OFFSET
0,1
COMMENTS
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
Also the number of induced cubes in the n-Lucas cube graph. - Eric W. Weisstein, Mar 13 2026
REFERENCES
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255.
Douglas Lind and Brian Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. (General material on subshifts of finite type)
LINKS
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 Num. Theor. Disc. Math. 27(2) (2021), 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, Applic. Appl. Math. Int'l J. 17(2) (AAM 2022), Art. 15, 558-570.
James Beyer, Point Counts of Cluster Varieties of Marked Surfaces Over Finite Fields, arXiv:2608.22208 [math.AG], 2026. See p. 12.
Tricia Brown, Alexander Stokolos, and Dmitry Dmitrishin, Convolutions of linear recurrent sequences and Gegenbauer polynomials, ResearchGate (2026). See p. 2.
Paula Catarino, Helena Campos, and Paulo Vasco, On the Mersenne sequence, Annales Mathematicae et Informaticae 46 (2016), 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), 27-39.
Fatih Erduvan and Refik Keskin, Fibonacci And Lucas Numbers Which Are Product Of Two Jacobsal-Lucas Numbers [sic], Appl. Math. E-Notes 23 (2023), 60-70.
M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes Num. Theor. Disc. Math. 20(4) (2014), 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. 4(1) (2025), 23-45. See p. 24.
A. F. Horadam, Jacobsthal and Pell Curves, Fib. Quart. 26 (1988), 79-83.
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34 (1996), 40-54.
Deepika Jhala, G. P. S. Rathore, and Kiran Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish J. Anal. Num. Theory 2(4) (2014), 119-124.
Thomas Koshy and Ralph P. Grimaldi, Ternary words and Jacobsthal numbers, Fib. Quart. 55(2) (2017), 129-136.
Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, J. Int. Seq. 28 (2025), Art. 25.8.8. See p. 5.
OEIS Wiki, Autosequence.
Kritkhajohn Onphaeng and Prapanpong Pongsriiam, Jacobsthal and Jacobsthal-Lucas Numbers and Sums Introduced by Jacobsthal and Tverberg, J. Int. Seq. 20 (2017), Art. 17.3.6.
Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, and Prapanpong Pongsriiam, Inequalities for Inclusion-Exclusion-Like Sums Involving the Ceiling and the Nearest Integer Functions, Integers 25 (2025), Art. 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. Int. Seq. 4 (2001), Art. 01.2.1.
Mourad Rahmani, The Akiyama-Tanigawa matrix and related combinatorial identities, Linear Algeb. Applic. 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.
Mario Raso and Daniele Venturi, On integer sequences in cryptography, J. Math. Cryptol. 20(1) (2026), Article 20240033.
Yüksel Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Adv. Res. 20(2) (2019), Art. AIR.51824, 1-15.
Yüksal Soykan, On Summing Formulas for Horadam Numbers, Asian J. Adv. Res. Reports 8(1) (2020), 45-61.
Yüksel Soykan, Generalized Fibonacci Numbers: Sum Formulas, J. Adv. Math. Comp. Sci. 35(1) (2020), 89-104.
Yüksel Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian J. Adv. Res. Reports 9(1) (2020), Art. AJARR.55441, 23-39.
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, Arch. Curr. Res. Int'l 20(2) (2020), 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 J. Math. Sci. 4(2) (2020), 297-331.
Yüksel Soykan, On Generalized (r, s)-numbers, Int. J. Adv. Appl. Math. Mech. 8(1) (2020), 1-14.
Yüksel Soykan, Erkan Taşdemir, and İnci Okumuş, On Dual Hyperbolic Numbers With Generalized Jacobsthal Numbers Components, Zonguldak Bülent Ecevit Univ., (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 76(2) (Poland 2022), 33-44.
Elif Tan, Luka Podrug, and Vesna Iršič Chenoweth, Horadam-Lucas Cubes, Axioms 13(12) (2024), 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, Cube Polynomial.
Eric Weisstein's World of Mathematics, Jacobsthal Number.
Eric Weisstein's World of Mathematics, Lucas Cube Graph.
Wikipedia, Lucas sequence.
Volkan Yildiz, Some divisibility properties of Jacobsthal numbers, arXiv:2212.08814 [math.CO], 2022.
Abdelmoumène Zekiri, Farid Bencherif, and Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq. 21 (2018), Art. 18.5.1.
Index entries for linear recurrences with constant coefficients, signature (1,2).
FORMULA
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(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
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 = 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.
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
For n >= 1: a(n) = hypergeom([(1-n)/2, -n/2], [1-n], -8). - Victor Petrescu, Nov 06 2025
E.g.f.: exp(-x) + exp(2*x). - Stefano Spezia, Dec 21 2025
MATHEMATICA
f[n_]:=2/(n+1); x=4; Table[x=f[x]; Denominator[x], {n, 0, 5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
(* Alternative: *)
nxt[{n_, a_}]:={n+1, 2a-3(-1)^(n+1)}; Transpose[NestList[nxt, {1, 2}, 40]] [[2]] (* Harvey P. Dale, May 27 2013 *)
(* Alternative: *)
LinearRecurrence[{1, 2}, {2, 1}, 40] (* Jean-François Alcover, Jan 07 2019 *)
(* Alternative: *)
Table[2^n + (-1)^n, {n, 0, 20}] (* Eric W. Weisstein, Mar 13 2026 *)
(* Alternative: *)
CoefficientList[Series[(-2 + x)/(-1 + x + 2 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 13 2026 *)
PROG
(SageMath) [lucas_number2(n, 1, -2) for n in range(0, 32)] # Zerinvary Lajos, Apr 30 2009
(PARI) a(n)=2^n+(-1)^n \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
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
(Magma) [2^n + (-1)^n: n in [0..30]]; // G. C. Greubel, Dec 17 2017
(Python)
def A014551(n): return (1<<n)+(-1 if n&1 else 1) # Chai Wah Wu, Oct 26 2025
CROSSREFS
Cf. A006995.
KEYWORD
nonn,nice,easy,changed
AUTHOR
STATUS
approved