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.

Previous Showing 11-16 of 16 results.

A163609 a(n) = ((5 + 2*sqrt(2))*(3 + sqrt(2))^n + (5 - 2*sqrt(2))*(3 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 19, 79, 341, 1493, 6571, 28975, 127853, 564293, 2490787, 10994671, 48532517, 214232405, 945666811, 4174374031, 18426576509, 81338840837, 359047009459, 1584910170895, 6996131959157, 30882420558677, 136321599637963
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

Comments

Binomial transform of A163608. Third binomial transform of A163888. Inverse binomial transform of A163610.

Links

Crossrefs

Cf. A163608, A163610, A163888.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(3+r)^n+(5-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009
  • Mathematica
    LinearRecurrence[{6, -7}, {5, 19}, 50] (* G. C. Greubel, Jul 29 2017 *)
  • PARI
    x='x+O('x^50); Vec((5-11*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 29 2017

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 5, a(1) = 19.
G.f.: (5-11*x)/(1-6*x+7*x^2).
a(n) = 5*A081179(n+1) - 11*A081179(n). - R. J. Mathar, Nov 08 2013
E.g.f.: exp(3*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A164072 a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 42, 238, 1316, 7196, 39144, 212408, 1151248, 6236272, 33772704, 182873824, 990172736, 5361148352, 29026768512, 157158071168, 850889810176, 4606905485056, 24942786537984, 135045615513088, 731165912572928
Offset: 0

Views

Author

Klaus Brockhaus, Aug 09 2009

Keywords

Comments

Binomial transform of A081179 without initial 0. Inverse binomial transform of A164031.

Links

Crossrefs

Cf. A081179, A164031.

Programs

  • Magma
    [ n le 2 select 6*n-5 else 8*Self(n-1)-14*Self(n-2): n in [1..21] ];
  • Mathematica
    CoefficientList[Series[(1 - x)/(1 - 8*x + 14*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{8,-14}, {1,7}, 50] (* G. C. Greubel, Sep 09 2017 *)
  • PARI
    x='x+O('x^50); Vec((1-x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Sep 09 2017

Formula

a(n) = ((2+3*sqrt(2))*(4+sqrt(2))^n + (2-3*sqrt(2))*(4-sqrt(2))^n)/4.
G.f.: (1-x)/(1-8*x+14*x^2).
E.g.f.: (cosh(sqrt(2)*x) + (3*sqrt(2)/2)*sinh(sqrt(2)*x))*exp(4*x). - G. C. Greubel, Sep 09 2017

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

A162270 a(n) = ((5+sqrt(2))*(3+sqrt(2))^n + (5-sqrt(2))*(3-sqrt(2))^n)/2.

Original entry on oeis.org

5, 17, 67, 283, 1229, 5393, 23755, 104779, 462389, 2040881, 9008563, 39765211, 175531325, 774831473, 3420269563, 15097797067, 66644895461, 294184793297, 1298594491555, 5732273396251, 25303478936621, 111694959845969
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

Comments

Third binomial transform of A162396.

Links

Crossrefs

Cf. A162396.

Programs

  • GAP
    a:=[5,17];; for n in [3..25] do a[n]:=6*a[n-1]-7*a[n-2]; od; a; # Muniru A Asiru, Sep 28 2018
  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+r)*(3+r)^n+(5-r)*(3-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
  • Magma
    I:=[5,17]; [n le 2 select I[n] else 6*Self(n-1) - 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 28 2018
  • Maple
    seq(coeff(series((5-13*x)/(1-6*x+7*x^2),x,n+1), x, n), n = 0..25); # Muniru A Asiru, Sep 28 2018
  • Mathematica
    LinearRecurrence[{6,-7},{5,17},30] (* Harvey P. Dale, Jun 04 2016 *)
  • PARI
    x='x+O('x^30); Vec((5-13*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 28 2018

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 5, a(1) = 17.
G.f.: (5-13*x)/(1-6*x+7*x^2).
a(n) = 5*A081179(n+1)-13*A081179(n). - R. J. Mathar, Feb 04 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009

A163348 a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 35, 161, 721, 3199, 14147, 62489, 275905, 1218007, 5376707, 23734193, 104768209, 462469903, 2041441955, 9011362409, 39778080769, 175588947751, 775087121123, 3421400092481, 15102790707025, 66666943594783
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Keywords

Comments

Binomial transform of A111566. Third binomial transform of A143095. Inverse binomial transform of A081180 without initial 0.

Links

Crossrefs

Cf. A111566, A143095 (1,4,2,8,4,16,...), A081180.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+2*r)*(3+r)^n+(1-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009
  • Mathematica
    LinearRecurrence[{6, -7}, {1, 7}, 50] (* G. C. Greubel, Dec 19 2016 *)
  • PARI
    Vec((1+x)/(1-6*x+7*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
a(n) = ((1+2*sqrt(2))*(3+sqrt(2))^n + (1-2*sqrt(2))*(3-sqrt(2))^n)/2.
G.f.: (1+x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 19 2016
a(n) = A081179(n)+A081179(n+1). - R. J. Mathar, Feb 04 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009

A171700 Triangle T : T(n,k)= A007318(n,k)*a(n-k) with a(0)=0 and a(n)= A077957(n-1) for n>0.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 2, 0, 3, 0, 0, 8, 0, 4, 0, 4, 0, 20, 0, 5, 0, 0, 24, 0, 40, 0, 6, 0, 8, 0, 84, 0, 70, 0, 7, 0, 0, 64, 0, 224, 0, 112, 0, 8, 0, 16, 0, 288, 0, 504, 0, 168, 0, 9, 0, 0, 160, 0, 960, 0, 1008, 0, 240, 0, 10, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 15 2009

Keywords

Comments

Diagonal sums : A001353(n+1) alternating with zeros.

Examples

			Triangle begins : 0 ; 1,0 ; 0,2,0 ; 2,0,3,0 ; 0,8,0,4,0 ; 4,0,20,0,5,0 ; ...

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n-1), A000129(n), A007070(n-1), A081179(n), A081180(n), A081182(n), A081183(n), A081184(n), A081185(n), A153593(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Previous Showing 11-16 of 16 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.