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 31-40 of 45 results. Next

A165511 a(0)=1, a(1)=10, a(n) = 90*a(n-2) - a(n-1).

Original entry on oeis.org

1, 10, 80, 820, 6380, 67420, 506780, 5561020, 40049180, 460442620, 3143983580, 38295852220, 244662669980, 3201964029820, 18817676268380, 269359086415420, 1424231777738780, 22818085999649020, 105362773996841180
Offset: 0

Views

Author

Philippe Deléham, Sep 21 2009

Keywords

Comments

a(n)/a(n-1) tends to -10.
First entry < 0: a(30) = -8009307078719785774426912420.

Links

Programs

  • Magma
    [(20*9^n-(-10)^n)/19: n in [0..50]]; // G. C. Greubel, Oct 21 2018
  • Mathematica
    LinearRecurrence[{-1,90},{1,10},20] (* or *) CoefficientList[Series[ (1+11x)/(1+x-90x^2),{x,0,20}],x]  (* Harvey P. Dale, Apr 30 2011 *)
  • PARI
    vector(50, n, n--; (20*9^n-(-10)^n)/19) \\ G. C. Greubel, Oct 21 2018

Formula

G.f.: (1+11*x)/(1+x-90*x^2).
a(n) = Sum_{k=0..n} A112555(n,k)*9^k.
a(n) = (20*9^n-(-10)^n)/19. - Klaus Brockhaus, Sep 26 2009
E.g.f.: (20*exp(9*x) - exp(-10*x))/19. - G. C. Greubel, Oct 21 2018

A165553 a(n) = (3/2)*(1+(-3)^(n-1)).

Original entry on oeis.org

1, 3, -3, 15, -39, 123, -363, 1095, -3279, 9843, -29523, 88575, -265719, 797163, -2391483, 7174455, -21523359, 64570083, -193710243, 581130735, -1743392199, 5230176603, -15690529803, 47071589415, -141214768239
Offset: 0

Views

Author

Philippe Deléham, Sep 21 2009

Keywords

Comments

a(n)/a(n-1) tends to -3.

Links

Crossrefs

Cf. A046717, A084247, A112555.

Programs

  • Mathematica
    3/2*(1 + (-3)^(Range[0, 29] - 1)) (* or *)
LinearRecurrence[{-2, 3}, {1, 3}, 30] (* Paolo Xausa, Apr 22 2024 *)

Formula

a(0)=1, a(1)=3, a(n)=3*a(n-2)-2*a(n-1).
G.f.: (1+5x)/(1+2x-3x^2).
a(n)= Sum_{k=0..n} A112555(n,k)*2^(n-k).

A165622 a(n)=(-4)*a(n-1)+8 with a(0)=1.

Original entry on oeis.org

1, 4, -8, 40, -152, 616, -2456, 9832, -39320, 157288, -629144, 2516584, -10066328, 40265320, -161061272, 644245096, -2576980376, 10307921512, -41231686040, 164926744168, -659706976664, 2638827906664, -10555311626648, 42221246506600, -168884986026392
Offset: 0

Views

Author

Philippe Deléham, Sep 22 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = -4. [From Klaus Brockhaus, Sep 26 2009]

Crossrefs

Cf. A122117,

Programs

Formula

G.f.: (1+7x)/(1+3x-4x^2). a(n)= Sum_{k, 0<=k<=n} A112555(n,k)*3^(n-k). a(0)=1, a(1)=4, a(n)=4*a(n-2)-3*a(n-1).
a(n) = (8-3*(-4)^n)/5. [From Klaus Brockhaus and Paolo P. Lava, Sep 26 2009]

Extensions

Corrected and extended by Harvey P. Dale, Jul 11 2015

A165746 a(n) = 3 - 2*3^n.

Original entry on oeis.org

1, -3, -15, -51, -159, -483, -1455, -4371, -13119, -39363, -118095, -354291, -1062879, -3188643, -9565935, -28697811, -86093439, -258280323, -774840975, -2324522931, -6973568799, -20920706403, -62762119215, -188286357651
Offset: 0

Views

Author

Philippe Deléham, Sep 26 2009

Keywords

Links

Crossrefs

Cf. A008776, A112555.

Programs

  • Magma
    [3-2*3^n : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011
  • Mathematica
    Table[3-2*3^n,{n,0,30}] (* or *) LinearRecurrence[{4,-3},{1,-3},30] (* Harvey P. Dale, Aug 29 2024 *)

Formula

G.f.: (1-7*x)/(1 - 4*x + 3*x^2).
a(n) = 3 - A008776(n).
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1; a(0)=1, a(1)=-3.
a(n) = Sum_{k=0..n} A112555(n,k)*(-4)^(n-k).
a(n) = 3*a(n-1) - 6. - Vincenzo Librandi, Nov 03 2011

A165748 a(n) = (8/9)*(2+7*(-8)^(n-1)).

Original entry on oeis.org

1, 8, -48, 400, -3184, 25488, -203888, 1631120, -13048944, 104391568, -835132528, 6681060240, -53448481904, 427587855248, -3420702841968, 27365622735760, -218924981886064, 1751399855088528, -14011198840708208
Offset: 0

Views

Author

Philippe Deléham, Sep 26 2009

Keywords

Links

Programs

  • Mathematica
    Table[(8/9)*(2 + 7*(-8)^(n - 1)), {n, 0, 100}] or
LinearRecurrence[{-7,8}, {1,8}, 100] (* G. C. Greubel, Apr 07 2016 *)
  • PARI
    x='x+O('x^99); Vec((1+15*x)/(1+7*x-8*x^2)) \\ Altug Alkan, Apr 07 2016

Formula

a(n) = (-8)*a(n-1) + 16 for n>=1, with a(0) = 1.
a(n) = 8*a(n-2) - 7*a(n-1), a(0)=1, a(1)=8.
G.f.: (1+15x)/(1+7x-8x^2).
a(n) = Sum_{0<=k<=n} A112555(n,k)*7^(n-k).
From G. C. Greubel, Apr 07 2016: (Start)
a(n) = -7*a(n-1) + 8*a(n-2).
E.g.f.: (1/9)*(16*exp(x) - 7*exp(-8*x)). (End)

A165749 a(n) = (9/5)*(1+4*(-9)^(n-1)).

Original entry on oeis.org

1, 9, -63, 585, -5247, 47241, -425151, 3826377, -34437375, 309936393, -2789427519, 25104847689, -225943629183, 2033492662665, -18301433963967, 164712905675721, -1482416151081471, 13341745359733257
Offset: 0

Views

Author

Philippe Deléham, Sep 26 2009

Keywords

Links

Programs

  • Mathematica
    Table[9/5 (1+4(-9)^(n-1)),{n,0,20}] (* or *) LinearRecurrence[{-8,9},{1,9},20] (* Harvey P. Dale, Nov 24 2011 *)
  • PARI
    x='x+O('x^99); Vec((1+17*x)/(1+8*x-9*x^2)) \\ Altug Alkan, Apr 07 2016
  • PARI
    a(n) = (9-4*(-9)^n)/5 \\ Charles R Greathouse IV, Apr 07 2016

Formula

a(n) = -8*a(n-1) + 9*a(n-2), a(0)=1, a(1)=9.
a(n) = (-9)*a(n-1) + 18 for n>=1, with a(0)=1.
G.f.: (1+17x)/(1+8x-9x^2).
a(n) = Sum_{0<=k<=n} A112555(n,k)*8^(n-k).
E.g.f.: (1/5)*(9*exp(x) - 4*exp(-9*x)). - G. C. Greubel, Apr 07 2016

A165750 a(n) = (10/11)*(2+9*(-10)^(n-1)).

Original entry on oeis.org

1, 10, -80, 820, -8180, 81820, -818180, 8181820, -81818180, 818181820, -8181818180, 81818181820, -818181818180, 8181818181820, -81818181818180, 818181818181820, -8181818181818180, 81818181818181820
Offset: 0

Views

Author

Philippe Deléham, Sep 26 2009

Keywords

Links

Programs

  • Mathematica
    Table[(10/11)*(2 + 9*(-10)^(n - 1)), {n, 0, 50}] (* or *)
LinearRecurrence[{-9,10}, {1,10}, 50] (* G. C. Greubel, Apr 07 2016 *)
  • PARI
    x='x+O('x^99); Vec((1+19*x)/(1+9*x-10*x^2)) \\ Altug Alkan, Apr 07 2016

Formula

a(n) = -9*a(n-1) + 10*a(n-2), a(0)=1, a(1)=10.
a(n) = (-10)*a(n-1) + 20 for n>=1, a(0)=1.
G.f.: (1+19x)/(1+9x-10x^2).
a(n)= Sum_{0<=k<=n} A112555(n,k)*9^(n-k).
E.g.f.: (1/11)*(20*exp(x) - 9*exp(-10*x)). - G. C. Greubel, Apr 07 2016

A165752 a(n) = (8-5*4^n)/3.

Original entry on oeis.org

1, -4, -24, -104, -424, -1704, -6824, -27304, -109224, -436904, -1747624, -6990504, -27962024, -111848104, -447392424, -1789569704, -7158278824, -28633115304, -114532461224, -458129844904, -1832519379624, -7330077518504
Offset: 0

Views

Author

Philippe Deléham, Sep 26 2009

Keywords

Links

Programs

  • Mathematica
    (8-5*4^Range[0,30])/3 (* or *) LinearRecurrence[{5,-4},{1,-4},30] (* Harvey P. Dale, Jan 10 2016 *)
  • PARI
    x='x+O('x^99); Vec((1-9*x)/(1-5*x+4*x^2)) \\ Altug Alkan, Apr 07 2016

Formula

a(n) = 4*a(n-1) - 8, a(0)=1.
a(n) = 5*a(n-1) - 4*a(n-2), a(0)=1, a(1)=-4, for n>1.
G.f.: (1-9x)/(1-5x+4x^2).
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-5)^(n-k).
a(n) = (-4)*A020989(n-1).
E.g.f.: (1/3)*(8*exp(x) - 5*exp(4*x)). - G. C. Greubel, Apr 07 2016

A165755 a(n) = (5-3*5^n)/2.

Original entry on oeis.org

1, -5, -35, -185, -935, -4685, -23435, -117185, -585935, -2929685, -14648435, -73242185, -366210935, -1831054685, -9155273435, -45776367185, -228881835935, -1144409179685, -5722045898435, -28610229492185
Offset: 0

Views

Author

Philippe Deléham, Sep 26 2009

Keywords

Links

Programs

  • Mathematica
    (5-3*5^Range[0,20])/2 (* or *) LinearRecurrence[{6,-5},{1,-5},20] (* Harvey P. Dale, Apr 18 2013 *)
  • PARI
    x='x+O('x^99); Vec((1-11*x)/(1-6*x+5*x^2)) \\ Altug Alkan, Apr 07 2016

Formula

a(n) = 5*a(n-1) - 10, a(0)=1.
a(n) = 6*a(n-1)-5*a(n-2), a(0)= 1, a(1)= -5, for n>1.
G.f.: (1-11x)/(1-6x+5x^2).
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-6)^(n-k).
a(n) = (-5)*A057651(n-1).
E.g.f.: (1/2)*(5*exp(x) - 3*exp(5*x)). - G. C. Greubel, Apr 07 2016

A165758 a(n) = (12-7*6^n)/5.

Original entry on oeis.org

1, -6, -48, -300, -1812, -10884, -65316, -391908, -2351460, -14108772, -84652644, -507915876, -3047495268, -18284971620, -109709829732, -658258978404, -3949553870436, -23697323222628, -142183939335780, -853103636014692
Offset: 0

Views

Author

Philippe Deléham, Sep 26 2009

Keywords

Links

Programs

  • Mathematica
    (12-7*6^Range[0,20])/5 (* or *) LinearRecurrence[{7,-6},{1,-6},20] (* Harvey P. Dale, Apr 30 2012 *)

Formula

a(n) = 6*a(n-1) - 12, a(0)=1.
a(n) = 7*a(n-1)-6*a(n-2), a(0)=1, a(1)= -6.
G.f.: (1-13x)/(1-7x+6x^2).
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-7)^(n-k).
a(n) = (-6)*A061801(n-1).
E.g.f.: (1/5)*(12*exp(x) - 7*exp(6*x)). - G. C. Greubel, Apr 07 2016

Extensions

Corrected by D. S. McNeil, Aug 20 2010
Previous Showing 31-40 of 45 results. Next

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

Content is available under The OEIS End-User License Agreement.