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.

A164640 a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

Original entry on oeis.org

1, 6, 8, 48, 64, 384, 512, 3072, 4096, 24576, 32768, 196608, 262144, 1572864, 2097152, 12582912, 16777216, 100663296, 134217728, 805306368, 1073741824, 6442450944, 8589934592, 51539607552, 68719476736, 412316860416
Offset: 1

Views

Author

Klaus Brockhaus, Aug 20 2009

Keywords

Comments

Interleaving of A001018 and A083233 without initial term 1.
Binomial transform is A164544. Third binomial transform is A038761.

Links

Crossrefs

Cf. A001018 (powers of 8), A083233 ((3*8^n+(0)^n)/4), A164544, A038761.

Programs

  • Magma
    [ n le 2 select 5*n-4 else 8*Self(n-2): n in [1..26] ];
  • Mathematica
    LinearRecurrence[{0,8},{1,6},40] (* Harvey P. Dale, Nov 06 2013 *)
  • PARI
    a(n)=(7-(-1)^n)*2^(1/4*(6*n-15+3*(-1)^n)) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (7-(-1)^n)*2^(1/4*(6*n -15+3*(-1)^n)).
G.f.: x*(1+6*x)/(1-8*x^2).

Extensions

G.f. corrected by Klaus Brockhaus, Sep 18 2009

A164545 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

Original entry on oeis.org

1, 8, 36, 176, 848, 4096, 19776, 95488, 461056, 2226176, 10748928, 51900416, 250597376, 1209991168, 5842354176, 28209381376, 136206942208, 657665294336, 3175488946176, 15332616962048, 74032423632896, 357460162379776
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A164544. Second binomial transform of A164640. Inverse binomial transform of A038761.

Links

Crossrefs

Cf. A038761, A164544, A164640.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+3*r)*(2+2*r)^n+(2-3*r)*(2-2*r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 19 2009
  • Mathematica
    LinearRecurrence[{4,4},{1,8},30] (* Harvey P. Dale, Dec 25 2011 *)
  • Sage
    [(2*i)^n*(chebyshev_U(n, -i) - 2*i*chebyshev_U(n-1, -i)) for n in (0..30)] # G. C. Greubel, Jul 17 2021

Formula

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
a(n) = ((2+3*sqrt(2))*(2+2*sqrt(2))^n + (2-3*sqrt(2))*(2-2*sqrt(2))^n)/4.
G.f.: (1 + 4*x)/(1 - 4*x - 4*x^2).
a(n) = (2*i)^n*( ChebyshevU(n, -i) - 2*i*ChebyshevU(n-1, -i) ). - G. C. Greubel, Jul 17 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 19 2009

A154346 a(n) = 12*a(n-1) - 28*a(n-2) for n > 1, with a(0)=1, a(1)=12.

Original entry on oeis.org

1, 12, 116, 1056, 9424, 83520, 738368, 6521856, 57587968, 508443648, 4488860672, 39629905920, 349870772224, 3088811900928, 27269361188864, 240745601040384, 2125405099196416, 18763984361226240, 165656469557215232
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009

Keywords

Comments

Binomial transform of A164547, second binomial transform of A164546, third binomial transform of A038761, fourth binomial transform of A164545, fifth binomial transform of A164544, sixth binomial transform of A164640.
Lim_{n -> infinity} a(n)/a(n-1) = 6 + 2*sqrt(2) = 8.8284271247....

Links

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A164547, A164546, A038761, A164545, A164544, A164640.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((6+2*r)^n-(6-2*r)^n)/(4*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009
  • Mathematica
    Join[{a=1,b=12},Table[c=12*b-28*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
LinearRecurrence[{12,-28},{1,12},20] (* Harvey P. Dale, May 23 2012 *)
Rest@ CoefficientList[Series[x/(1 - 12 x + 28 x^2), {x, 0, 19}], x] (* Michael De Vlieger, Sep 13 2016 *)
  • PARI
    a(n)=([0,1; -28,12]^(n-1)*[1;12])[1,1] \\ Charles R Greathouse IV, Sep 13 2016
  • Sage
    [lucas_number1(n,12,28) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = 12*a(n-1) - 28*a(n-2) for n > 1. - Philippe Deléham, Jan 12 2009
a(n) = ( (6 + 2*sqrt(2))^n - (6 - 2*sqrt(2))^n )/(4*sqrt(2)).
G.f.: x/(1 - 12*x + 28*x^2). - Klaus Brockhaus, Jan 12 2009, corrected Oct 08 2009
E.g.f.: (1/(2*sqrt(2)))*exp(6*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 13 2016
a(n) =2^(n-1)*A081179(n). - R. J. Mathar, Feb 04 2021

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 12 2009
Edited by Klaus Brockhaus, Oct 08 2009
Offset corrected. - R. J. Mathar, Jun 19 2021

A330390 G.f.: (1 + 15*x) / (1 - 2*x - 17*x^2).

Original entry on oeis.org

1, 17, 51, 391, 1649, 9945, 47923, 264911, 1344513, 7192513, 37241747, 196756215, 1026622129, 5398099913, 28248776019, 148265250559, 776759693441, 4074028646385, 21352972081267, 111964431151079, 586929387683697, 3077254104935737, 16132307800494323
Offset: 0

Views

Author

Kyle MacLean Smith, Dec 13 2019

Keywords

Links

Crossrefs

Cf. A164544, A328606.

Programs

  • Mathematica
    CoefficientList[Series[(1+15x)/(1-2x-17x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,17},{1,17},30] (* Harvey P. Dale, Jul 31 2021 *)
  • PARI
    Vec((1 + 15*x) / (1 - 2*x - 17*x^2) + O(x^25)) \\ Colin Barker, Jan 25 2020

Formula

a(n) = 2*a(n-1) + 17*a(n-2) for n>1.
a(n)/a(n-1) ~ 1 + 3*sqrt(2).
a(n) = ((1 - 3*sqrt(2))^n*(-16+3*sqrt(2)) + (1+3*sqrt(2))^n*(16 + 3*sqrt(2))) / (6*sqrt(2)). - Colin Barker, Dec 14 2019
Showing 1-4 of 4 results.

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