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 1-3 of 3 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

A172250 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 2, -1, 0, 0, 1, 1, -1, 0, 0, 0, 3, -2, 0, 0, 0, 0, 1, 3, -4, 1, 0, 0, 0, 0, 4, -2, -2, 1, 0, 0, 0, 0, 1, 6, -9, 3, 0, 0, 0, 0, 0, 0, 5, 0, -9, 6, -1, 0, 0, 0, 0, 0, 1, 10, -15, 3, 3, -1, 0, 0, 0, 0, 0, 0, 6, 5, -24, 18, -4, 0, 0, 0, 0, 0, 0, 0, 1, 15, -20, -6, 18, -8, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 29 2010

Keywords

Examples

			Triangle begins:
  1;
  0,  1;
  0,  1,  0;
  0,  0,  2, -1;
  0,  0,  1,  1, -1;
  0,  0,  0,  3, -2,  0;
  0,  0,  0,  1,  3, -4,  1;
  0,  0,  0,  0,  4, -2, -2,  1; ...

Crossrefs

Cf. A101950.

Formula

T(n,k) = T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(n,k) = 0 if k > n or if k < 0.
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A088139(n+1), A001607(n+1), A000007(n), A000012(n), A099087(n), A190960(n+1) for x = -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 15 2012
G.f.: 1/(1-y*x+y*(y-1)*x^2). - Philippe Deléham, Feb 15 2012

A202603 Triangle T(n,k), read by rows, given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, -1, 1, 1, -3, -5, -2, 0, 1, 1, -4, -7, -2, 2, 1, 1, 1, -5, -9, -1, 7, 5, 1, 1, 1, -6, -11, 1, 15, 12, 3, 0, 1, 1, -7, -13, 4, 26, 21, 3, -3, -1, 1, 1, -8, -15, 8, 40, 31, -3, -15, -7, -1
Offset: 0

Views

Author

Philippe Deléham, Dec 21 2011

Keywords

Comments

Mirror image of triangle in A129267.

Examples

			Triangle begins :
1
1, 1
1, 1, 0
1, 1, -1, -1
1, 1, -2, -3, -1
1, 1, -3, -5, -2, 0
1, 1, -4, -7, -2, 2, 1
1, 1, -5, -9, -1, 7, 5, 1

Crossrefs

Cf. A129267, A010892, A005449

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) - T(n-2,l-2) with T(0,0)= T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000012(n), A099087(n), A190960(n+1) for x = 0, 1, 2 respectively.
G.f.: 1/(1-(1+y)*x+(y+y^2)*x^2).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.