A105723 -id:A105723 - 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

A062510 a(n) = 2^n + (-1)^(n+1).

Original entry on oeis.org

0, 3, 3, 9, 15, 33, 63, 129, 255, 513, 1023, 2049, 4095, 8193, 16383, 32769, 65535, 131073, 262143, 524289, 1048575, 2097153, 4194303, 8388609, 16777215, 33554433, 67108863, 134217729, 268435455, 536870913, 1073741823, 2147483649

Offset: 0

Jason Earls, Jun 24 2001

The identity 2 = 2^2/3 + 2^3/(3*3) - 2^4/(3*3*9) - 2^5/(3*3*9*15) + + - - can be viewed as a generalized Engel-type expansion of the number 2 to the base 2. Compare with A014551. - Peter Bala, Nov 13 2013

D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc. Boston, MA, 1976, p. 29.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A102345, A105723.

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

LinearRecurrence[{1,2},{0,3}, 30] (* or *) Table[2^n - (-1)^n, {n,0,30}] (* G. C. Greubel, Jan 15 2018 *)

PARI
```
for(n=0,22,print(2^n+(-1)^(n+1)))
```

a(n) = 3*A001045(n). - Paul Curtz, Jan 17 2008
G.f.: 3*x / ( (1+x)*(1-2*x) )
G.f.: Q(0) where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Apr 13 2013
E.g.f.: (exp(3*x) - 1)*exp(-x). - Ilya Gutkovskiy, Nov 20 2016

More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001

A085058 a(n) = A001511(n+1) + 1.

Original entry on oeis.org

2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 6, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 7, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 6, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 8, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 6, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 7, 2, 3, 2, 4, 2, 3, 2, 5, 2

Offset: 0

N. J. A. Sloane, Aug 11 2003

Number of divisors of 2n+2 of the form 2^k. - Giovanni Teofilatto, Jul 25 2007
Number of steps for iteration of map x -> (3/2)*ceiling(x) to reach an integer when started at 2*n+1.
Also number of steps for iteration of map x -> (3/2)*floor(x) to reach an integer when started at 2*n+3. - Benoit Cloitre, Sep 27 2003
The first time that a(n) = e+1 is when n is of the form 2^e - 1. - Robert G. Wilson v, Sep 28 2003
Let 2^k(n) = largest power of 2 dividing tangent number A000182(n). Then a(n-1) = 2*n - k(n). - Yasutoshi Kohmoto, Dec 23 2006
a(n) is the number of integers generated by b(i+1) = (3+2n)*(b(i) + b(i-1))/2, following these two initial values, b(0) = b(1) = 1. Thereafter only non-integers are generated. - Richard R. Forberg, Nov 09 2014
a(n) is the 2-adic valuation of 4*n+4, which is equal to the number of trailing 1-bits of 4*n+3 in binary. - Ruud H.G. van Tol, Sep 11 2023

Antti Karttunen, Table of n, a(n) for n = 0..16383
J. C. Lagarias and N. J. A. Sloane, Approximate squaring, Experimental Math., 13 (2004), 113-128.

Cf. A001511, A085060, A007814, A105723, A000182.

[Valuation(n+1, 2)+2: n in [0..100]]; // Vincenzo Librandi, Jan 16 2016

f := x->(3/2)*ceil(x); g := proc(n) local t1,c; global f; t1 := f(n); c := 1; while not type(t1, 'integer') do c := c+1; t1 := f(t1); od; RETURN([c,t1]); end;
a := n ->  A001511(n+1) + 1: A001511 := n -> padic[ordp](2*n, 2): seq(a(n), n=0..104); # Johannes W. Meijer, Dec 22 2012

g = 3 Ceiling[ # ]/2 &; f[n_?OddQ] := Length @ NestWhileList[ g, g[n], !IntegerQ[ # ] & ]; Table[ f[n], {n, 1, 210, 2}]

A085058(n)=if(n<0,0,c=2*n+7/2; x=0; while(frac(c)>0,c=3/2*floor(c); x++); x) \\ Benoit Cloitre, Sep 27 2003

A085058(n)=if(n<0,0,c=(2*n+1)*3/2; x=1; while(frac(c)>0,c=3/2*ceil(c); x++); x) \\ Benoit Cloitre, Sep 27 2003

a(n) = valuation(n+1,2)+2; \\ Michel Marcus, Jan 15 2016

def A085058(n): return (~(n+1) & n).bit_length()+2 # Chai Wah Wu, Apr 14 2023

a(n) = A007814(3^(n+1) - (-1)^(n+1)) = A007814(A105723(n+1)). - Reinhard Zumkeller, Apr 18 2005
a(n) = A001511(n+1) + 1 = A001511(2*n+2). - Ray Chandler, Jul 29 2007
a(n) = A007814(5^(n+1) - 1). - Ivan Neretin, Jan 15 2016
a(n) = A007814(4*(n+1)) = A007814(n+1) + 2. - Ruud H.G. van Tol, Sep 11 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. - Amiram Eldar, Sep 13 2024

Edited by Franklin T. Adams-Watters, Dec 09 2013

A053524 a(n) = (6^n - (-2)^n)/8.

Original entry on oeis.org

0, 1, 4, 28, 160, 976, 5824, 35008, 209920, 1259776, 7558144, 45349888, 272097280, 1632587776, 9795518464, 58773127168, 352638730240, 2115832446976, 12694994550784, 76169967566848, 457019804876800, 2742118830309376, 16452712979759104

Offset: 0

N. J. A. Sloane, Barry E. Williams, Jan 15 2000

The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.1(b).
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (4,12).

Cf. A015518.

[2^n/8*(3^n-(-1)^n): n in [0..30]]; // Vincenzo Librandi, Mar 11 2014

A053524:=n->(6^n-(-2)^n)/8; seq(A053524(n), n=0..30); # Wesley Ivan Hurt, Mar 11 2014

Table[(6^n -(-2)^n)/8, {n, 0, 30}] (* Vincenzo Librandi, Mar 11 2014 *)

a(n) = (6^n-(-2)^n)/8; \\ Joerg Arndt, Mar 11 2014

Vec(-x/((2*x+1)*(6*x-1)) + O(x^30)) \\ Colin Barker, Mar 11 2014

[lucas_number1(n,4,-12) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

E.g.f.: (exp(6*x) - exp(-2*x))/8.
a(n) = 2^(n-3) * (3^n - (-1)^n) = 2^(n-3)*A105723(n).
a(n) = 4*a(n-1) + 12*a(n-2), with a(0)=0, a(1)=1.
G.f.: x / ((1+2*x)*(1-6*x)). - Colin Barker, Mar 11 2014

A102345 a(n) = 3^n + (-1)^n.

Original entry on oeis.org

2, 2, 10, 26, 82, 242, 730, 2186, 6562, 19682, 59050, 177146, 531442, 1594322, 4782970, 14348906, 43046722, 129140162, 387420490, 1162261466, 3486784402, 10460353202, 31381059610, 94143178826, 282429536482, 847288609442

Offset: 0

Graeme McRae, Feb 16 2005

a(n) = A105723(n) + 2*(-1)^n; (a(n) + A105723(n))/2 = A000244(n). - Reinhard Zumkeller, Apr 18 2005

Michael De Vlieger, Table of n, a(n) for n = 0..2095
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,3).

Apart from leading term, same as A084182.

Cf. A000244, A046717, A105723.

Table[3^n+(-1)^n,{n,0,30}] (* or *) LinearRecurrence[{2,3},{2,2},30] (* Harvey P. Dale, Jun 19 2016 *)

[lucas_number2(n,2,-3) for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

a(n) = 2*a(n-1) + 3*a(n-2).
From Elmo R. Oliveira, Dec 18 2023: (Start)
G.f.: 2*(1-x)/((1+x)*(1-3*x)).
E.g.f.: exp(-x) + exp(3*x).
a(n) = 2*A046717(n). (End)

A164907 a(n) = (3*3^n-(-1)^n)/2.

Original entry on oeis.org

1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165

Offset: 0

Klaus Brockhaus, Aug 31 2009

Interleaving of A096053 and A083884 without initial term 1.
Partial sums are (essentially) in A080926.
First differences are (essentially) in A105723.
a(n)+a(n+1) = A008776(n+1) = A099856(n+1) = A110593(n+2).
Binomial transform of A056450. Inverse binomial transform of A164908.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,3).

Equals A046717 without initial term 1 and A080925 without initial term 0. Equals A084182 / 2 from second term onward.

Cf. A096053, A083884, A080926, A105723, A008776, A099856, A110593, A056450, A164908.

Magma
```
[ (3*3^n-(-1)^n)/2: n in [0..25] ];
```

A164907:=n->(3*3^n - (-1)^n)/2; seq(A164907(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

Table[(3*3^n - (-1)^n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 21 2014 *)
LinearRecurrence[{2,3},{1,5},50] (* Harvey P. Dale, Oct 31 2018 *)

a(n) = 2*a(n-1)+3*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1+3*x)/((1+x)*(1-3*x)).
a(n) = 3*a(n-1)+2*(-1)^n. - Carmine Suriano, Mar 21 2014

A274073 a(n) = 6^n-(-1)^n.

Original entry on oeis.org

0, 7, 35, 217, 1295, 7777, 46655, 279937, 1679615, 10077697, 60466175, 362797057, 2176782335, 13060694017, 78364164095, 470184984577, 2821109907455, 16926659444737, 101559956668415, 609359740010497, 3656158440062975, 21936950640377857, 131621703842267135

Offset: 0

Colin Barker, Jun 09 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,6).

Cf. A015540.

Sequences of the type k^n-(-1)^n: A062157 (k=0), A010673 (k=1), A062510 (k=2), A105723 (k=3), A247281 (k=4), A274072 (k=5), this sequence (k=6).

concat(0, Vec(7*x/((1+x)*(1-6*x)) + O(x^30)))

O.g.f.: 7*x/((1+x)*(1-6*x)).
E.g.f.: exp(6*x) - exp(-x).
a(n) = 5*a(n-1) + 6*a(n-2) for n>1.
a(n) = 7*A015540(n).

A274072 a(n) = 5^n-(-1)^n.

Original entry on oeis.org

0, 6, 24, 126, 624, 3126, 15624, 78126, 390624, 1953126, 9765624, 48828126, 244140624, 1220703126, 6103515624, 30517578126, 152587890624, 762939453126, 3814697265624, 19073486328126, 95367431640624, 476837158203126, 2384185791015624, 11920928955078126

Offset: 0

Colin Barker, Jun 09 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A015531.

Sequences of the type k^n-(-1)^n: A062157 (k=0), A010673 (k=1), A062510 (k=2), A105723 (k=3), A247281 (k=4), this sequence (k=5), A274073 (k=6).

LinearRecurrence[{4, 5}, {0, 6}, 30] (* Paolo Xausa, Oct 21 2024 *)

concat(0, Vec(6*x/((1+x)*(1-5*x)) + O(x^30)))

O.g.f.: 6*x/((1+x)*(1-5*x)).
E.g.f.: exp(5*x) - exp(-x).
a(n) = 4*a(n-1) + 5*a(n-2) for n>1.
a(n) = 6*A015531(n).

cp's OEIS Frontend

A014551 Jacobsthal-Lucas numbers.

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A085058 a(n) = A001511(n+1) + 1.

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A053524 a(n) = (6^n - (-2)^n)/8.

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A102345 a(n) = 3^n + (-1)^n.

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A164907 a(n) = (3*3^n-(-1)^n)/2.

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