A179606 -id:A179606 - cp's OEIS frontend

A072264 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=1.

1, 1, 8, 29, 127, 526, 2213, 9269, 38872, 162961, 683243, 2864534, 12009817, 50352121, 211105448, 885076949, 3710758087, 15557659006, 65226767453, 273468597389, 1146539629432, 4806961875241, 20153583772883, 84495560694854, 354254600948977, 1485241606321201

Offset: 0

Miklos Kristof, Jul 08 2002

			a(5)=3*a(4)+5*a(3): 127=3*29+5*8=87+40.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A015523, A072263, A152187, A179606, A197189.

a:=[1,1];; for n in [3..30] do a[n]:=3*a[n-1]+5*a[n-2]; od; a; # G. C. Greubel, Jan 14 2020

[n le 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..26]];  // Bruno Berselli, Oct 11 2011

seq(coeff(series((1-2*x)/(1-3*x-5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 14 2020

LinearRecurrence[{3,5},{1,1},30] (* Harvey P. Dale, Feb 17 2018 *)

my(x='x+O('x^30)); Vec((1-2*x)/(1-3*x-5*x^2)) \\ G. C. Greubel, Jan 14 2020

def A072264_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x)/(1-3*x-5*x^2) ).list()
A072264_list(30) # G. C. Greubel, Jan 14 2020

G.f.: (1-2*x)/(1-3*x-5*x^2). - Jaume Oliver Lafont, Mar 06 2009
G.f.: G(0)*(1-2*x)/(2-3*x), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
a(n) = 5^((n-1)/2)*( sqrt(5)*Fibonacci(n+1, 3/sqrt(5)) - 2*Fibonacci(n, 3/sqrt(5)) ). - G. C. Greubel, Jan 14 2020

Offset changed and more terms added by Bruno Berselli, Oct 11 2011

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).

Original entry on oeis.org

1, 4, 16, 60, 228, 864, 3276, 12420, 47088, 178524, 676836, 2566080, 9728748, 36884484, 139839696, 530172540, 2010036708, 7620627744, 28891993356, 109537863300, 415289569968, 1574482299804, 5969315609316, 22631393727360, 85802128010028, 325300565212164

Offset: 0

N. J. A. Sloane, Nov 20 2006

From Johannes W. Meijer, Aug 14 2010: (Start)
A berserker sequence, see A180141. For the corner squares 16 A[5] vectors with decimal values between 3 and 384 lead to this sequence. These vectors lead for the side squares to A180142 and for the central square to A155116.
This sequence belongs to a family of sequences with GF(x) = (1+x+k*x^2)/(1-3*x+(k-4)*x^2). Berserker sequences that are members of this family are 4*A055099(n) (k=2; with leading 1 added), A123620 (k=1; this sequence), A000302 (k=0), 4*A179606 (k=-1; with leading 1 added) and A180141 (k=-2). Some other members of this family are 4*A003688 (k=3; with leading 1 added), 4*A003946 (k=4; with leading 1 added), 4*A002878 (k=5; with leading 1 added) and 4*A033484 (k=6; with leading 1 added).
(End)
a(n) is the number of length n sequences on an alphabet of 4 letters that do not contain more than 2 consecutive equal letters. For example, a(3)=60 because we count all 4^3=64 words except: aaa, bbb, ccc, ddd. - Geoffrey Critzer, Mar 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 205
Index entries for linear recurrences with constant coefficients, signature (3,3).

Column 4 in A265584.

[1] cat [Round(((2^(1-n)*(-(3-Sqrt(21))^(1+n) + (3+Sqrt(21))^(1+n))))/(3*Sqrt(21))): n in [1..50]]; // G. C. Greubel, Oct 26 2017

nn=25;CoefficientList[Series[(1-z^(m+1))/(1-r z +(r-1)z^(m+1))/.{r->4,m->2},{z,0,nn}],z] (* Geoffrey Critzer, Mar 12 2014 *)
CoefficientList[Series[(1 + x + x^2)/(1 - 3 x - 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{3,3},{1,4,16},30] (* Harvey P. Dale, Jul 14 2023 *)

my(x='x+O('x^50)); Vec((1+x+x^2)/(1-3*x-3*x^2)) \\ G. C. Greubel, Oct 16 2017

a(0)=1, a(1)=4, a(2)=16, a(n)=3*a(n-1)+3*a(n-2) for n>2. - Philippe Deléham, Sep 18 2009

a(n) = ((2^(1-n)*(-(3-sqrt(21))^(1+n) + (3+sqrt(21))^(1+n)))) / (3*sqrt(21)) for n>0. - Colin Barker, Oct 17 2017

cp's OEIS Frontend

A072264 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=1.

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A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).

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cp's OEIS Frontend

A072264 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A123620 Expansion of (1 + x + x^2) / (1 - 3*x - 3*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A072264 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=1.

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).