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 1-2 of 2 results.

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

Comments

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Links

Crossrefs

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
1..A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
2..A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
3..A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
4..A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
5..A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
6..A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
7..A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
8..A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
9..A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
10.A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011
  • Mathematica
    LinearRecurrence[{2,-10}, {0,1}, 50]
  • PARI
    a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016
  • SageMath
    [lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

Formula

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A003698 Number of 2-factors in C_4 X P_n.

Original entry on oeis.org

1, 9, 53, 341, 2169, 13825, 88093, 561357, 3577121, 22794425, 145252485, 925589701, 5898117961, 37584466929, 239498796653, 1526153708861, 9725080775409, 61970950592425, 394896331045333, 2516390514947637
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Crossrefs

Cf. A190973.

Programs

  • GAP
    a:=[1,9,53];; for n in [4..30] do a[n]:=6*a[n-1]+3*a[n-2]-4*a[n-3]; od; a; # G. C. Greubel, Dec 24 2019
  • Magma
    I:=[1,9,53]; [n le 3 select I[n] else 6*Self(n-1) +3*Self(n-2) -4*Self(n-3): n in [1..20]]; // G. C. Greubel, Dec 24 2019
  • Maple
    seq( simplify( (-1)^n + 2^n*Chebyshev(n,7/4) - 2^(n+1)*ChebyshevU(n-1,7/4))/2 ), n=1..30); # G. C. Greubel, Dec 24 2019
  • Mathematica
    Rest@CoefficientList[Series[x*(1-x)*(1+4*x)/((1+x)*(1-7*x+4*x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 13 2013 *)
Table[((-1)^n + 2^n*ChebyshevU[n, 7/4] - 2^(n+1)*ChebyshevU[n-1, 7/4])/2, {n, 30}] (* G. C. Greubel, Dec 24 2019 *)
  • PARI
    vector(30, n, ((-1)^n + 2^n*polchebyshev(n, 2, 7/4) - 2^(n+1)*polchebyshev(n-1, 2, 7/4))/2 ) \\ G. C. Greubel, Dec 24 2019
  • Sage
    [((-1)^n + 2^n*chebyshev_U(n, 7/4) - 2^(n+1)*chebyshev_U(n-1, 7/4))/2 for n in (1..30)] # G. C. Greubel, Dec 24 2019

Formula

a(n) = 6*a(n-1) + 3*a(n-2) - 4*a(n-3), n>3.
G.f.: x*(1-x)*(1+4*x)/((1+x)*(1-7*x+4*x^2)). - Colin Barker, Aug 30 2012
Showing 1-2 of 2 results.

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