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 1-5 of 5 results.

A140966 a(n) = (5 + (-2)^n)/3.

Original entry on oeis.org

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063
Offset: 0

Views

Author

Paul Curtz, Jul 27 2008

Keywords

Comments

Inverse binomial transform of A048573.
This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).
The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.
Sequences with k=2 are A094554 and A094555.
Sequences with k=3 are A084175, A108924, and A139818.

Links

Crossrefs

Cf. A048573.
Cf. A097163, A135520, A136326, A136336, A137208.
Cf. A094554, A094555.
Cf. A084175, A108924, A139818.
Cf. A000079, A001045, A078008, A083582, A083584, A163834.

Programs

Formula

a(n) = -a(n-1) + 2*a(n-2).
G.f.: (2+3*x)/((1-x)*(1+2*x)).
a(n+1) - a(n) = (-1)^(n+1)*A000079(n).
a(n+3) = (-1)^n*A083582(n).
a(n+1) - 2*a(n) = -a(n+2).
a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).
a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014
E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Extensions

Definition simplified by R. J. Mathar, Sep 11 2009

A214101 T(n,k)=Number of 0..2 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 1, 3, 3, 2, 9, 5, 19, 4, 27, 11, 30, 121, 8, 81, 21, 143, 180, 771, 16, 243, 43, 322, 2041, 1080, 4913, 32, 729, 85, 1179, 5068, 29540, 6480, 31307, 64, 2187, 171, 3110, 37441, 79968, 428383, 38880, 199497, 128, 6561, 341, 10183, 121588, 1241355, 1262128
Offset: 1

Views

Author

R. H. Hardin Jul 04 2012

Keywords

Comments

Table starts
..1..1....3....5.....11......21.......43........85........171.........341
..3..2...19...30....143.....322.....1179......3110......10183.......28842
..9..4..121..180...2041....5068....37441....121588.....722009.....2720828
.27..8..771.1080..29540...79968..1241355...4807928...54733587...263068168
.81.16.4913.6480.428383.1262128.41634729.190532944.4254090231.25595530224

Examples

			Some solutions for n=4 k=1
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..0....2..0....1..0....1..2....1..2....1..2....1..2....2..0....1..0....1..2
..0..1....1..2....0..1....0..1....2..0....2..0....2..0....0..2....2..1....0..1
..1..2....2..0....2..0....2..0....0..2....1..2....0..1....1..0....1..2....1..0

Links

Crossrefs

Column 3 is A138977
Column 4 is A052934
Row 1 is A001045
Row 2 is A094554(n+1)

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 2*a(n-1)
k=3: a(n) = 7*a(n-1) -4*a(n-2)
k=4: a(n) = 6*a(n-1)
k=5: a(n) = 19*a(n-1) -71*a(n-2) +86*a(n-3) -24*a(n-4)
k=6: a(n) = 18*a(n-1) -36*a(n-2) +16*a(n-3)
k=7: a(n) = 54*a(n-1) -820*a(n-2) +4906*a(n-3) -11803*a(n-4) +11888*a(n-5) -4672*a(n-6) +576*a(n-7)
Empirical for row n:
n=1: a(k)=a(k-1)+2*a(k-2)
n=2: a(k)=2*a(k-1)+5*a(k-2)-6*a(k-3)
n=3: a(k)=3*a(k-1)+15*a(k-2)-33*a(k-3)-22*a(k-4)+38*a(k-5)+8*a(k-6)-8*a(k-7)
n=4: (order 11)
n=5: (order 29)
n=6: (order 40)

A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).

Original entry on oeis.org

0, 1, 1, 6, 11, 46, 111, 386, 1051, 3366, 9671, 29866, 87891, 267086, 794431, 2396946, 7163531, 21545206, 64526391, 193797626, 580955971, 1743741726, 5229477551, 15691927906, 47068793211, 141220360646, 423633119911, 1270955283786
Offset: 0

Views

Author

Paul Barry, May 11 2004

Keywords

Comments

Average of binomial and inverse binomial transforms of the Jacobsthal numbers A001045. - Paul Barry, Jan 04 2005

Links

Crossrefs

Cf. A001045, A094554, A094556.

Programs

  • Mathematica
    LinearRecurrence[{2, 5, -6}, {0, 1, 1, 6}, 30] (* Greg Dresden, Jun 19 2021 *)
  • PARI
    a(n) = if(n==0, 0, (3^n - (-2)^n + 1)/6) \\ Andrew Howroyd, Jun 15 2021

Formula

G.f.: x*(1 - x - x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 3^n/6 - (-2)^n/6 + 1/6 - 0^n/6.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*A001045(n-2k). - Paul Barry, Jan 04 2005
E.g.f.: exp(-2*x)*(exp(5*x) + exp(3*x) - exp(2*x) - 1)/6. - Stefano Spezia, Dec 26 2021

A094556 Number of walks of length n between opposite vertices on a triangular prism.

Original entry on oeis.org

0, 1, 0, 7, 8, 51, 100, 407, 1008, 3451, 9500, 30207, 87208, 268451, 791700, 2402407, 7152608, 21567051, 64482700, 193885007, 580781208, 1744091251, 5228778500, 15693326007, 47065997008, 141225953051, 423621935100, 1270977653407
Offset: 0

Views

Author

Paul Barry, May 11 2004

Keywords

Links

Crossrefs

Cf. A094554, A094555.

Programs

  • Mathematica
    LinearRecurrence[{2,5,-6},{0,1,0,7},30] (* or *) CoefficientList[ Series[ x (1-2x+2x^2)/((1-x)(1+2x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, Jul 13 2011 *)
  • PARI
    a(n) = if(n==0, 0, (3^n - 2*(-2)^n - 1)/6) \\ Andrew Howroyd, Jun 15 2021

Formula

G.f.: x*(1 - 2*x + 2*x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 3^n/6 - (-2)^n/3 - 1/6 + 0^n/3.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
E.g.f.: exp(-x)*(2+exp(3*x))*sinh(x)/3. - Stefano Spezia, Sep 26 2023

A344747 a(n) = (1/6)*(3^n + (-2)^n - 1).

Original entry on oeis.org

0, 2, 3, 16, 35, 132, 343, 1136, 3195, 10012, 29183, 89256, 264355, 799892, 2386023, 7185376, 21501515, 64613772, 193622863, 581305496, 1743042675, 5230875652, 15689131703, 47074385616, 141209175835, 423655489532, 1270910544543, 3812843481736, 11438306748995
Offset: 1

Views

Author

Ryan Brooks, Jun 14 2021

Keywords

Examples

			a(4) = (1/6)*(3^4 + (-2)^4 - 1) = (1/6)*(81+16-1) = 16.

Links

Crossrefs

Potentially related to A094554, A094555, and A094556 (which have the same recurrence).

Programs

  • Mathematica
    LinearRecurrence[{2, 5, -6}, {0, 2, 3}, 30] (* Greg Dresden, Jun 19 2021 *)

Formula

G.f.: x*(2 - x)/((1 - x)*(1 + 2*x)*(1 - 3*x)). - Andrew Howroyd, Jun 15 2021
a(n) = A094554(n-1) + 2*A094556(n-1). - Greg Dresden, Jun 19 2021
Showing 1-5 of 5 results.

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