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.

A084155 A Pell-related fourth-order recurrence.

Original entry on oeis.org

0, 1, 4, 19, 88, 401, 1804, 8051, 35760, 158401, 700564, 3095731, 13673224, 60375953, 266559388, 1176763859, 5194762080, 22931453953, 101225940772, 446836798675, 1972442421688, 8706804701201, 38433749994028
Offset: 0

Views

Author

Paul Barry, May 16 2003

Keywords

Comments

Binomial transform of A084154.

Links

Crossrefs

Cf. A001333, A083878.

Programs

  • GAP
    a:=[0,1,4,19];; for n in [5..25] do a[n]:=8*a[n-1]-18*a[n-2]+8*a[n-3]+7*a[n-4]; od; a; # Muniru A Asiru, Oct 18 2018
  • Magma
    I:=[0,1,4,19]; [n le 4 select I[n] else 8*Self(n-1) -18*Self(n-2) +8*Self(n-3) +7*Self(n-4): n in [1..40]]; // G. C. Greubel, Oct 17 2018
  • Maple
    seq(coeff(series(x*(1-4*x+5*x^2)/((1-2*x-x^2)*(1-6*x+7*x^2)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 18 2018
  • Mathematica
    LinearRecurrence[{8,-18,8,7},{0,1,4,19},30] (* Harvey P. Dale, Aug 16 2015 *)
  • PARI
    m=40; v=concat([0,1,4,19], vector(m-4)); for(n=5, m, v[n] = 8*v[n-1] -18*v[n-2] +8*v[n-3] +7*v[n-4]); v \\ G. C. Greubel, Oct 17 2018

Formula

a(n) = (A083878(n) - A001333(n))/2.
a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) + 7*a(n-4), a(0)=0, a(1)=1, a(2)=4, a(3)=19.
a(n) = ((3+sqrt(2))^n +(3-sqrt(2))^n -(1+sqrt(2))^n -(1-sqrt(2))^n)/4.
G.f.: x*(1-4*x+5*x^2)/((1-2*x-x^2)*(1-6*x+7*x^2)).
E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(2)*x).

A084156 Binomial transform of sinh(x)*cosh(sqrt(3)*x).

Original entry on oeis.org

0, 1, 2, 13, 44, 181, 662, 2521, 9368, 35113, 130922, 489061, 1824836, 6811741, 25420670, 94875313, 354076208, 1321442641, 4931681234, 18405321661, 68689566044, 256353060613, 956722558310, 3570537526921, 13325427195080, 49731172316281, 185599261007162
Offset: 0

Views

Author

Paul Barry, May 16 2003

Keywords

Links

Crossrefs

Cf. A001075, A084154, A084157, A254006.

Programs

  • Magma
    I:=[0,1,2,13]; [n le 4 select I[n] else 4*Self(n-1)+2*Self(n-2)-12*Self(n-3)+3*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 13 2014
  • Maple
    seq((2*simplify(ChebyshevT(n,2)) - (1+(-1)^n)*3^(n/2))/4, n = 0..30); # G. C. Greubel, Oct 10 2022
  • Mathematica
    LinearRecurrence[{4,2,-12,3},{0,1,2,13},30] (* Harvey P. Dale, Feb 01 2014 *)
  • SageMath
    def A084156(n): return (chebyshev_T(n, 2) - ((n+1)%2)*3^(n/2))/2
[A084156(n) for n in range(31)] # G. C. Greubel, Oct 10 2022

Formula

a(n) = 4*a(n-1) + 2*a(n-2) - 12*a(n-3) + 3*a(n-4).
a(n) = ((2+sqrt(3))^n + (2-sqrt(3))^n - (sqrt(3))^n - (-sqrt(3))^n)/4.
G.f.: x*(1-2*x+3*x^2)/((1-3*x^2)(1-4*x+x^2)).
E.g.f. : exp(x)*sinh(x)*cosh(sqrt(3)*x).
a(n) = (2*ChebyshevT(n, 2) - (1+(-1)^n)*3^(n/2))/4 = (A001075(n) - A254006(n))/2. - G. C. Greubel, Oct 10 2022
Showing 1-2 of 2 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.