A007482 -id:A007482 - cp's OEIS frontend

A285398 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.

1, 19, 452, 10948, 266300, 6484372, 157936172, 3847025764, 93707895260, 2282596837492, 55601016789068, 1354367059315396, 32990588541122684, 803607076375862356, 19574804963320797548, 476816346057854861860, 11614615234500986326556, 282916657894827156657460

Offset: 0

Peter Karpov, Apr 23 2017

Cell configuration converges to a fractal with dimension 2.906...

Colin Barker, Table of n, a(n) for n = 0..700
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (32,-195,216).

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285397, A285399, A285400.

I:=[19, 452, 10948]; [1] cat [n le 3 select I[n] else 32*Self(n-1) - 195*Self(n-2) + 216*Self(n-3) : n in [1..41]]; // G. C. Greubel, Dec 09 2021

{1}~Join~LinearRecurrence[{32, -195, 216}, {19, 452, 10948}, 17]

Vec((1 - x)*(1 - 3*x)*(1 - 9*x) / (1 - 32*x + 195*x^2 - 216*x^3) + O(x^20)) \\ Colin Barker, Apr 23 2017

def A285398_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-13*x+39*x^2-27*x^3)/(1-32*x+195*x^2-216*x^3) ).list()
A285398_list(40) # G. C. Greubel, Dec 09 2021

a(0) = 1, a(1) = 19, a(2) = 452, a(3) = 10948, a(n) = 28*a(n-1) - 195*a(n-2) + 216*a(n-3).

G.f.: (1-13*x+39*x^2-27*x^3)/(1-32*x+195*x^2-216*x^3).

A285399 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 2; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 13, 182, 2548, 35672, 499408, 6991712, 97883968, 1370375552, 19185257728, 268593608192, 3760310514688, 52644347205632, 737020860878848, 10318292052303872, 144456088732254208, 2022385242251558912, 28313393391521824768, 396387507481305546752

Offset: 0

Peter Karpov, Apr 23 2017

Cell configuration converges to a fractal with dimension 2.402...

Colin Barker, Table of n, a(n) for n = 0..850
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (14).

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285397, A285398, A285400.

[1] cat [13*14^(n-1): n in [1..40]]; // G. C. Greubel, Dec 09 2021

A285399:=n->13*14^(n-1): 1,seq(A285399(n), n=1..30); # Wesley Ivan Hurt, Apr 23 2017

{1}~Join~LinearRecurrence[{14}, {13}, 18]

Vec((1-x) / (1-14*x) + O(x^20)) \\ Colin Barker, Apr 23 2017

[1]+[13*14^(n-1) for n in (1..40)] # G. C. Greubel, Dec 09 2021

a(0) = 1, a(1) = 13, a(n) = 14*a(n-1).
G.f.: (1-x)/(1-14*x).
a(n) = 13 * 14^(n-1) for n>0. - Colin Barker, Apr 23 2017
E.g.f.: (1 + 13*exp(14*x))/14. - G. C. Greubel, Dec 09 2021

A285400 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 3; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 18, 378, 7938, 166698, 3500658, 73513818, 1543790178, 32419593738, 680811468498, 14297040838458, 300237857607618, 6304995009759978, 132404895204959538, 2780502799304150298, 58390558785387156258, 1226201734493130281418, 25750236424355735909778

Offset: 0

Peter Karpov, Apr 23 2017

Cell configuration converges to a fractal with dimension 2.771...

Colin Barker, Table of n, a(n) for n = 0..750
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (21).

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285397, A285398, A285399.

[1] cat [18*21^(n-1): n in [1..40]]; // G. C. Greubel, Dec 09 2021

{1}~Join~LinearRecurrence[{21}, {18}, 17]

Vec((1-3*x) / (1-21*x) + O(x^20)) \\ Colin Barker, Apr 23 2017

[1]+[18*21^(n-1) for n in (1..40)] # G. C. Greubel, Dec 09 2021

a(0) = 1, a(1) = 18, a(n) = 21*a(n-1).
G.f.: (1-3*x)/(1-21*x).
a(n) = 2 * 3^(n+1) * 7^(n-1) for n>0. - Colin Barker, Apr 23 2017
E.g.f.: (1 + 6*exp(21*x))/7. - G. C. Greubel, Dec 09 2021

A135030 Generalized Fibonacci numbers: a(n) = 6a(n-1) + 2a(n-2).

Original entry on oeis.org

0, 1, 6, 38, 240, 1516, 9576, 60488, 382080, 2413456, 15244896, 96296288, 608267520, 3842197696, 24269721216, 153302722688, 968355778560, 6116740116736, 38637152257536, 244056393778688, 1541612667187200

Offset: 0

Rolf Pleisch, Feb 10 2008, Feb 14 2008

For n>0, a(n) equals the number of words of length n-1 over {0,1,...,7} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017

Joshua Zucker and Robert Israel, Table of n, a(n) for n = 0..1000 (n=0..51 from Joshua Zucker).
Index entries for linear recurrences with constant coefficients, signature (6, 2).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A180250.

[n le 2 select n-1 else 6*Self(n-1) + 2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Sep 18 2016

A:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(n) = 2*(3*a(n-1) + a(n-2))},a(n),remember):
seq(A(n),n=1..30); # Robert Israel, Sep 16 2014

Join[{a=0,b=1},Table[c=6*b+2*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,2},{0,1},30] (* or *) CoefficientList[Series[ -(x/(2x^2+6x-1)),{x,0,30}],x] (* Harvey P. Dale, Jun 20 2011 *)

a(n)=([0,1; 2,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,6,-2) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(0) = 0; a(1) = 1; a(n) = 2*(3*a(n-1) + a(n-2)).
a(n) = 1/(2*sqrt(11))*( (3 + sqrt(11))^n - (3 - sqrt(11))^n ).
G.f.: x/(1 - 6*x - 2*x^2). - Harvey P. Dale, Jun 20 2011
a(n+1) = Sum_{k=0..n} A099097(n,k)*2^k. - Philippe Deléham, Sep 16 2014
E.g.f.: (1/sqrt(11))*exp(3*x)*sinh(sqrt(11)*x). - G. C. Greubel, Sep 17 2016

More terms from Joshua Zucker, Feb 23 2008

A007484 a(n) = 3a(n-1) + 2a(n-2), with a(0)=2, a(1)=7.

Original entry on oeis.org

2, 7, 25, 89, 317, 1129, 4021, 14321, 51005, 181657, 646981, 2304257, 8206733, 29228713, 104099605, 370756241, 1320467933, 4702916281, 16749684709, 59654886689, 212464029485, 756701861833, 2695033644469, 9598504657073, 34185581260157, 121753753094617, 433632421804165

Offset: 0

N. J. A. Sloane

Number of subsequences of [1,...,2n+1] in which each even number has an odd neighbor.
Same as Pisot sequence E(2,7) (see A008776).
8*a(n) = A007482(n+2) + A007483(n+1) (conjectured, see A104934 for related formula). - Creighton Dement, Apr 15 2005
Conjecture verified using generating functions. - Robert Israel, Jul 12 2018
a(n) = sum of the elements of the matrix M^n, where M = {{1, 2}, {2, 2}}. - Griffin N. Macris, Mar 25 2016
a(3) = 25 is the only composite among the first 8 terms, but then the density of primes decreases, dropping below 50% at the 27th term. - M. F. Hasler, Jul 12 2018
a(n) is also the number of dominating sets in the (2n+1)-triangular snake graph for n > 0. - Eric W. Weisstein, Jun 09 2019

			G.f. = 2 + 7*x + 25*x^2 + 89*x^3 + 317*x^4 + 1129*x^5 + ... - _Michael Somos_, Jul 19 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
R. K. Guy and W. O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly 34:2 (1996), pp. 152-155.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Triangular Snake Graph
Index entries for linear recurrences with constant coefficients, signature (3,2).

Cf. A007455, A007481, A007483, A007484, A104934.

See A008776 for definitions of Pisot sequences.

a007484 n = a007484_list !! n
a007484_list = 2 : 7 : zipWith (+)
               (map (* 3) $ tail a007484_list) (map (* 2) a007484_list)
-- Reinhard Zumkeller, Nov 02 2015

A007484:=[2, 7]; [n le 2 select A007484[n] else 3*Self(n-1)+2*Self(n-2): n in [1..40]]; // Wesley Ivan Hurt, Jan 24 2017

A007484 := proc(n) option remember; if n=0 then 2; elif n=1 then 7; else 3*A007484(n-1)+2*A007484(n-2); fi; end;

LinearRecurrence[{3, 2}, {2, 7}, 40] (* Harvey P. Dale, Apr 24 2012 *)
Table[(2^-n ((3 - Sqrt[17])^n (-4 + Sqrt[17]) + (3 + Sqrt[17])^n (4 + Sqrt[17])))/Sqrt[17], {n, 0, 20}] // Expand (* Eric W. Weisstein, Jun 09 2019 *)
CoefficientList[Series[(2+x)/(1 -3x -2x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 09 2019 *)
a[ n_] := MatrixPower[{{1, 2}, {2, 2}}, n]//Flatten//Total; (* Michael Somos, Jul 19 2021 *)

a(n)=([0,1; 2,3]^n*[2;7])[1,1] \\ Charles R Greathouse IV, Mar 25 2016

A007484_vec(N)=Vec((2+x)/(1-3*x-2*x^2)+O(x^n)) \\ M. F. Hasler, Jul 12 2018

[(i*sqrt(2))^(n-1)*( i*2*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) + chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..30)] # G. C. Greubel, Jul 18 2021

a(n) = nearest integer to (and converges rapidly to) (1+4/sqrt(17))*((3+sqrt(17))/2)^n. - N. J. A. Sloane, Jul 30 2016
If p[i] = Fibonacci(i+2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
G.f.: (2 + x)/(1 - 3*x - 2*x^2). - M. F. Hasler, Jul 12 2018
From G. C. Greubel, Jul 18 2021: (Start)
a(n) = (i*sqrt(2))^(n-1)*( i*2*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + ChebyshevU(n-1, -3*i/(2*sqrt(2))) ).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*((7*n-8*k)/(n-k))*2^k*3^(n-2*k-1) with a(0) = 2. (End)
If we extend the definition of A007483(m) to negative m by using the recurrence, then a(n) = A007483(-3-n)*(-2)^n holds for all n in Z. - Michael Somos, Jul 19 2021
E.g.f.: 2*exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 4*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, May 24 2024

Definition edited by N. J. A. Sloane, Jul 30 2016

A052913 a(n+2) = 5a(n+1) - 2a(n), with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 18, 82, 374, 1706, 7782, 35498, 161926, 738634, 3369318, 15369322, 70107974, 319801226, 1458790182, 6654348458, 30354161926, 138462112714, 631602239718, 2881086973162, 13142230386374, 59948977985546, 273460429154982, 1247404189803818, 5690100090709126

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Main diagonal of the array: m(1,j)=3^(j-1), m(i,1)=1; m(i,j) = m(i-1,j) + m(i,j-1): 1 3 9 27 81 ... / 1 4 13 40 ... / 1 5 18 58 ... / 1 6 24 82 ... - Benoit Cloitre, Aug 05 2002
a(n) is also the number of 3 X n matrices of integers for which the upper-left hand corner is a 1, the rows and columns are weakly increasing, and two adjacent entries differ by at most 1. - Richard Stanley, Jun 06 2010
a(n) is the number of compositions of n when there are 4 types of 1 and 2 types of other natural numbers. - Milan Janjic, Aug 13 2010
If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, or A162909/A162910, or A071766/A229742, or A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n>=0), and the products numerator*denominator, term by term, are summed at each level n, then the resulting sequence of integers is a(n). - Yosu Yurramendi, May 23 2015
Number of 1’s in the substitution system {0 -> 110, 1 -> 11110} at step n from initial string "1" (1 -> 11110 -> 11110111101111011110110 -> ...) . - Ilya Gutkovskiy, Apr 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 894
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-2).

Cf. A007482 (inverse binomial transform).

a:=[1,4];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 16 2019

I:=[1,4]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, May 24 2015

R:=PowerSeriesRing(Integers(), 25); Coefficients(R!((1-x)/(1-5*x+2*x^2))); // Marius A. Burtea, Oct 16 2019

spec := [S,{S=Sequence(Union(Prod(Sequence(Z),Union(Z,Z)),Z,Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series((1-x)/(1-5*x+2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019

Transpose[NestList[{Last[#],5Last[#]-2First[#]}&, {1,4},20]][[1]] (* Harvey P. Dale, Mar 12 2011 *)
LinearRecurrence[{5, -2}, {1, 4}, 25] (* Jean-François Alcover, Jan 08 2019 *)

Vec((1-x)/(1-5*x+2*x^2) + O(x^30)) \\ Michel Marcus, Mar 05 2015

def A052913_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-5*x+2*x^2)).list()
A052913_list(30) # G. C. Greubel, Oct 16 2019

G.f.: (1-x)/(1-5*x+2*x^2).
a(n) = Sum_{alpha=RootOf(1 - 5*z + 2*z^2)} (1/17)*(3+alpha)*alpha^(-1-n).
a(n) = ((17+3*sqrt(17))/34)*((5+sqrt(17))/2)^n + ((17-3*sqrt(17))/34)*((5-sqrt(17))/2)^n. - N. J. A. Sloane, Jun 03 2002
a(n) = A107839(n) - A107839(n-1). - R. J. Mathar, May 21 2015
a(n) = 2*A020698(n-1), n>1. - R. J. Mathar, Nov 23 2015
E.g.f.: (1/17)*exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 16 2019
a(n) = 3*A107839(n-1) + (-1)^n*A152594(n) with A107839(-1) = 0. - Klaus Purath, Jul 29 2020

Typo in definition corrected by Bruno Berselli, Jun 07 2010

A154692 Triangle read by rows: T(n, k) = (2^(n-k)3^k + 2^k3^(n-k))*binomial(n, k).

Original entry on oeis.org

2, 5, 5, 13, 24, 13, 35, 90, 90, 35, 97, 312, 432, 312, 97, 275, 1050, 1800, 1800, 1050, 275, 793, 3492, 7020, 8640, 7020, 3492, 793, 2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315, 6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

			Triangle begins
     2;
     5,     5;
    13,    24,    13;
    35,    90,    90,     35;
    97,   312,   432,    312,     97;
   275,  1050,  1800,   1800,   1050,    275;
   793,  3492,  7020,   8640,   7020,   3492,   793;
  2315, 11550, 26460,  37800,  37800,  26460, 11550,  2315;
  6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000225, A007482, A013620, A088137, A119309.
Sums include: A010673 (alternating sign row), A020699 (row), A020729 (row).
Related sequences: A007318, A154690,

A154692:= func< n,k | (2^(n-k)*3^k + 2^k*3^(n-k))*Binomial(n,k) >;
[A154692(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154692 := proc(n,m)
        (2^(n-m)*3^m+2^m*3^(n-m))*binomial(n,m) ;
end proc:
seq(seq(A154692(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Oct 24 2011

p=2; q=3;
T[n_, m_]= (p^(n-m)*q^m + p^m*q^(n-m))*Binomial[n,m];
Table[T[n,m], {n,0,10}, {m,0,n}]//Flatten

from sage.all import *
def A154692(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*binomial(n,k)
print(flatten([[A154692(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Sum_{k=0..n} T(n, k) = A020729(n) = A020699(n+1).
T(n,m) = A013620(n,m) + A013620(m,n). - R. J. Mathar, Oct 24 2011
From G. C. Greubel, Jan 18 2025: (Start)
T(2*n, n) = A119309(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1) + A007482(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A088137(n+1) + A000225(n+1). (End)

A180250 a(n) = 5a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 1, 5, 35, 225, 1475, 9625, 62875, 410625, 2681875, 17515625, 114396875, 747140625, 4879671875, 31869765625, 208145546875, 1359425390625, 8878582421875, 57987166015625, 378721654296875, 2473479931640625, 16154616201171875, 105507880322265625

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 16 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (5,10).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Join[{a=0,b=1},Table[c=5*b+10*a;a=b;b=c,{n,100}]]
LinearRecurrence[{5,10}, {0,1}, 30] (* G. C. Greubel, Jan 16 2018 *)

a(n)=([0,1;10,5]^(n-1))[1,2] \\ Charles R Greathouse IV, Oct 03 2016

my(x='x+O('x^30)); concat([0], Vec(x^2/(1-5*x-10*x^2))) \\ G. C. Greubel, Jan 16 2018

A180250= BinaryRecurrenceSequence(5,10,0,1)
[A180250(n-1) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

a(n) = ((5+sqrt(65))^(n-1) - (5-sqrt(65))^(n-1))/(2^(n-1)*sqrt(65)). - Rolf Pleisch, May 14 2011
G.f.: x^2/(1-5*x-10*x^2).
a(n) = (i*sqrt(10))^(n-1) * ChebyshevU(n-1, -i*sqrt(5/8)). - G. C. Greubel, Jul 21 2023

A015551 Expansion of x/(1 - 6x - 5x^2).

Original entry on oeis.org

0, 1, 6, 41, 276, 1861, 12546, 84581, 570216, 3844201, 25916286, 174718721, 1177893756, 7940956141, 53535205626, 360916014461, 2433172114896, 16403612761681, 110587537144566, 745543286675801, 5026197405777636

Offset: 0

Olivier Gérard

Let the generator matrix for the ternary Golay G_12 code be [I|B], where the elements of B are taken from the set {0,1,2}. Then a(n)=(B^n)1,2 for instance. - _Paul Barry, Feb 13 2004

Pisano period lengths: 1, 2, 4, 4, 1, 4, 42, 8, 12, 2, 10, 4, 12, 42, 4, 16, 96, 12, 360, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina, Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Index entries for linear recurrences with constant coefficients, signature (6,5).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A057089, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

Join[{a=0,b=1},Table[c=6*b+5*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
CoefficientList[Series[x/(1-6x-5x^2),{x,0,20}],x] (* or *) LinearRecurrence[ {6,5},{0,1},30] (* Harvey P. Dale, Oct 30 2017 *)

a(n)=([0,1; 5,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,6,-5) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 5*a(n-2).

a(n) = sqrt(14)*(3+sqrt(14))^n/28 - sqrt(14)*(3-sqrt(14))^n/28. - Paul Barry, Feb 13 2004

A007481 Number of subsequences of [ 1,...,n ] in which each even number has an odd neighbor.

Original entry on oeis.org

1, 2, 3, 7, 11, 25, 39, 89, 139, 317, 495, 1129, 1763, 4021, 6279, 14321, 22363, 51005, 79647, 181657, 283667, 646981, 1010295, 2304257, 3598219, 8206733, 12815247, 29228713, 45642179, 104099605, 162557031, 370756241, 578955451, 1320467933

Offset: 0

N. J. A. Sloane

A055099(n) = a(2*n+1) - a(2*n) = a(2*(n+1)) - a(2*n+1). - Reinhard Zumkeller, Oct 25 2015

			For n=2, there are the following three subsequences of [1,2] with the desired property: empty, [1], [1,2].
For n=3, there are the following seven subsequences of [1,2,3] with the desired property: empty, [1], [3], [1,2], [2,3], [1,3], [1,2,3].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..400
R. K. Guy and W. O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly 34:2 (1996), pp. 152-155.
Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, 2).

Cf. A007455, A007482, A007484, A055099.

a007481 n = a007481_list !! n
a007481_list = 1 : 2 : 3 : 7 : zipWith (+)
               (map (* 3) $ drop 2 a007481_list) (map (* 2) a007481_list)
-- Reinhard Zumkeller, Oct 25 2015

LinearRecurrence[{0,3,0,2},{1,2,3,7},40] (* Harvey P. Dale, Feb 29 2012 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 2,0,3,0]^n*[1;2;3;7])[1,1] \\ Charles R Greathouse IV, Mar 02 2016

a(n) = 3*a(n-2) + 2*a(n-4).

G.f.: (x^3+2*x+1)/(-2*x^4-3*x^2+1). - Harvey P. Dale, Feb 29 2012

More terms from James Sellers, Dec 24 1999

cp's OEIS Frontend

A285398 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.

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A285399 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 2; a(n) is the number of cells after n iterations.

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A285400 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 3; a(n) is the number of cells after n iterations.

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A135030 Generalized Fibonacci numbers: a(n) = 6*a(n-1) + 2*a(n-2).

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A007484 a(n) = 3*a(n-1) + 2*a(n-2), with a(0)=2, a(1)=7.

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A052913 a(n+2) = 5*a(n+1) - 2*a(n), with a(0) = 1, a(1) = 4.

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A135030 Generalized Fibonacci numbers: a(n) = 6a(n-1) + 2a(n-2).

A007484 a(n) = 3a(n-1) + 2a(n-2), with a(0)=2, a(1)=7.

A052913 a(n+2) = 5a(n+1) - 2a(n), with a(0) = 1, a(1) = 4.

A154692 Triangle read by rows: T(n, k) = (2^(n-k)3^k + 2^k3^(n-k))*binomial(n, k).

A180250 a(n) = 5a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.

A015551 Expansion of x/(1 - 6x - 5x^2).