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.

A164560 Partial sums of A164532.

Original entry on oeis.org

1, 5, 11, 35, 71, 215, 431, 1295, 2591, 7775, 15551, 46655, 93311, 279935, 559871, 1679615, 3359231, 10077695, 20155391, 60466175, 120932351, 362797055, 725594111, 2176782335, 4353564671, 13060694015, 26121388031, 78364164095
Offset: 1

Views

Author

Klaus Brockhaus, Aug 16 2009

Keywords

Comments

Interleaving of A164559 and A024062 without initial term 0.

Links

Crossrefs

Cf. A164532, A164123 (partial sums of A162436), A164559 (6^n/3-1), A024062 (6^n-1), A026549.

Programs

  • Magma
    T:=[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..28] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

Formula

a(n) = 6*a(n-2)+5 for n > 2; a(1) = 1, a(2) = 5.
a(n) = (3-(-1)^n)*6^(1/4*(2*n-1+(-1)^n))/2-1.
G.f.: x*(1+4*x)/((1-x)*(1-6*x^2)).
a(n) = A026549(n) - 1.

A123011 a(n) = 2*a(n-1) + 5*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 15, 55, 185, 645, 2215, 7655, 26385, 91045, 314015, 1083255, 3736585, 12889445, 44461815, 153370855, 529050785, 1824955845, 6295165615, 21715110455, 74906048985, 258387650245, 891305545415, 3074549342055
Offset: 0

Views

Author

Roger L. Bagula, Sep 23 2006

Keywords

Links

Crossrefs

Cf. A002532, A164532, A164549.

Programs

  • Magma
    [ n le 2 select 4*n-3 else 2*Self(n-1)+5*Self(n-2): n in [1..24] ]; // Klaus Brockhaus, Aug 15 2009
  • Mathematica
    LinearRecurrence[{2,5}, {1,5}, 31] (* G. C. Greubel, Jul 13 2021 *)
  • Sage
    [1]+[(sqrt(5)*i)^(n-1)*(sqrt(5)*i*chebyshev_U(n, -i/sqrt(5)) + 3*chebyshev_U(n-1, -i/sqrt(5))) for n in (1..30)] # G. C. Greubel, Jul 13 2021

Formula

a(n) = ((3+2*sqrt(6))*(1+sqrt(6))^n + (3-2*sqrt(6))*(1-sqrt(6))^n)/6. - Klaus Brockhaus, Aug 15 2009
From Klaus Brockhaus, Aug 15 2009: (Start)
G.f.: (1+3*x)/(1-2*x-5*x^2).
Binomial transform of A164532.
Inverse binomial transform of A164549. (End)
a(n) = (sqrt(5)*i)^(n-1)*(sqrt(5)*i*ChebyshevU(n, -i/sqrt(5)) + 3*ChebyshevU(n-1, -i/sqrt(5))) for n > 0 with a(0) = 1. - G. C. Greubel, Jul 13 2021

Extensions

Edited by N. J. A. Sloane, Aug 27 2009, using simpler definition suggested by Klaus Brockhaus, Aug 15 2009

A154235 a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

Original entry on oeis.org

1, 8, 54, 352, 2276, 14688, 94744, 611072, 3941136, 25418368, 163935584, 1057300992, 6819052096, 43979406848, 283644733824, 1829363802112, 11798463078656, 76094066608128, 490767902078464, 3165202550546432
Offset: 1

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Keywords

Comments

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(6) = 6.4494897427....
Binomial transform of A164550, second binomial transform of A164549, third binomial transform of A123011, fourth binomial transform of A164532.
Binomial transform is A164551, second binomial transform is A164552, third binomial transform is A164553.

Links

Crossrefs

Cf. A010464 (decimal expansion of square root of 6), A123011, A164532, A164549, A164550, A164551, A164552, A164553.

Programs

  • GAP
    a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-10*a[n-2]; od; a; # G. C. Greubel, May 21 2019
  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009
  • Mathematica
    LinearRecurrence[{8, -10}, {1, 8}, 30] (* or *) Table[Simplify[((4 + Sqrt[6])^n -(4-Sqrt[6])^n)/(2*Sqrt[6])], {n, 30}] (* G. C. Greubel, Sep 06 2016 *)
  • PARI
    a(n)=([0,1; -10,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 07 2016
  • PARI
    my(x='x+O('x^30)); Vec(x/(1-8*x+10*x^2)) \\ G. C. Greubel, May 21 2019
  • Sage
    [lucas_number1(n,8,10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 8*a(n-1) - 10*a(n-2) for n > 1, where a(0)=0, a(1)=1.
G.f.: x/(1 - 8*x + 10*x^2). (End)

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009
Edited by Klaus Brockhaus, Oct 04 2009
Showing 1-3 of 3 results.

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

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