A021001 -id:A021001 - cp's OEIS frontend

A183603 T(n,k)=1/6 the number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock containing all three values.

6, 27, 27, 120, 204, 120, 534, 1479, 1479, 534, 2376, 10797, 17174, 10797, 2376, 10572, 78729, 201770, 201770, 78729, 10572, 47040, 574185, 2366412, 3831324, 2366412, 574185, 47040, 209304, 4187499, 27768032, 72592860, 72592860, 27768032, 4187499

Offset: 1

R. H. Hardin Jan 05 2011

Table starts
.......6.........27..........120.............534..............2376
......27........204.........1479...........10797.............78729
.....120.......1479........17174..........201770...........2366412
.....534......10797.......201770.........3831324..........72592860
....2376......78729......2366412........72592860........2220901926
...10572.....574185.....27768032......1376627608.......68026656422
...47040....4187499....325834456.....26108240340.....2083981368760
..209304...30539367...3823553752....495215814566....63854542807704
..931296..222722937..44868561272...9393691937690..1956721189681174
.4143792.1624313781.526526214848.178194216929170.59964037003007332

			Some solutions with a(1,1)=0 for 4X3
..0..1..0....0..0..1....0..2..0....0..1..1....0..1..0....0..1..0....0..0..0
..2..2..0....1..2..1....0..1..0....2..0..2....2..2..1....1..2..1....1..2..1
..1..0..1....2..0..2....1..2..0....1..1..2....1..0..2....0..2..0....1..0..0
..2..2..1....1..1..0....0..2..1....2..0..1....2..2..1....2..1..1....0..2..1

R. H. Hardin, Table of n, a(n) for n = 1..179

Column 1 is 3*A021001(n-1)

A107979 a(n) = 4a(n-1) + 2a(n-2) for n>1, with a(0)=2, a(1)=9.

Original entry on oeis.org

2, 9, 40, 178, 792, 3524, 15680, 69768, 310432, 1381264, 6145920, 27346208, 121676672, 541399104, 2408949760, 10718597248, 47692288512, 212206348544, 944209971200, 4201252581888, 18693430269952, 83176226243584

Offset: 0

Emeric Deutsch, Jun 12 2005

Kekulé numbers for certain benzenoids.

This is the case r=2 of the generalized Pell numbers as defined in Bród. - Michel Marcus, Oct 28 2020

			G.f. = 2 + 9*x + 40*x^2 + 178*x^3 + 792*x^4 + 3524*x^5 + 15680*x^6 + 69768*x^7 + ...

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 78).

Dorota Bród, On a New One Parameter Generalization of Pell Numbers, Annales Mathematicae Silesianae 33 (2019), 66-76.
Index entries for linear recurrences with constant coefficients, signature (4,2).

Cf. A021001. - R. J. Mathar, Aug 24 2008

Cf. A000129, A090017.

a[0]:=2: a[1]:=9: for n from 2 to 26 do a[n]:=4*a[n-1]+2*a[n-2] od: seq(a[n],n=0..26);

LinearRecurrence[{4,2},{2,9},30] (* or *) CoefficientList[Series[(-x-2)/(2x^2+4x-1),{x,0,30}],x] (* Harvey P. Dale, Jun 21 2011 *)
a[ n_] := With[{m = n + 2}, If[ m < 0, -(-2)^m, 1] SeriesCoefficient[ x / (2 - 8 x - 4 x^2), {x, 0, Abs@m}]]; (* Michael Somos, May 23 2014 *)
a[ n_] := With[{m = n + 2, r = Sqrt[6]}, If[ m < 0, -(-2)^m, Sign@m] Expand[(2 + r)^(Abs@m) / (2 r)][[1]]]; (* Michael Somos, May 23 2014 *)

{a(n) = my(m = n+2); if( m<0, -(-2)^m, 1) * polcoeff( x / (2 - 8*x - 4*x^2) + x * O(x^abs(m)), abs(m))}; /* Michael Somos, May 23 2014 */

{a(n) = my(r = 2 + quadgen(24)); imag( (1 + 2*r) * r^n)}; /* Michael Somos, May 23 2014 */

a(n)=([0,1; 2,4]^n*[2;9])[1,1] \\ Charles R Greathouse IV, Feb 07 2017

From R. J. Mathar, Aug 24 2008: (Start)
O.g.f.: (2+x)/(1-4x-2x^2).
a(n) = 2*A090017(n) + A090017(n-1). (End)
a(n) = 1/12*((sqrt(6)-3)(-(2-sqrt(6))^n) + (3+sqrt(6))(2+sqrt(6))^n). - Harvey P. Dale, Jun 21 2011
a(n) = A000129(n+2) + Sum_{k=1..n} A000129(k+1)*a(n-k). - Ralf Stephan, May 23 2014

A268409 a(n) = 4a(n - 1) + 2a(n - 2) for n>1, a(0)=3, a(1)=5.

Original entry on oeis.org

3, 5, 26, 114, 508, 2260, 10056, 44744, 199088, 885840, 3941536, 17537824, 78034368, 347213120, 1544921216, 6874111104, 30586286848, 136093369600, 605546052096, 2694370947584, 11988575894528, 53343045473280, 237349333682176, 1056083425675264

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m, is (k - (k*r - m)*x)/(1 - r*x - s*x^2). This recurrence gives the closed form b(n) = (2^(-n - 1)*((k*r - 2*m)*(r - sqrt(r^2 + 4*s))^n + (2*m - k*r)*(sqrt(r^2 + 4*s) + r)^n + k*sqrt(r^2 + 4*s)*(r - sqrt(r^2 + 4*s))^n + k*sqrt(r^2 + 4*s)*(sqrt(r^2 + 4*s) + r)^n))/sqrt(r^2 + 4*s).

Index entries for linear recurrences with constant coefficients, signature (4,2).

Cf. A021001, A084059, A090017, A107979, A164549, A176213, A189741.

[n le 2 select 2*n+1 else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == 4 a[n - 1] + 2 a[n - 2]}, a, {n, 23}]
LinearRecurrence[{4, 2}, {3, 5}, 24]
Table[((18 + Sqrt[6]) (2 - Sqrt[6])^n - (Sqrt[6] - 18) (2 + Sqrt[6])^n)/12, {n, 0, 23}]

Vec((3 - 7*x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016

G.f.: (3 - 7*x)/(1 - 4*x - 2*x^2).
a(n) = ((18 + sqrt(6))*(2 - sqrt(6))^n - (sqrt(6) - 18)*(2 + sqrt(6))^n)/12.
Lim_{n -> infinity} a(n + 1)/a(n) = 2 + sqrt(6) = A176213.
a(n) = 3*A090017(n+1) -7*A090017(n). - R. J. Mathar, Mar 12 2017

cp's OEIS Frontend

A183603 T(n,k)=1/6 the number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock containing all three values.

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A107979 a(n) = 4a(n-1) + 2a(n-2) for n>1, with a(0)=2, a(1)=9.

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A268409 a(n) = 4a(n - 1) + 2a(n - 2) for n>1, a(0)=3, a(1)=5.

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cp's OEIS Frontend

A183603 T(n,k)=1/6 the number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock containing all three values.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A107979 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, with a(0)=2, a(1)=9.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A268409 a(n) = 4*a(n - 1) + 2*a(n - 2) for n>1, a(0)=3, a(1)=5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A107979 a(n) = 4a(n-1) + 2a(n-2) for n>1, with a(0)=2, a(1)=9.

A268409 a(n) = 4a(n - 1) + 2a(n - 2) for n>1, a(0)=3, a(1)=5.