047 A Chebyshev transform of the Fibonacci numbers.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul Barry, Oct 31 2004

Keywords

Comments

Multiplicative with a(p^e) = (-1)^(e+1) if p = 2, 0 if p = 5, 1 if p == 1 or 9 (mod 10), (-1)^e if p == 3 or 7 (mod 10). - David W. Wilson, Jun 10 2005
This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 1, P2 = -1, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 24 2014
From Peter Bala, Mar 24 2014: (Start)
This is the particular case P1 = 1, P2 = -1, Q = 1 of the following results:
Let P1, P2 and Q be integers. Let alpha and beta denote the roots of the quadratic equation x^2 - 1/2*P1*x + 1/4*P2 = 0. Let T(n,x;Q) denote the bivariate Chebyshev polynomial of the first kind defined by T(n,x;Q) = 1/2*( (x + sqrt(x^2 - Q))^n + (x - sqrt(x^2 - Q))^n ) (when Q = 1, T(n,x;Q) reduces to the ordinary Chebyshev polynomial of the first kind T(n,x)). Then we have
1) The sequence A(n) := ( T(n,alpha;Q) - T(n,beta;Q) )/(alpha - beta) is a linear divisibility sequence of the fourth order.
2) A(n) belongs to the 3-parameter family of fourth-order divisibility sequences found by Williams and Guy.
3) The o.g.f. of the sequence A(n) is the rational function x*(1 - Q*x^2)/(1 - P1*x + (P2 + 2*Q)*x^2 - P1*Q*x^3 + Q^2*x^4).
4) The o.g.f. is the Chebyshev transform of the rational function x/(1 - P1*x + P2*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - Q*x^2)/(1 + Q*x^2)*A(x/(1 + Q*x^2)).
5) Let q = sqrt(Q) and set a = sqrt( q + (P2)/(4*q) + (P1)/2 ) and b = sqrt( q + (P2)/(4*q) - (P1)/2 ). Then the o.g.f. of the sequence A(n) is the Hadamard product of the rational functions x/(1 - (a + b)*x + q*x^2) and x/(1 - (a - b)*x + q*x^2). Thus A(n) is the product of two (usually, non-integer) Lucas-type sequences.
6) A(n) = the bottom left entry of the 2 X 2 matrix 2*T(n,1/2*M;Q), where M is the 2 X 2 matrix [0, -P2; 1, P1].
For examples of the above see A006238, A054493, A078070, A092184, A098306, A100048, A108196, A138573, A152090 and A218134. (End)

Examples

			A Chebyshev transform of the Fibonacci numbers A000045: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
The denominator is the 10th cyclotomic polynomial.
G.f. = x + x^2 - x^3 - x^4 - x^6 - x^7 + x^8 + x^9 + x^11 + x^12 - x^13 + ...

Links

Crossrefs

Other Chebyshev transforms: A011655, A099443, A100048.
Cf. A006238, A054493, A078070, A092184, A098306, A108196, A138573, A152090, A218134.
Cf. also A011558, A080891, A244895, A226162.

Programs

  • Maple
    a := n -> (-1)^iquo(n, 5)*signum(mods(n, 5)):
seq(a(n), n=0..89); # after Michael Somos, Peter Luschny, Dec 30 2018
  • Mathematica
    a[ n_] := {1, 1, -1, -1, 0, -1, -1, 1, 1, 0}[[Mod[ n, 10, 1]]]; (* Michael Somos, May 24 2015 *)
a[ n_] := (-1)^Quotient[ n, 5] Sign[ Mod[ n, 5, -2]]; (* Michael Somos, May 24 2015 *)
a[ n_] := (-1)^Quotient[n, 5] {1, 1, -1, -1, 0}[[Mod[ n, 5, 1]]]; (* Michael Somos, May 24 2015 *)
LinearRecurrence[{1, -1, 1, -1}, {0, 1, 1, -1}, 90] (* Jean-François Alcover, Jun 11 2019 *)
  • PARI
    {a(n) = (-1)^(n\5) * [0, 1, 1, -1, -1][n%5+1]}; /* Michael Somos, May 24 2015 */
  • PARI
    {a(n) = (-1)^(n\5) * sign( centerlift( Mod(n, 5)))}; /* Michael Somos, May 24 2015 */

Formula

G.f.: x*(1 - x^2)/(1 - x + x^2 - x^3 + x^4).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k *binomial(n-k, k)*A000045(n-2*k)/(n -k).
From Peter Bala, Mar 24 2014: (Start)
a(n) = (T(n,alpha) - T(n,beta))/(alpha - beta), where alpha = (1 + sqrt(5))/4 and beta = (1 - sqrt(5))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = bottom left entry of the matrix T(n, M), where M is the 2 X 2 matrix [0, 1/4; 1, 1/2].
The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/2*(sqrt(5) + 1)*x + x^2) and x/(1 - 1/2*(sqrt(5) - 1)*x + x^2). (End)
Euler transform of length 10 sequence [ 1, -2, 0, 0, -1, 0, 0, 0, 0, 1]. - Michael Somos, May 24 2015
a(n) = a(-n) = -a(n + 5) for all n in Z. - Michael Somos, May 24 2015
|A011558(n)| = |A080891(n)| = |a(n)| = A244895(n). - Michael Somos, May 24 2015

A071101 Expansion of (5 + 6*x + 3*x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.

Original entry on oeis.org

5, 16, 45, 130, 377, 1088, 3145, 9090, 26269, 75920, 219413, 634114, 1832625, 5296384, 15306833, 44237570, 127848949, 369490320, 1067846845, 3086134658, 8919094697, 25776662080, 74495936025, 215297250946, 622220603405, 1798250918672, 5197041610021
Offset: 0

Views

Author

N. J. A. Sloane, May 28 2002

Keywords

Comments

Number of tilings of the 2-mod-4 pillow of order n is a perfect square times a(n). [Propp, 1999, p. 272]

Examples

			G.f. = 5 + 16*x + 45*x^2 + 130*x^3 + 377*x^4 + 1088*x^5 + 3145*x^6 + 9090*x^7 + ...

References

  • J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 12).

Links

Crossrefs

Cf. A112835, A138573.

Programs

  • GAP
    a:=[5,16,45,130];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2] +2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018
  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4) )); // G. C. Greubel, Jul 29 2019
  • Maple
    seq(coeff(series((5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018
  • Mathematica
    Table[Abs[Fibonacci[n+3, 1+I]]^2, {n,0,30}] (* Vladimir Reshetnikov, Oct 05 2016 *)
CoefficientList[Series[(5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 30}], x] (* Stefano Spezia, Sep 12 2018 *)
LinearRecurrence[{2,2,2,-1},{5,16,45,130},30] (* Harvey P. Dale, Oct 03 2024 *)
  • PARI
    {a(n) = my(m = abs(n+3)); polcoeff( (x - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + x * O(x^m), m)};  /* Michael Somos, Dec 15 2011 */
  • PARI
    x='x+O('x^33); Vec((5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4)) \\ Altug Alkan, Sep 12 2018
  • Python
    from math import log
a0,a1,a2,a3,n = 130,45,16,5,3
print(0,a3)
print(1,a2)
print(2,a1)
print(3,a0)
while log(a0)/log(10) < 1000:
    a0,a1,a2,a3,n = 2*(a0+a1+a2)-a3,a0,a1,a2,n+1
    print(n,a0) # A.H.M. Smeets, Sep 12 2018
  • Sage
    ((5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019

Formula

G.f.: (5 + 6*x + 3*x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4).
a(-n) = a(-6 + n). a(-1) = 2, a(-2) = 1, a(-3) = 0. a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4). - Michael Somos, Dec 15 2011
A112835(2*n + 3) = a(n).
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

A140824 Expansion of (x-x^3)/(1-3*x+2*x^2-3*x^3+x^4).

Original entry on oeis.org

0, 1, 3, 6, 15, 41, 108, 281, 735, 1926, 5043, 13201, 34560, 90481, 236883, 620166, 1623615, 4250681, 11128428, 29134601, 76275375, 199691526, 522799203, 1368706081, 3583319040, 9381251041, 24560434083, 64300051206, 168339719535, 440719107401, 1153817602668
Offset: 0

Views

Author

N. J. A. Sloane, Sep 07 2009, based on email from R. K. Guy, Mar 09 2009

Keywords

Comments

Case P1 = 3, P2 = 0, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

Links

Crossrefs

Cf. A006238, A005248, A054493, A078070, A092184, A098306, A100047, A100048, A108196, A138573, A152090, A218134.

Programs

  • Mathematica
    LinearRecurrence[{3, -2, 3, -1}, {0, 1, 3, 6}, 50] (* G. C. Greubel, Aug 08 2017 *)
  • PARI
    x='x+O('x^50); concat([0], Vec((x-x^3)/(1-3*x+2*x^2-3*x^3+x^4))) \\ G. C. Greubel, Aug 08 2017

Formula

a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 6, a(n) - 3 a(n + 1) + 2 a(n + 2) - 3 a(n + 3) + a(n + 4) = 0.
From Peter Bala, Mar 25 2014: (Start)
a(n) = 2/3*( T(n,3/2) - T(n,0) ), where T(n,x) is a Chebyshev polynomial of the first kind.
a(n) = 1/3 * (A005248(n) - (i^n + (-i)^n)) = 1/3 * (Fibonacci(2*n-1) + Fibonacci(2*n+1) - (i^n + (-i)^n)).
a(n) = bottom left entry of the 2 X 2 matrix 2*T(n, 1/2*M), where M is the 2 X 2 matrix [0, 0; 1, 3].
The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/sqrt(2)*(sqrt(5) + i)*x + x^2) and x/(1 - 1/sqrt(2)*(sqrt(5) - i)*x + x^2). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = A099483(n) - A099483(n-2). - R. J. Mathar, Feb 10 2016

A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) in powers of x.

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 5, 13, 16, 37, 45, 109, 130, 313, 377, 905, 1088, 2617, 3145, 7561, 9090, 21853, 26269, 63157, 75920, 182525, 219413, 527509, 634114, 1524529, 1832625, 4405969, 5296384, 12733489, 15306833, 36800465, 44237570, 106355317
Offset: 0

Views

Author

Michael Somos, Feb 10 2012

Keywords

Comments

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

Examples

			G.f. = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 13*x^7 + 16*x^8 + 37*x^9 + ...

References

  • J. A. Sjogren, Cycles and spanning trees. Math. Comput. Modelling 15, No.9, 87-102 (1991).

Links

Crossrefs

Cf. A071100 (bisection), A071101 (bisection), A112835, A138573.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ))); // G. C. Greubel, Aug 12 2018
  • Mathematica
    CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)
  • PARI
    {a(n) = my(m = abs(n)); polcoeff( x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) + x * O(x^m), m)};
  • PARI
    {a(n) = my(m = abs(n), v); v = polroots( Pol([ 1, 2, 4, 2, 1])); sqrtint( round( prod( k=1, 4, v[k]^m - 1, 2^(m%2) / 20)))};

Formula

G.f.: x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8).
a(n) = a(-n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8) for all n in Z.
a(2*n + 5) = A071100(n). a(2*n + 6) = A071101(n). a(n + 3) = A112835(n). a(2*n) = A138573(n).
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