A062510 -id:A062510 - cp's OEIS frontend

A014551 Jacobsthal-Lucas numbers.

2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 2147483647, 4294967297, 8589934591

Offset: 0

Eric W. Weisstein

Also gives the number of points of period n in the subshift of finite type corresponding to the square matrix A=[1,2;1,0] (this is then given by trace(A^n)). - Thomas Ward, Mar 07 2001
Sequence is identical to its signed inverse binomial transform (autosequence of the second kind). - Paul Curtz, Jul 11 2008
a(n) can be expressed in terms of values of the Fibonacci polynomials F_n(x), computed at x=1/sqrt(2). - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008
Pisano period lengths: 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, 2, 8, 6, 18, 4, ... - R. J. Mathar, Aug 10 2012
Let F(x) = Product_{n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number 1 + F(-1/2) = 2.83717 78068 73232 99799 ... = 2 + 1/(1 + 1/(5 + 1/(7 + 1/(17 + ...)))). See A111317. - Peter Bala, Dec 26 2012
With different signs,  2, -1, 5, -7, 17, -31, 65, -127, 257, -511, 1025, -2047, ... is the Lucas V(-1,-2) sequence. - R. J. Mathar, Jan 08 2013
The identity 2 = 2/2 + 2^2/(2*1) - 2^3/(2*1*5) - 2^4/(2*1*5*7) + 2^5/(2*1*5*7*17) + 2^6/(2*1*5*7*17*31) - - + + can be viewed as a generalized Engel-type expansion of the number 2 to the base 2. Compare with A062510. - Peter Bala, Nov 13 2013
For n >= 2, a(n) is the number of ways to tile a 2 X n strip, where the first two columns have an extra cell at the top, with 1 X 2 dominoes and 2 X 2 squares. Shown here is one of the a(7)=127 ways for the n=7 case:
  ._.
  |_|_________.
  | |   | |_| |
  ||__|_|_|_|. - Greg Dresden, Sep 26 2021
Named by Horadam (1988) after the German mathematician Ernst Jacobsthal (1882-1965) and the French mathematician Édouard Lucas (1842-1891). - Amiram Eldar, Oct 02 2023

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255.
Lind and Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. (General material on subshifts of finite type)
Kritkhajohn Onphaeng and Prapanpong Pongsriiam. Jacobsthal and Jacobsthal-Lucas Numbers and Sums Introduced by Jacobsthal and Tverberg. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.6.
Abdelmoumène Zekiri, Farid Bencherif, Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.

T. D. Noe, Table of n, a(n) for n = 0..200
Kunle Adegoke, Robert Frontczak, and Taras Goy, Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression, Notes on Num. Theor. and Disc. Math. (2021) Vol. 27, No. 2, 54-63.
Tewodros Amdeberhan, A note on Fibonacci-type polynomials, arXiv:0811.4652 [math.NT], 2008.
Hacène Belbachir, Amine Belkhir, and Ihab-Eddin Djellas, Permanent of Toeplitz-Hessenberg Matrices with Generalized Fibonacci and Lucas entries, Applications and Applied Mathematics: An International Journal (AAM 2022), Vol. 17, Iss. 2, Art. 15, 558-570.
Paula Catarino, Helena Campos, and Paulo Vasco. On the Mersenne sequence. Annales Mathematicae et Informaticae, 46 (2016), pp. 37-53.
Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013), pp. 27-39.
Fatih Erduvan and Refik Keskin, Fibonacci And Lucas Numbers Which Are Product Of Two Jacobsal-Lucas Numbers [sic], Appl. Math. E-Notes (2023) Vol. 23, 60-70.
M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
Élis Gardel da Costa Mesquita, Eudes Antonio Costa, Paula M. M. C. Catarino, and Francisco R. V. Alves, Jacobsthal-Mulatu Numbers, Latin Amer. J. Math. (2025) Vol. 4, No. 1, 23-45. See p. 24.
A. F. Horadam, Jacobsthal and Pell Curves, Fib. Quart. 26, 79-83, 1988.
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34, 40-54, 1996.
D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
Thomas Koshy and Ralph P. Grimaldi, Ternary words and Jacobsthal numbers, Fib. Quart., 55 (No. 2, 2017), 129-136.
Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, and Prapanpong Pongsriiam, Inequalities for Inclusion-Exclusion-Like Sums Involving the Ceiling and the Nearest Integer Functions, Integers (2025) Vol. 25, Art. No. A45. See p. 3.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
M. Rahmani, The Akiyama-Tanigawa matrix and related combinatorial identities, Linear Algebra and its Applications 438 (2013) 219-230. - From _N. J. A. Sloane_, Dec 26 2012
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 41.
Yüksel Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824.
Yüksal Soykan, On Summing Formulas for Horadam Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 8, Issue 1, 45-61.
Yüksel Soykan, Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.
Yüksel Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.
Yüksel Soykan, Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Sum_{k=0..n} W_k^3 and Sum_{k=1..n} W_(-k)^3, Archives of Current Research International (2020) Vol. 20, Issue 2, 58-69.
Yüksel Soykan, A Study on Generalized Fibonacci Numbers: Sum Formulas Sum_{k=0..n} k * x^k * W_k^3 and Sum_{k=1..n} k * x^k W_-k^3 for the Cubes of Terms, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.
Yüksel Soykan, On Generalized (r, s)-numbers, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 8, No. 1, 1-14.
Yüksel Soykan, Erkan Taşdemir, and İnci Okumuş, On Dual Hyperbolic Numbers With Generalized Jacobsthal Numbers Components, Zonguldak Bülent Ecevit University, (Zonguldak, Turkey, 2019).
Anetta Szynal-Liana, Iwona Włoch, and Mirosław Liana, Generalized commutative quaternion polynomials of the Fibonacci type, Annales Math. Sect. A, Univ. Mariae Curie-Skłodowska (Poland 2022) Vol. 76, No. 2, 33-44.
Elif Tan, Luka Podrug, and Vesna Iršič Chenoweth, Horadam-Lucas Cubes, Axioms (2024) Vol. 13, No. 12, 837.
Kai Wang, On Horadam Sequences and Related Infinite Series, (2020).
Kai Wang, General Identities for Horadam Sequences, (2020).
Eric Weisstein's World of Mathematics, Jacobsthal Number.
Wikipedia, Lucas sequence.
OEIS Wiki, Autosequence.
Volkan Yildiz, Some divisibility properties of Jacobsthal numbers, arXiv:2212.08814 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,2).
Index entries for Lucas sequences.
Index entries for sequences related to Chebyshev polynomials.

Cf. A001045 (companion "autosequence"), A019322, A066845, A111317.
Cf. A135440 (first differences), A166920 (partial sums).
Cf. A000079, A033999, A102345, A105723.
Cf. A006995.

a014551 n = a000079 n + a033999 n
a014551_list = map fst $ iterate (\(x,s) -> (2 * x - 3 * s, -s)) (2, 1)
-- Reinhard Zumkeller, Jan 02 2013

[2^n + (-1)^n: n in [0..30]]; // G. C. Greubel, Dec 17 2017

f[n_]:=2/(n+1);x=4;Table[x=f[x];Denominator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
nxt[{n_,a_}]:={n+1,2a-3(-1)^(n+1)}; Transpose[NestList[nxt,{1,2},40]] [[2]] (* Harvey P. Dale, May 27 2013 *)
LinearRecurrence[{1, 2}, {2, 1}, 40] (* Jean-François Alcover, Jan 07 2019 *)

a(n)=2^n+(-1)^n \\ Charles R Greathouse IV, Nov 20 2012

[lucas_number2(n,1,-2) for n in range(0, 32)] # Zerinvary Lajos, Apr 30 2009

a(n+1) = 2 * a(n) - (-1)^n * 3.
From Len Smiley, Dec 07 2001: (Start)
a(n) = 2^n + (-1)^n.
G.f.: (2-x)/(1-x-2*x^2). (End)
E.g.f.: exp(x) + exp(-2*x) produces a signed version. - Paul Barry, Apr 27 2003
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-1, 2*k)*3^(2*k)/2^(n-2). - Paul Barry, Feb 21 2003
0, 1, 5, 7 ... is 2^n - 2*0^n + (-1)^n, the 2nd inverse binomial transform of (2^n-1)^2 (A060867). - Paul Barry, Sep 05 2003
a(n) = 2*T(n, i/(2*sqrt(2))) * (-i*sqrt(2))^n with i^2=-1. - Paul Barry, Nov 17 2003
a(n) = A078008(n) + A001045(n+1). - Paul Barry, Feb 12 2004
a(n) = 2*A001045(n+1) - A001045(n). - Paul Barry, Mar 22 2004
a(0)=2, a(1)=1, a(n) = a(n-1) + 2*a(n-2) for n > 1. - Philippe Deléham, Nov 07 2006
a(2*n+1) = Product_{d|(2*n+1)} cyclotomic(d,2). a(2^k*(2*n+1)) = Product_{d|(2*n+1)} cyclotomic(2*d,2^(2^k)). - Miklos Kristof, Mar 12 2007
a(n) = 2^{(n-1)/2}F_{n-1}(1/sqrt(2)) + 2^{(n+2)/2}F_{n-2}(1/sqrt(2)). - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008
E.g.f.: U(0) where U(k) = 1 + (-1)^k/(2^k - 4^k*x*2/(2*x*2^k + (-1)^k*(k+1)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 02 2012
G.f.: U(0) where U(k) = 1 + (-1)^k/(2^k - 4^k*x*2/(2*x*2^k + (-1)^k/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 02 2012
a(n) = sqrt(9*(A001045)^2 + (-1)^n*2^(n+2)). - Vladimir Shevelev, Mar 13 2013
G.f.: 2 + G(0)*x*(1+4*x)/(2-x), where G(k) = 1 + 1/(1 - x*(9*k-1)/( x*(9*k+8) - 2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 13 2013
a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 9*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
For n >= 1: a(n) = A006995(2^((n+2)/2)) when n is even, a(n) = A006995(3*2^((n-1)/2) - 1) when n is odd. - Bob Selcoe, Sep 04 2017
a(n) = J(n) + 4*J(n-1), a(0)=2, where J is A001045. - Yuchun Ji, Apr 23 2019
For n >= 0, 1/(2*a(n+1)) = Sum_{m>=n} a(m)/(a(m+1)*a(m+2)). - Kai Wang, Mar 03 2020
For 4 > h >= 0, k >= 0, a(4*k+h) mod 5 = a(h) mod 5. - Kai Wang, May 06 2020
From Kai Wang, May 30 2020: (Start)
(2 - a(n+1)/a(n))/9 = Sum_{m>=n} (-2)^m/(a(m)*a(m+1)).
a(n) = 2*A001045(n+1) - A001045(n).
a(n)^2 = a(2*n) + 2*(-2)^n.
a(n)^2 = 9*A001045(n)^2 + 4*(-2)^n.
a(2*n) = 9*A001045(n)^2 + 2*(-2)^n.
2*A001045(m+n) = A001045(m)*a(n) + a(m)*A001045(n).
2*(-2)^n*A001045(m-n) = A001045(m)*a(n) - a(m)*A001045(n).
A001045(m+n) + (-2)^n*A001045(m-n) = A001045(m)*a(n).
A001045(m+n) - (-2)^n*A001045(m-n) = a(m)*A001045(n).
2*a(m+n) = 9*A001045(m)*A001045(n) + a(m)*a(n).
2*(-2)^n*a(m-n) = a(m)*a(n) - 9*A001045(m)*A001045(n).
a(m+n) - (-2)^n*a(m-n) = 9*A001045(m)*A001045(n).
a(m+n) + (-2)^n*a(m-n) = a(m)*a(n).
a(m+n)*a(m-n) - a(m)*a(m) = 9*(-2)^(m-n)*A001045(n)^2.
a(m+1)*a(n) - a(m)*a(n+1) = 9*(-2)^n*A001045(m-n). (End)
a(n) = F(n+1) + F(n-1) + Sum_{k=0..(n-2)} a(k)*F(n-1-k) for F(n) the Fibonacci numbers and for n > 1. - Greg Dresden, Jun 03 2020

A029858 a(n) = (3^n - 3)/2.

Original entry on oeis.org

0, 3, 12, 39, 120, 363, 1092, 3279, 9840, 29523, 88572, 265719, 797160, 2391483, 7174452, 21523359, 64570080, 193710243, 581130732, 1743392199, 5230176600, 15690529803, 47071589412, 141214768239

Offset: 1

Christian G. Bower

Also the number of 2-block covers of a labeled n-set. a(n) = A055154(n,2). Generally, number of k-block covers of a labeled n-set is T(n,k) = (1/k!)*Sum_{i = 1..k + 1} Stirling1(k + 1,i)*(2^(i - 1) - 1)^n. In particular, T(n,2) = (1/2!)*(3^n - 3), T(n,3) = (1/3!)*(7^n - 6*3^n + 11), T(n,4) = (1/4)!*(15^n - 10*7^n + 35*3^n - 50), ... - Vladeta Jovovic, Jan 19 2001
Conjectured to be the number of integers from 0 to 10^(n-1) - 1 that lack 0, 1, 2, 3, 4, 5 and 6 as a digit. - Alexandre Wajnberg, Apr 25 2005. This is easily verified to be true. - Renzo Benedetti, Sep 25 2008
Number of monic irreducible polynomials of degree 1 in GF(3)[x1,...,xn]. - Max Alekseyev, Jan 23 2006
Also, the greatest number of identical weights among which an odd one can be identified and it can be decided if the odd one is heavier or lighter, using n weighings with a comparing balance. If the odd one only needs to be identified, the sequence starts 4, 13, 40 and is A003462 (3^n - 1)/2, n > 1. - Tanya Khovanova, Dec 11 2006; corrected by Samuel E. Rhoads, Apr 18 2016
Binomial transform yields A134057. Inverse binomial transform yields A062510 with one additional 0 in front. - R. J. Mathar, Jun 18 2008
Numbers k where the recurrence s(0)=0, if s(k-1) >= k then s(k) = s(k-1) - k otherwise s(k) = s(k-1) + k produces s(k) = 0. - Hugo Pfoertner, Jan 05 2012
For n > 1: A008344(a(n)) = a(n). - Reinhard Zumkeller, May 09 2012
Also the number of edges in the (n-1)-Hanoi graph. - Eric W. Weisstein, Jun 18 2017
A level 1 Sierpiński triangle graph is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.  a(n) is the number of degree 4 vertices in the level n Sierpinski triangle graph. - Allan Bickle, Jul 30 2020
Also the number of minimum vertex cuts in the n-Apollonian network. - Eric W. Weisstein, Dec 20 2020
Also the minimum number of turns in n-dimensional Euclidean space needed to visit all 3^n points of the grid {0, 1, 2}^n, moving in straight lines between turns (repeated visits and direction changes at non-grid points are allowed). - Marco Ripà, Aug 06 2025

			For the Sierpiński triangle, Level 1 is a triangle, so a(1) = 0.
Level 2 has three corners (degree 2) and three degree 4 vertices, so a(2) = 3.
The level 2 Hanoi graph has 3 triangles joined by 3 edges, so a(2+1) = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
Alex Born, Cor A. J. Hukrnes, and Gerhard J. Woeginger, How to detect a counterfeit coin: adaptive versus non-adaptive solutions, Inf. Proc. Lett. 86 (2003) 137-141.
Gary Darby, The Counterfeit Coin
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See p. 17.
Lorenz Halbeisen and Norbert Hungerbuhler, The general counterfeit coin problem, Discr. Math 147 (1-3) (1995) 139-150, Theorem 1 with b=1.
Andreas Hinz, Sandi Klavzar, and Sara Sabrina Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Bennet Manvel, Counterfeit coin problems, Math. Mag. 50 (2) (1977) 90-92, theorem 2.
Marco Ripà, Solving the 106 years old 3^k points problem with the clockwise-algorithm, Journal of Fundamental Mathematics and Applications, 2020, 3(2), 84-97.
Allen Stenger and Jack Wert, The Twelve Coins (or Twelve bags of Gold)
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Minimum Vertex Cut
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A055154, A003462, A007051, A034472, A024023.
Cf. A000203, A008776.
Cf. A007283, A029858, A067771, A233774, A233775, A246959 (Sierpiński triangle graphs).
Cf. A000225, A029858, A058809, A375256 (Hanoi graphs).

a029858 = (`div` 2) . (subtract 3) . (3 ^)
a029858_list = iterate ((+ 3) . (* 3)) 0
-- Reinhard Zumkeller, May 09 2012

[(3^n-3)/2: n in [1..30]]; // Vincenzo Librandi, Jun 05 2011

a:=n->sum(3^j,j=1..n): seq(a(n),n=0..23); # Zerinvary Lajos, Jun 27 2007

Table[(3^n - 3)/2, {n, 24}] (* Alonso del Arte, Dec 29 2014 *)

a(n)=(3^n-3)\2 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = 3*a(n-1) + 3. - Alexandre Wajnberg, Apr 25 2005
O.g.f: 3*x^2/((1-x)*(1-3*x)). - R. J. Mathar, Jun 18 2008
a(n) = 3^(n-1) + a(n-1) (with a(1)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = 3*A003462(n-1). - R. J. Mathar, Sep 10 2015
E.g.f.: 3*(-1 + exp(2*x))*exp(x)/2. - Ilya Gutkovskiy, Apr 19 2016
a(n) = A067771(n-1) - 3. - Allan Bickle, Jul 30 2020
a(n) = sigma(A008776(n-2)) for n>=2. - Flávio V. Fernandes, Apr 20 2021

Corrected by T. D. Noe, Nov 07 2006

A051049 Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one.

Original entry on oeis.org

1, 1, 4, 7, 16, 31, 64, 127, 256, 511, 1024, 2047, 4096, 8191, 16384, 32767, 65536, 131071, 262144, 524287, 1048576, 2097151, 4194304, 8388607, 16777216, 33554431, 67108864, 134217727, 268435456, 536870911, 1073741824

Offset: 0

Eric W. Weisstein

Might be called the "Purkiss sequence", after Henry John Purkiss who in 1865 found that this is the number of moves for the accelerated Chinese Rings puzzle (baguenaudier). [Email from Andreas M. Hinz, Feb 15 2017, who also pointed out that there was an error in the definition in this entry]. - N. J. A. Sloane, Feb 18 2017
The row sums of triangle A166692. - Paul Curtz, Oct 20 2009
The inverse binomial transform equals (-1)^n*A062510(n) with an extra leading term 1. - Paul Curtz, Oct 20 2009
This is the sequence A(1,1;1,2;1) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by Rules 261, 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 37, and 381, based on the 5-celled von Neumann neighborhood. - Robert Price, Jan 02 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Andreas M. Hinz, The Lichtenberg sequence, Fib. Quart., 55 (2017), 2-12.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 56. Book's website
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From _Wolfdieter Lang_, Oct 18 2010]
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Baguenaudier
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000069, A000079, A000225, A000975, A001045.
Cf. A001969, A026147, A062510, A135221.
Row sums of A131086.
Row sums of A166692.

[2^n -(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Aug 14 2011

A051049:= proc(n): 2^n -(1-(-1)^n)/2 end: seq(A051049(n), n=0..40); # Johannes W. Meijer, Jun 24 2011

a[n_?EvenQ]:= 2^(n-1) -1; a[n_?OddQ]:= 2^(n-1); Table[a[n], {n,50}]
LinearRecurrence[{2,1,-2},{1,1,4},40] (* Jean-François Alcover, Jan 08 2019 *)

a(n)=2^(n-1)-(n%2==0) \\ Charles R Greathouse IV, Mar 22 2013

[2^n -(n%2) for n in range(41)] # G. C. Greubel, Apr 23 2023

a(n) = (2^(n+1) - (1 + (-1)^(n+1)))/2. - Paul Barry, Apr 24 2003
a(n+2) = a(n+1) + 2*a(n) + 1, a(0)=a(1)=1. - Paul Barry, May 01 2003
From Paul Barry, Sep 19 2003: (Start)
G.f.: (1 - x + x^2)/((1 - x^2)*(1 - 2*x));
e.g.f.: exp(2*x) - sinh(x). (End)
a(n) = ((Sum_{k=0..n} 2^k) + (-1)^n)/2 = (A000225(n+1) + (-1)^n)/2. - Paul Barry, May 27 2003
(a(n+1) - a(n))/3 = A001045(n). - Paul Barry, May 27 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n+1, 2*k). - Paul Barry, May 27 2003
a(n) = (Sum_{k=0..n} binomial(n,k) + (-1)^(n-k)) - 1. - Paul Barry, Jul 21 2003
a(n) = Sum_{k=0..n} Sum_{j=0..n-k, (j-k) mod 2 = 0} binomial(n-k, j). - Paul Barry, Jan 25 2005
Row sums of triangle A135221. - Gary W. Adamson, Nov 23 2007
a(n) = A001045(n+1) + A000975(n+1) - A000079(n). - Paul Curtz, Oct 20 2009
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0) = a(1) = 1, a(2) = 4. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010
a(n) = 3*a(n-1) - 2*a(n-2) + 3*(-1)^n. - Gary Detlefs, Dec 21 2010
a(n) = 3* A000975(n-1) + 1, n > 0. - Gary Detlefs, Dec 21 2010
a(n+2) = A001969(2^n+1) + A000069(2^n); evil + odious. - Johannes W. Meijer, Jun 24 2011, Jun 26 2011
E.g.f.: exp(2x) - sinh(x) = Q(0); Q(k) = 1 - k!*x^(k+1)/((2*k + 1)!*2^k - 2*(((2*k + 1)!*2^k)^2)/((2*k + 1)!*2^(k+1) - x^k*(k + 1)!/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 16 2011
a(n) = Sum_{k=0..n} Sum_{i=0..n} C(k-1,i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = A000975(n+1) - A001045(n). - Yuchun Ji, Jul 08 2018
a(n) = A026147(2^(n-1)) for n > 0. - Chunqing Liu, Dec 18 2022

Edited and information added by Johannes W. Meijer, Jun 24 2011

A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 7, 13, 27, 53, 107, 213, 427, 853, 1707, 3413, 6827, 13653, 27307, 54613, 109227, 218453, 436907, 873813, 1747627, 3495253, 6990507, 13981013, 27962027, 55924053, 111848107, 223696213, 447392427, 894784853, 1789569707, 3579139413, 7158278827, 14316557653

Offset: 0

Michael Somos, Jun 17 1999

Number of positive integers requiring exactly n signed bits in the modified non-adjacent form representation. - Ralf Stephan, Aug 02 2003
The n-th entry (n>1) of the sequence is equal to the 1,1-entry of the n-th power of the unnormalized 4 X 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini, Oct 27 2004
Pisano period lengths:  1, 1, 6, 2, 2, 6, 6, 2, 18, 2, 10, 6, 12, 6, 6, 2, 8, 18, 18, 2, ... - R. J. Mathar, Aug 10 2012
For n >= 1, a(n) is the number of ways to tile a strip of length n+2 with blue squares and blue and red dominos, with the restriction that the first two tiles must be the same color. - Guanji Chen and Greg Dresden, Jul 15 2024

			G.f. = 2 + 3*x + 7*x^2 + 13*x^3 + 27*x^4 + 53*x^5 + 107*x^6 + 213*x^7 + 427*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wieb Bosma, Signed bits and fast exponentiation, Journal de théorie des nombres de Bordeaux, Vol. 13, No. 1 (2001), pp. 27-41.
Karl Dilcher and Larry Ericksen, Continued fractions and Stern polynomials, Ramanujan Journal 45.3 (2018): 659-681. See Table 2.
Karl Dilcher and Hayley Tomkins, Square classes and divisibility properties of Stern polynomials, Integers, Vol. 18 (2018), Article #A29.
Petro Kosobutskyy, The Collatz problem as a reverse n->0 problem on a graph tree formed from theta*2^n Jacobsthal-type numbers, arXiv:2306.14635 [math.GM], 2023.
Petro Kosobutskyy and Dariia Rebot, Collatz conjecture 3n+/-1 as a Newton binomial problem, Comp. Des. Sys. Theor. Prac., Lviv Nat'l Polytech. Univ. (Ukraine 2023) Vol. 5, No. 1, 137-145. See p. 140.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Sam Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, Vol. 117, No. 7 (2010), pp. 581-598.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A084214 (first differences).
Cf. A001045, A014551, A024493, A062510.
Cf. A007283, A007679, A078008, A081254.
Cf. A083322, A083944, A083581, A084170.
Cf. A102713, A127978, A130755, A139763.
Cf. A140360, A140966, A166249, A168642.
Cf. A280560, A290604, A340627.
Cf. A112387, A135318.

[(5*2^n+(-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

LinearRecurrence[{1,2},{2,3},40] (* Harvey P. Dale, Dec 11 2017 *)

{a(n) = if( n<0, 0, (5*2^n + (-1)^n) / 3)};

{a(n) = if (n<0 ,0, if( n<2, n+2, a(n-1) + 2*a(n-2)))};

[(5*2^n+(-1)^n)/3 for n in range(35)] # G. C. Greubel, Apr 10 2019

G.f.: (2 + x) / (1 - x - 2*x^2).
a(n) = (5*2^n + (-1)^n) / 3.
a(n) = 2^(n+1) - A001045(n).
a(n) = A084170(n)+1 = abs(A083581(n)-3) = A081254(n+1) - A081254(n) = A084214(n+2)/2.
a(n) = 2*A001045(n+1) + A001045(n) (note that 2 is the limit of A001045(n+1)/A001045(n)). - Paul Barry, Sep 14 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=-charpoly(A,-1). - Milan Janjic, Jan 27 2010
Equivalently, with different offset, a(n) = b(n+1) with b(0)=1 and b(n) = Sum_{i=0..n-1} (-1)^i (1 + (-1)^i b(i)). - Olivier Gérard, Jul 30 2012
a(n) = A000975(n-2)*10 + 5 + 2*(-1)^(n-2), a(0)=2, a(1)=3. - Yuchun Ji, Mar 18 2019
a(n+1) = Sum_{i=0..n} a(i) + 1 + (1-(-1)^n)/2, a(0)=2. - Yuchun Ji, Apr 10 2019
a(n) = 2^n + J(n+1) = J(n+2) + J(n+1) - J(n), where J is A001045. - Yuchun Ji, Apr 10 2019
a(n) = A001045(n+2) + A078008(n) = A062510(n+1) - A078008(n+1) = (A001045(n+2) + A062510(n+1))/2 = A014551(n) + 2*A001045(n). - Paul Curtz, Jul 14 2021
From Thomas Scheuerle, Jul 14 2021: (Start)
a(n) = A083322(n) + A024493(n).
a(n) = A127978(n) - A102713(n).
a(n) = A130755(n) - A166249(n).
a(n) = A007679(n) + A139763(n).
a(n) = A168642(n) XOR A007283(n).
a(n) = A290604(n) + A083944(n). (End)
From Paul Curtz, Jul 21 2021: (Start)
a(n) = 5*A001045(n) - A280560(n+1) = abs(A140360(n+1)) - A280560(n+1).
a(n) = 2^n + A001045(n+1) = A001045(n+3) - A000079(n).
a(n) = A001045(n+4) - A340627(n). (End)
a(n) = A001045(n+5) - A005010(n).
a(n+1) + a(n) = a(n+2) - a(n) = 5*2^n. - Michael Somos, Feb 22 2023
a(n) = A135318(2*n) + A135318(2*n+1) = A112387(2*n) + A112387(2*n+1). - Paul Curtz, Jun 26 2024
E.g.f.: (cosh(x) + 5*cosh(2*x) - sinh(x) + 5*sinh(2*x))/3. - Stefano Spezia, May 18 2025

Formula of Milan Janjic moved here from wrong sequence by Paul D. Hanna, May 29 2010

A230184 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or antidiagonally next to at least one element with value 2-x(i,j).

Original entry on oeis.org

0, 3, 0, 3, 9, 0, 9, 27, 33, 0, 15, 159, 231, 129, 0, 33, 825, 3411, 1971, 513, 0, 63, 4395, 44487, 73857, 16815, 2049, 0, 129, 23307, 596973, 2432241, 1603431, 143451, 8193, 0, 255, 123729, 7957785, 82359177, 133265847, 34825803, 1223799, 32769, 0, 513

Offset: 1

R. H. Hardin, Oct 11 2013

Table starts
.0.....3........3...........9.............15................33
.0.....9.......27.........159............825..............4395
.0....33......231........3411..........44487............596973
.0...129.....1971.......73857........2432241..........82359177
.0...513....16815.....1603431......133265847.......11393567289
.0..2049...143451....34825803.....7303192425.....1576417829097
.0..8193..1223799...756450105...400233701367...218117038953009
.0.32769.10440387.16430979183.21933865129257.30179260908320577

			Some solutions for n=3 k=4
..2..0..1..1....0..2..1..1....2..0..0..1....0..2..2..0....1..1..2..2
..0..2..2..0....0..0..2..2....0..2..1..1....0..1..2..0....2..0..0..0
..0..2..1..1....1..1..0..2....1..1..0..2....1..2..0..2....0..2..2..0

R. H. Hardin, Table of n, a(n) for n = 1..238

Column 2 is A084508(n+1)

Row 1 is A062510(n-1)

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 5*a(n-1) -4*a(n-2)
k=3: a(n) = 9*a(n-1) -4*a(n-2)
k=4: a(n) = 26*a(n-1) -97*a(n-2) +89*a(n-3) -18*a(n-4) +a(n-5)
k=5: [order 6] for n>8
k=6: [order 23] for n>24
k=7: [order 34] for n>37
Empirical for row n:
n=1: a(n) = a(n-1) +2*a(n-2)
n=2: a(n) = 4*a(n-1) +7*a(n-2) +a(n-3) -6*a(n-4) -5*a(n-5) for n>6
n=3: [order 21] for n>23
n=4: [order 93] for n>96

A230675 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally, diagonally or antidiagonally next to at least one element with value 2-x(i,j).

Original entry on oeis.org

0, 3, 0, 3, 15, 0, 9, 87, 81, 0, 15, 513, 1173, 423, 0, 33, 3387, 18915, 17271, 2247, 0, 63, 21933, 340563, 730503, 251595, 11925, 0, 129, 141411, 6081561, 37034355, 28368687, 3669765, 63291, 0, 255, 913245, 108108231, 1844154933, 4022141121, 1100265795

Offset: 1

R. H. Hardin, Oct 27 2013

Table starts
.0......3.........3.............9...............15...................33
.0.....15........87...........513.............3387................21933
.0.....81......1173.........18915...........340563..............6081561
.0....423.....17271........730503.........37034355...........1844154933
.0...2247....251595......28368687.......4022141121.........559929294657
.0..11925...3669765....1100265795.....436827078531......169908632412447
.0..63291..53519163...42681522741...47444197636239....51564141085334199
.0.335943.780527655.1655698076319.5152946201086569.15648664251672721725

			Some solutions for n=3 k=4
..0..2..0..2....2..0..0..2....2..0..1..2....0..0..0..2....1..1..1..1
..0..0..1..1....0..1..0..2....1..1..0..1....2..2..0..1....1..2..1..2
..2..2..0..1....1..2..1..2....1..1..1..2....2..0..1..2....2..0..0..2

R. H. Hardin, Table of n, a(n) for n = 1..126

Column 2 is A229841

Row 1 is A062510(n-1)

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 5*a(n-1) +2*a(n-2) -a(n-3) -5*a(n-4)
k=3: a(n) = 14*a(n-1) +9*a(n-2) -9*a(n-3) +30*a(n-4) -19*a(n-5)
k=4: [order 32]
k=5: [order 80]
Empirical for row n:
n=1: a(n) = a(n-1) +2*a(n-2)
n=2: a(n) = 5*a(n-1) +6*a(n-2) +20*a(n-3) +12*a(n-4) +8*a(n-5)
n=3: [order 15] for n>16
n=4: [order 64] for n>65

A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

Original entry on oeis.org

1, 0, 3, 2, 9, 12, 31, 54, 117, 224, 459, 906, 1825, 3636, 7287, 14558, 29133, 58248, 116515, 233010, 466041, 932060, 1864143, 3728262, 7456549, 14913072, 29826171, 59652314, 119304657, 238609284, 477218599, 954437166, 1908874365

Offset: 0

Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000

Growth of happy bug population in GCSE math course work assignment.
The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. - Wolfdieter Lang, Jun 25 2010
With offset 1: a(n) = -2^n*Sum_{k=0..n} k^p*q^k for p=1, q=-1/2. See also A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2). - Stanislav Sykora, Nov 27 2013
From Paul Curtz, Nov 02 2021 (Start)
a(n-2) difference table (from 0, 0, a(n)):
    0    0    1    0    3    2    9    12    31    54  ...
    0    1   -1    3   -1    7    3    19    23    63  ...
    1   -2    4   -4    8   -4   16     4    40    44  ...
   -3    6   -8   12  -12   20  -12    36     4    84  ...
    9  -14   20  -24   32  -32   48   -32    80     0  ...
  -23   34  -44   56  -64   80  -80   112   -80   176  ...
   57  -78  100 -120  144 -160  192  -192   256  -192  ...
   ... .
The signature is valid for every row.
a(n-2) + a(n-1) = A001045(n).
a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).
a(n-2) + a(n+3) = see A144472(n+1).
Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).
First subdiagonal: -A036895(n) = -2*A001787(n).
Main diagonal: A001787(n) = -first and -third upper diagonals.
Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Iwan Duursma, Xiao Li, and Hsin-Po Wang, Multilinear Algebra for Distributed Storage, arXiv:2006.08911 [cs.IT], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A232603, A232604.

Cf. A001045, A001787, A001792, A014480, A036895, A062510, A045891, A172160.

CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* Michael De Vlieger, Sep 30 2019 *)

c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ Stanislav Sykora, Nov 27 2013

G.f.: 1 / (1-3*x^2-2*x^3).
With offset 1: a(1)=1; a(n) = 2*a(n-1) - (-1)^n*n; a(n) = (1/9)*(2^(n+1) - (-1)^n*(3*n+2)). - Benoit Cloitre, Nov 02 2002
a(n) = Sum_{k=0..floor(n/2)} A078008(n-2k). - Paul Barry, Nov 24 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k*(2/3)^(n-2k). - Paul Barry, Oct 16 2004
a(n) = Sum_{k=0..n} A078008(k)*(1 - (-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005
a(n) = ( 2^(n+2) + (-1)^n*(3*n+5) )/9 (see also the B. Cloitre comment above). From the o.g.f. 1/(1-3*x^2-2*x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. - Wolfdieter Lang, Jun 25 2010
From Wolfdieter Lang, Aug 26 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n > 1, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in a 08/25/10 e-mail to the author.) (End)
G.f.: Sum_{n>=0} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - Paul D. Hanna, Mar 03 2012
E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - Stefano Spezia, Sep 27 2019

More terms from James Sellers, Feb 28 2000 and Christian G. Bower, Feb 29 2000

A183554 T(n,k)=Number of nXk 0..2 arrays with each element equal to either the maximum or the minimum of its horizontal and vertical neighbors.

Original entry on oeis.org

0, 3, 3, 3, 15, 3, 9, 71, 71, 9, 15, 299, 583, 299, 15, 33, 1325, 5287, 5287, 1325, 33, 63, 5845, 46841, 95101, 46841, 5845, 63, 129, 25785, 417809, 1728533, 1728533, 417809, 25785, 129, 255, 113841, 3721925, 31363527, 63868351, 31363527, 3721925, 113841

Offset: 1

R. H. Hardin Jan 05 2011

Table starts
...0.......3..........3.............9...............15..................33
...3......15.........71...........299.............1325................5845
...3......71........583..........5287............46841..............417809
...9.....299.......5287.........95101..........1728533............31363527
..15....1325......46841.......1728533.........63868351..........2359134621
..33....5845.....417809......31363527.......2359134621........177385265619
..63...25785....3721925.....569461759......87176303417......13344696858771
.129..113841...33169105...10339213765....3221377019429....1003889179688773
.255..502523..295565367..187720469393..119037100116183...75519989933088273
.513.2218399.2633809823.3408294462361.4398697422258081.5681183354145927001

			Some solutions for 4X3
..0..0..1....0..0..1....0..0..0....0..0..1....1..0..0....1..1..0....1..1..1
..2..0..1....0..2..1....2..2..0....2..2..1....1..1..0....0..0..0....1..2..1
..2..0..0....2..2..2....2..1..2....2..1..1....0..1..0....0..0..1....1..2..1
..2..2..2....0..0..0....2..1..2....2..2..1....0..1..1....2..2..1....1..2..1

R. H. Hardin, Table of n, a(n) for n = 1..127

Column 1 is A062510(n-1)

A221967 T(n,k)=Number of -k..k arrays of length n with the sum ahead of each element differing from the sum following that element by k or less.

Original entry on oeis.org

3, 5, 9, 7, 25, 15, 9, 49, 65, 33, 11, 81, 175, 225, 63, 13, 121, 369, 833, 705, 129, 15, 169, 671, 2241, 3647, 2305, 255, 17, 225, 1105, 4961, 12609, 16513, 7425, 513, 19, 289, 1695, 9633, 34111, 73089, 73983, 24065, 1023, 21, 361, 2465, 17025, 78273, 241153

Offset: 1

R. H. Hardin Feb 01 2013

Table starts
....3.......5.........7..........9..........11...........13............15
....9......25........49.........81.........121..........169...........225
...15......65.......175........369.........671.........1105..........1695
...33.....225.......833.......2241........4961.........9633.........17025
...63.....705......3647......12609.......34111........78273........159615
..129....2305.....16513......73089......241153.......653185.......1535745
..255....7425.....73983.....419841.....1690623......5407233......14661375
..513...24065....332801....2419713....11888129.....44890625.....140355585
.1023...77825...1495039...13930497....83512319....372332545....1342437375
.2049..251905...6719489...80230401...586864641...3089205249...12843782145
.4095..815105..30195711..462012417..4123582463..25628045313..122870296575
.8193.2637825.135700481.2660655105.28975366145.212618141697.1175482548225

			Some solutions for n=6 k=4
..4...-2....4....1...-4...-1...-2....1...-2...-1....1....3....4....1...-1...-1
.-4....4...-4....0....4....4....3....2....3....2...-2...-4...-2...-3....3....3
..1...-3....3...-2...-1...-2...-3...-2...-2....2....0....3....1....2....0...-1
..0....2...-1....3...-2....0....2....2....3...-3....4....1...-3...-2...-2....1
..3...-4...-2...-3....3....3...-2....1...-1....0...-1...-3....0....3...-3...-2
..1....1....2....1...-1...-2...-1....1...-1....1....0....1....2...-4....4....2

R. H. Hardin, Table of n, a(n) for n = 1..334

Column 1 is A062510(n+1)
Column 2 is A189318
Row 2 is A016754
Row 3 is A005917(n+1)
Row 4 is A142993

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2)
k=2: a(n) = 3*a(n-1) +2*a(n-2) -4*a(n-3)
k=3: a(n) = 3*a(n-1) +8*a(n-2) -4*a(n-3) -8*a(n-4)
k=4: a(n) = 5*a(n-1) +8*a(n-2) -20*a(n-3) -8*a(n-4) +16*a(n-5)
k=5: a(n) = 5*a(n-1) +18*a(n-2) -20*a(n-3) -48*a(n-4) +16*a(n-5) +32*a(n-6)
k=6: a(n) = 7*a(n-1) +18*a(n-2) -56*a(n-3) -48*a(n-4) +112*a(n-5) +32*a(n-6) -64*a(n-7)
k=7: a(n) = 7*a(n-1) +32*a(n-2) -56*a(n-3) -160*a(n-4) +112*a(n-5) +256*a(n-6) -64*a(n-7) -128*a(n-8)
Empirical for row n:
n=1: a(n) = 2*n + 1
n=2: a(n) = 4*n^2 + 4*n + 1
n=3: a(n) = 4*n^3 + 6*n^2 + 4*n + 1
n=4: a(n) = (16/3)*n^4 + (32/3)*n^3 + (32/3)*n^2 + (16/3)*n + 1
n=5: a(n) = (20/3)*n^5 + (50/3)*n^4 + 20*n^3 + (40/3)*n^2 + (16/3)*n + 1
n=6: a(n) = (128/15)*n^6 + (128/5)*n^5 + (112/3)*n^4 + 32*n^3 + (272/15)*n^2 + (32/5)*n + 1
n=7: a(n) = (488/45)*n^7 + (1708/45)*n^6 + (2912/45)*n^5 + (602/9)*n^4 + (2072/45)*n^3 + (952/45)*n^2 + (32/5)*n + 1

A229847 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or vertically next to at least one element with value 2-x(i,j).

Original entry on oeis.org

0, 3, 3, 3, 15, 3, 9, 81, 81, 9, 15, 423, 987, 423, 15, 33, 2247, 13401, 13401, 2247, 33, 63, 11925, 178311, 453837, 178311, 11925, 63, 129, 63291, 2381469, 15269193, 15269193, 2381469, 63291, 129, 255, 335943, 31783299, 514371027, 1296261759, 514371027

Offset: 1

R. H. Hardin, Oct 01 2013

Table starts
...0......3.........3............9..............15..................33
...3.....15........81..........423............2247...............11925
...3.....81.......987........13401..........178311.............2381469
...9....423.....13401.......453837........15269193...........514371027
..15...2247....178311.....15269193......1296261759........110203382331
..33..11925...2381469....514371027....110203382331......23643209345079
..63..63291..31783299..17324310663...9366963651897....5071373978989773
.129.335943.424240833.583513743897.796197119067381.1087831682400037245

			Some solutions for n=3 k=4
..0..1..2..0....1..1..0..1....1..1..2..0....1..1..1..1....1..1..1..2
..2..1..1..1....2..2..2..1....1..2..1..1....1..0..2..0....0..2..0..0
..1..1..2..0....0..0..2..1....1..0..0..2....0..2..1..1....2..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A062510(n-1)

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2)
k=2: a(n) = 5*a(n-1) +2*a(n-2) -a(n-3) -5*a(n-4)
k=3: [order 15]
k=4: [order 53]

cp's OEIS Frontend

A014551 Jacobsthal-Lucas numbers.

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A029858 a(n) = (3^n - 3)/2.

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A051049 Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one.

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A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

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A230184 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or antidiagonally next to at least one element with value 2-x(i,j).

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A230675 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally, diagonally or antidiagonally next to at least one element with value 2-x(i,j).

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A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.