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.

A164607 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

Original entry on oeis.org

1, 10, 44, 216, 1040, 5024, 24256, 117120, 565504, 2730496, 13184000, 63657984, 307367936, 1484103680, 7165886464, 34599960576, 167063388160, 806653394944, 3894867132416, 18806082109440, 90803796967424, 438439516307456
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

Comments

Binomial transform of A083100. Second binomial transform of A164683. Inverse binomial transform of A054490.

Links

Crossrefs

Cf. A083100, A164683, A054490.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+4*r)*(2+2*r)^n+(2-4*r)*(2-2*r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 22 2009
  • Mathematica
    LinearRecurrence[{4,4},{1,10},40] (* Harvey P. Dale, Jun 28 2011 *)
  • PARI
    x='x+O('x^50); Vec((1+6*x)/(1-4*x-4*x^2)) \\ G. C. Greubel, Aug 10 2017

Formula

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 10.
a(n) = ((2+4*sqrt(2))*(2+2*sqrt(2))^n + (2-4*sqrt(2))*(2-2*sqrt(2))^n)/4.
G.f.: (1+6*x)/(1-4*x-4*x^2).
G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - (1-x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
E.g.f.: exp(2*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 22 2009

A164608 Expansion of (1+4*x)/(1-8*x+8*x^2).

Original entry on oeis.org

1, 12, 88, 608, 4160, 28416, 194048, 1325056, 9048064, 61784064, 421888000, 2880831488, 19671547904, 134325731328, 917233467392, 6263261888512, 42768227368960, 292039723843584, 1994171971796992, 13617057983627264
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

Comments

Binomial transform of A054490. Fourth binomial transform of A164683. Inverse binomial transform of A164609.

Links

Crossrefs

Cf. A054490, A164683, A164609.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+4*r)*(4+2*r)^n+(2-4*r)*(4-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 22 2009
  • Mathematica
    LinearRecurrence[{8,-8}, {1,12,88}, 50] (* G. C. Greubel, Aug 10 2017 *)
  • PARI
    x='x+O('x^50); Vec((1+4*x)/(1-8*x+8*x^2)) \\ G. C. Greubel, Aug 10 2017

Formula

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 12.
a(n) = A057084(n) + 4*A057084(n-1).
a(n) = ((2+4*sqrt(2))*(4+2*sqrt(2))^n + (2-4*sqrt(2))*(4-2*sqrt(2))^n)/4.
E.g.f.: exp(4*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 22 2009

A164609 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 13.

Original entry on oeis.org

1, 13, 113, 909, 7169, 56237, 440497, 3448941, 27000961, 211377613, 1654759793, 12954178509, 101410868609, 793887651437, 6214891748017, 48652827405741, 380875114341121, 2981653077513613, 23341653831337073, 182728435995639309
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

Comments

Binomial transform of A164608. Fifth binomial transform of A164683.

Links

Crossrefs

Cf. A164608, A164683.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+4*r)*(5+2*r)^n+(2-4*r)*(5-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 22 2009
  • Mathematica
    LinearRecurrence[{10,-17},{1,13},20] (* Harvey P. Dale, Nov 05 2014 *)
  • PARI
    x='x+O('x^50); Vec((1+3*x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 10 2017

Formula

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 13.
a(n) = ((2+4*sqrt(2))*(5+2*sqrt(2))^n + (2-4*sqrt(2))*(5-2*sqrt(2))^n)/4.
G.f.: (1+3*x)/(1-10*x+17*x^2).
E.g.f.: exp(5*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 22 2009

A154348 a(n) = 16*a(n-1) - 56*a(n-2) for n>1, with a(0)=1, a(1)=16.

Original entry on oeis.org

1, 16, 200, 2304, 25664, 281600, 3068416, 33325056, 361369600, 3915710464, 42414669824, 459354931200, 4974457389056, 53867442077696, 583309459456000, 6316374594945024, 68396663789584384, 740629643316428800
Offset: 0

Views

Author

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

Keywords

Comments

Third binomial transform of A164609, fourth binomial transform of A164608, fifth binomial transform of A054490, sixth binomial transform of A164607, seventh binomial transform of A083100, eighth binomial transform of A164683.
lim_{n -> infinity} a(n)/a(n-1) = 8 + 2*sqrt(2) = 10.8284271247....

Links

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A164609, A164608, A054490, A164607, A083100, A164683.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((8+2*r)^n-(8-2*r)^n)/(4*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009
  • Mathematica
    Join[{a=1,b=16},Table[c=16*b-56*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
LinearRecurrence[{16,-56},{1,16},30] (* Harvey P. Dale, Aug 31 2016 *)

Formula

a(n) = 16*a(n-1) - 56*a(n-2) for n>1. - Philippe Deléham, Jan 12 2009
a(n) = ( (8 + 2*sqrt(2))^n - (8 - 2*sqrt(2))^n )/(4*sqrt(2)).
G.f.: 1/(1 - 16*x + 56*x^2). - Klaus Brockhaus, Jan 12 2009; corrected Oct 08 2009
E.g.f.: (1/(2*sqrt(2)))*exp(8*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 13 2016

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
Showing 1-4 of 4 results.

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