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.

A084241 a(n) = -5*a(n-1)-4*a(n-2) with n>1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, -5, 21, -85, 341, -1365, 5461, -21845, 87381, -349525, 1398101, -5592405, 22369621, -89478485, 357913941, -1431655765, 5726623061, -22906492245, 91625968981, -366503875925, 1466015503701, -5864062014805, 23456248059221, -93824992236885
Offset: 0

Views

Author

Paul Barry, May 21 2003

Keywords

Comments

Binomial transform is (0,1,-3,9,-27,...).

Links

Crossrefs

Apart from signs, identical to A002450.
Cf. A084240.

Programs

  • Mathematica
    LinearRecurrence[{-5,-4},{0,1},40] (* Harvey P. Dale, Dec 20 2014 *)

Formula

a(n) = ((-1)^n-(-4)^n)/3.
a(n) = Sum_{k=1..n} (-1)^(n+k)*binomial(n, k)*(-3)^(k-1).
G.f.: x/((1+x)*(1+4*x)).
E.g.f.: (exp(-x)-exp(-4x))/3.
abs(a(n)) = A002450(n) = A001045(2n).

A166978 a(n) = 4*( 1-(-1)^n) -2^n.

Original entry on oeis.org

-1, 6, -4, 0, -16, -24, -64, -120, -256, -504, -1024, -2040, -4096, -8184, -16384, -32760, -65536, -131064, -262144, -524280, -1048576, -2097144, -4194304, -8388600, -16777216, -33554424, -67108864, -134217720, -268435456, -536870904, -1073741824, -2147483640
Offset: 0

Views

Author

Paul Curtz, Oct 26 2009

Keywords

Links

Programs

  • Magma
    [4*( 1-(-1)^n) -2^n: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011
  • Mathematica
    LinearRecurrence[{2,1,-2}, {-1, 6, -4}, 50] (* or *) Table[4*(1-(-1)^n) - 2^n, {n,0,25}] (* G. C. Greubel, May 30 2016 *)

Formula

a(n) = A166956(n+1)-3*A166956(n).
a(2n) = -A000302(n). a(2n+1) = 6*(-1)^n*A084240(n).
a(n+1) - 2*a(n) = 4*( 3*(-1)^n-1) = 8 *(-1)^n*A000034(n).
G.f.: -(5*x-1)*(3*x-1) / ( (x-1)*(2*x-1)*(1+x) ). - R. J. Mathar, Jul 01 2011
E.g.f.: 8*sinh(x) - exp(2*x). - G. C. Greubel, May 30 2016

A167193 a(n) = (1/3)*(1 - (-2)^n + 3*(-1)^n ) = (-1)^(n+1)*A167030(n).

Original entry on oeis.org

1, 0, 0, 2, -4, 10, -20, 42, -84, 170, -340, 682, -1364, 2730, -5460, 10922, -21844, 43690, -87380, 174762, -349524, 699050, -1398100, 2796202, -5592404, 11184810, -22369620, 44739242, -89478484, 178956970, -357913940, 715827882, -1431655764, 2863311530, -5726623060, 11453246122, -22906492244, 45812984490
Offset: 0

Views

Author

Paul Curtz, Oct 30 2009

Keywords

Comments

This is the inverse binomial transform of 1, 1, 1, 3, 5, 11,.. (continued as in A001045 and conjectured to be equal to A152046).
Any sequence (like this one) which obeys a(n)= -2a(n-1)+a(n-2)+2a(n-3) also obeys a(n)=5a(n-2)-4a(n-4), proved by telescoping; see A101622.

Links

Programs

  • Magma
    [( 1-(-1)^n*2^n)/3+(-1)^n: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011
  • Mathematica
    LinearRecurrence[{-2,1,2},{1,0,0}, 25] (* or *) Table[(1/3)*(1 + 3*(-1)^n - (-2)^n), {n,0,25}] (* G. C. Greubel, Jun 04 2016 *)

Formula

a(2n) = (-1)^n* A084240(n). a(2n+1) = A020988(n).
G.f.: ( -1 - 2*x + x^2 ) / ( (x-1)*(1+2*x)*(1+x) ).
a(n) = -a(n-1) + 2*a(n-2) - 2*(-1)^n.
a(n) = -2*a(n-1) + a(n-2) + 2*a(n-3).
E.g.f.: (1/3)*(exp(x) + 3*exp(-x) - exp(-2*x)). - G. C. Greubel, Jun 04 2016
Showing 1-3 of 3 results.

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