A154879 -id:A154879 - cp's OEIS frontend

A115341 a(n) = abs(A154879(n+1)).

2, 4, 0, 8, 8, 24, 40, 88, 168, 344, 680, 1368, 2728, 5464, 10920, 21848, 43688, 87384, 174760, 349528, 699048, 1398104, 2796200, 5592408, 11184808, 22369624, 44739240, 89478488, 178956968, 357913944, 715827880, 1431655768, 2863311528

Offset: 0

Roger L. Bagula, Mar 06 2006

General form: a(n)=2^n-a(n-1). - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

For n>=1, a(n) is a(n) is the number of generalized compositions of n+3 when there are i^2-2*i-1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A078008, A097073. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[2] cat [(2^(n+1)-8*(-1)^n)/3: n in [1..30]]; // G. C. Greubel, Dec 30 2017

g0[n_] = 2 - Sum[(-1)^(i + 1)/Sqrt[2]^(2*i), {i, 0, n}] f[x_] = ZTransform[g0[n], n, x] g[n_] = InverseZTransform[f[1/x], x, n] a0 = Table[Abs[g[n]], {n, 1, 25}]
k=0;lst={k};Do[k=2^n-k;AppendTo[lst, k], {n, 3, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[If[n==0, 2, (2^(n+1)-8*(-1)^n)/3], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=0,30, print1(if(n==0, 2, (2^(n+1)-8*(-1)^n)/3), ", ")) \\ G. C. Greubel, Dec 30 2017

a(n) = (2^(n+1)-8*(-1)^n)/3, n>0.
a(n) = a(n-1) + 2*a(n-2), n>2.
G.f.: 2+4*x*(1-x)/((1+x)*(1-2*x)).

Edited by the Associate Editors of the OEIS, Aug 21 2009

A155734 Binomial transform of A154879.

Original entry on oeis.org

3, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329

Offset: 0

Paul Curtz, Jan 26 2009

Binomial transform of the third differences of A001045.
The binomial transform of the first differences of A001045 is in A133494.
The binomial transform of the 2nd differences of A001045 is in A133494, with the sign of A133494(0) flipped.
The binomial transform of the p-th differences of A001045 is the number A077925(p-1) followed by A000244.

Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A154879, A078008. Essentially the same as A140429 and A000244.

read("transforms") ; A001045 := proc(n) option remember ; if n <= 1 then n; else procname(n-1)+2*procname(n-2) ; fi; end:
a001045 := [seq(A001045(n),n=0..80) ] ; a154879 := DIFF(DIFF(DIFF(a001045))) ; BINOMIAL(a154879) ; # R. J. Mathar, Jul 23 2009

From Colin Barker, Apr 05 2012: (Start)
a(n) = 3*a(n-1) for n > 1.
G.f.: (3-8*x)/(1-3*x). (End)
G.f.: (1 - 2/G(0))/x where G(k) = 1 + 2^k/(1 - 2*x/(2*x + 2^k/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012

Edited and extended by R. J. Mathar, Jul 23 2009

A007283 a(n) = 3*2^n.

Original entry on oeis.org

3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002
Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - Reinhard Zumkeller, Feb 25 2003
Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003
The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006
Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006
Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007
Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - Milan Janjic, May 10 2007
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - Jaroslav Krizek, Aug 17 2009
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012
a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - Jon Perry, Oct 10 2012
If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ...  which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013
a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - L. Edson Jeffery, Jan 11 2014
If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - Ross La Haye, Jan 19 2014
Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)
Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E.  Moreover, the best possible k is 3 * 2^(n-1).
The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016
For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - Ivan N. Ianakiev, Dec 09 2016
Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - Jeffrey Shallit, Dec 02 2019
Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - Marco Ripà, Aug 22 2022
The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023
The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - Felix Huber, Feb 15 2024
A level 1 Sierpiński triangle 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.  For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
John Elias, Illustration: 2^n+1 hexagram perimeters
Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Tanya Khovanova, Recursive Sequences
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.
Essentially same as A003945 and A042950.
Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Cf. A000079, A100540, A124508, A221718.
Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
Cf. A133466, A225546, A163511, A243071.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959.

a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)

A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

PARI
```
a(n)=3*2^n
```

a(n)=3<Charles R Greathouse IV, Oct 10 2012

def A007283(n): return 3<Chai Wah Wu, Feb 14 2023

(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019

G.f.: 3/(1-2*x).
a(n) = 2*a(n - 1), n > 0; a(0) = 3.
a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - Benoit Cloitre, Aug 20 2002
a(n) = A118416(n + 1, 2) for n > 1. - Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n) + A000079(n + 1). - Zerinvary Lajos, May 12 2007
a(n) = A000079(n)*3. - Omar E. Pol, Dec 16 2008
From Paul Curtz, Feb 05 2009: (Start)
a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)
a(n) = 2^n + 2^(n + 1). - Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n + 1, n) = A173787(n + 2, n). - Reinhard Zumkeller, Feb 28 2010
A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - Reinhard Zumkeller, Sep 01 2012
a(n) = (A000225(n) + 1)*3. - Martin Ettl, Nov 11 2012
E.g.f.: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016
A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 2016
a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - Ezhilarasu Velayutham, Sep 01 2019
a(A048672(n)) = A225546(A133466(n)). - Michel Marcus and Peter Munn, Nov 29 2019
Sum_{n>=1} 1/a(n) = 2/3. - Amiram Eldar, Oct 28 2020

A062092 a(n) = 2*a(n-1) - (-1)^n for n > 0, a(0)=2.

Original entry on oeis.org

2, 5, 9, 19, 37, 75, 149, 299, 597, 1195, 2389, 4779, 9557, 19115, 38229, 76459, 152917, 305835, 611669, 1223339, 2446677, 4893355, 9786709, 19573419, 39146837, 78293675, 156587349, 313174699, 626349397, 1252698795, 2505397589

Offset: 0

Amarnath Murthy, Jun 16 2001

Let A be the Hessenberg matrix of order n, defined by: A[1,j] = A[i,i] = 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n>=1, a(n-1) = charpoly(A,3). - Milan Janjic, Jan 24 2010

T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.

Harry J. Smith, Table of n, a(n) for n = 0..200
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.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000032, A001045, A002487, A005009, A078008, A154879, A201630.

Cf. A171160 (first differences).

[(7*2^n-(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Apr 04 2023

LinearRecurrence[{1, 2}, {2, 5}, 40] (* Jean-François Alcover, Aug 02 2021 *)

a(n) = (7*2^n - (-1)^n)/3; \\ Harry J. Smith, Aug 01 2009

[(7*2^n-(-1)^n)/3 for n in range(41)] # G. C. Greubel, Apr 04 2023

a(n) = a(n-1) + 2*a(n-2).
a(n) = (7*2^n - (-1)^n)/3.
a(n) = 2^(n+1) + A001045(n).
A002487(a(n)) = A000032(n+1).
G.f.: (2+3*x)/(1-x-2*x^2).
E.g.f.: (7*exp(2*x) - exp(-x))/3.
a(n) = Sum_{j=0..2} A001045(n-j) (sum of 3 consecutive elements of the Jacobsthal sequence). - Alexander Adamchuk, May 16 2006
From Paul Curtz, Jun 03 2022: (Start)
a(n) = A001045(n+3) - A078008(n).
a(n) = A078008(n+3) - A001045(n).
a(n) = A005009(n-1) - a(n-1) for n >= 1.
a(n) = a(n-2) + A005009(n-2) for n >= 2.
a(n) = A154879(n-2) + 3*A201630(n-2) for n >= 2. (End)

More terms from Jason Earls, Jun 18 2001

Additional comments from Michael Somos, Jun 24 2002

A155701 a(n) = (4^n + 8)/3.

Original entry on oeis.org

3, 4, 8, 24, 88, 344, 1368, 5464, 21848, 87384, 349528, 1398104, 5592408, 22369624, 89478488, 357913944, 1431655768, 5726623064, 22906492248, 91625968984, 366503875928, 1466015503704, 5864062014808, 23456248059224, 93824992236888, 375299968947544

Offset: 0

Paul Curtz, Jan 25 2009

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

[(4^n+8)/3: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

A155701 := proc(n) (4^n+8)/3 ; end: seq(A155701(n),n=0..80) ; # R. J. Mathar, Jul 23 2009

A155701[n_]:=(4^n+8)/3;Array[A155701,50,0] (* Paolo Xausa, Dec 05 2023 *)

a(n) = 3 + A002450(n).
a(n) = 5*a(n-1) - 4*a(n-2) = 4*a(n-1) - 8.
a(n) = A154879(2n) = A154890(2n).
a(n+1) - a(n) = A000302(n).
a(n+1) = 4*A047849(n) = 4*A078008(2n).
G.f.: (3-11*x)/((4*x-1)*(x-1)). - R. J. Mathar, Jul 23 2009

Edited and extended by R. J. Mathar, Jul 23 2009

A156591 First differences of A154570.

Original entry on oeis.org

2, -7, 6, -8, 4, -12, -4, -28, -36, -92, -164, -348, -676, -1372, -2724, -5468, -10916, -21852, -43684, -87388, -174756, -349532, -699044, -1398108, -2796196, -5592412, -11184804, -22369628, -44739236, -89478492, -178956964, -357913948, -715827876

Offset: 0

Paul Curtz, Feb 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 2).

LinearRecurrence[{1,2},{2,-7,6},40] (* Harvey P. Dale, Feb 01 2025 *)

a(n)+a(n+1) = -A000079(n-1), n>0.
a(n+1) = 4*(-1)^(n+1)- A154879(n).
G.f.: (1-3*x)*(2-3*x)/((1+x)*(1-2*x)). a(n) = -2^n/6+20*(-1)^n/3, n>0. - R. J. Mathar, Feb 25 2009

Edited by R. J. Mathar, Feb 25 2009

A156605 a(n) = (4^n + 20)/3.

Original entry on oeis.org

7, 8, 12, 28, 92, 348, 1372, 5468, 21852, 87388, 349532, 1398108, 5592412, 22369628, 89478492, 357913948, 1431655772, 5726623068, 22906492252, 91625968988, 366503875932, 1466015503708, 5864062014812, 23456248059228, 93824992236892, 375299968947548

Offset: 0

Paul Curtz, Feb 11 2009

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

Cf. A002450, A154879, A156591.

[(4^n+20)/3: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

(4^Range[0,40] + 20)/3 (* G. C. Greubel, Jun 25 2021 *)

[(4^n + 20)/3 for n in (0..40)] # G. C. Greubel, Jun 25 2021

a(n) = -A156591(2n+1).
a(n) = 4 + A154879(2n) = 7 + A002450(n).
a(n) = 4*a(n-1) - 20, n > 0.
G.f.: (7 - 27*x)/((1-x)*(1-4*x)). - R. J. Mathar, Feb 23 2009
E.g.f.: (1/3)*(20*exp(x) + exp(4*x)). - G. C. Greubel, Jun 25 2021

Edited and extended by R. J. Mathar, Feb 23 2009

A171160 a(n) = a(n-1) + 2*a(n-2) with a(0)=3, a(1)=4.

Original entry on oeis.org

3, 4, 10, 18, 38, 74, 150, 298, 598, 1194, 2390, 4778, 9558, 19114, 38230, 76458, 152918, 305834, 611670, 1223338, 2446678, 4893354, 9786710, 19573418, 39146838, 78293674, 156587350, 313174698, 626349398, 1252698794, 2505397590, 5010795178, 10021590358

Offset: 0

Paul Curtz, Dec 04 2009

J. Mulder, Table of n, a(n) for n = 0..2999
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A078008, A175805.
Cf. A062510, A083582, (-1)^n*A140966.
Cf. A092297, A154879.
Cf. A062092, A083595.

f[n_]:=2/(n+1);x=5;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{3,4},40] (* Harvey P. Dale, Sep 04 2013 *)

Vec(-(x+3)/((x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Feb 10 2015

a(n) = (1/3)*(2*(-1)^n + 7*2^n), with n>=0. - Paolo P. Lava, Dec 14 2009
G.f.: -(x+3) / ((x+1)*(2*x-1)). - Colin Barker, Feb 10 2015
From Paul Curtz, Jun 03 2022: (Start)
a(n) = A078008(n) + A078008(n+1) + A078008(n+2).
a(n) = 2^(n+1) + A078008(n).
a(n) = A001045(n+3) - A001045(n).
(a(n) + a(n+1) = a(n+2) - a(n) = A005009(n).)
a(n) + a(n+3) = A175805(n).
a(n) = A062510(n) + A083582(n-1) with A083582(-1) = 3.
a(n) = A092297(n) + A154879(n). (End)
a(n) = 2*A062092(n-1), for n>0; 2*a(n) = A083595(n+1). - Paul Curtz, Jun 08 2022

Edited by N. J. A. Sloane, Dec 05 2009
More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010
More terms from Max Alekseyev, Apr 24 2010

A173197 a(0)=1, a(n)= 2+2^n/6+4*(-1)^n/3, n>0.

Original entry on oeis.org

1, 1, 4, 2, 6, 6, 14, 22, 46, 86, 174, 342, 686, 1366, 2734, 5462, 10926, 21846, 43694, 87382, 174766, 349526, 699054, 1398102, 2796206, 5592406, 11184814, 22369622, 44739246, 89478486, 178956974, 357913942, 715827886, 1431655766, 2863311534, 5726623062, 11453246126, 22906492246, 45812984494, 91625968982, 183251937966, 366503875926, 733007751854

Offset: 0

Paul Curtz, Feb 12 2010

Linked to Jacobsthal numbers (expansion of tan(x), a.k.a. Zag numbers) A000182=1,2,16,272,...: a(n+1)-2a(n) = -(-1)^n*(A000182(n) mod 10) = (-1,2,-6,2,-6,2,-6,...).
Cf. A173114, related to Euler (or secant, or Zig) numbers, A000364. a(n+1)+A010684=A001045.
First differences: 0,3,-2,4,0,8,8,24,... = 0,A154879 (third differences of A001045).
Main diagonal: A003945; first upper diagonal: -A171449; second: 4*A011782.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

a(n) = A093380(n+4), n>3.
a(n) = +2*a(n-1) +a(n-2) -2*a(n-3), n>3.
G.f.: 1-x*(-1-2*x+7*x^2)/((x-1)*(2*x-1)*(1+x)).
a(2n+2)+a(2n+3)=6*A047689.
a(2n)-a(2n-2) = 3,1,2,4,8,16,... = 3,A000079.

A344109 a(n) = (52^n + 7(-1)^n)/3.

Original entry on oeis.org

4, 1, 9, 11, 29, 51, 109, 211, 429, 851, 1709, 3411, 6829, 13651, 27309, 54611, 109229, 218451, 436909, 873811, 1747629, 3495251, 6990509, 13981011, 27962029, 55924051, 111848109, 223696211, 447392429, 894784851, 1789569709, 3579139411, 7158278829, 14316557651, 28633115309, 57266230611, 114532461229, 229064922451

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Paul Curtz, May 09 2021

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 pp. 24, 26, 43.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A020714, A110286, A154879, A321421.

Cf. A014551, A156550, A084214.

LinearRecurrence[{1,2}, {4,1}, 28] (* Amiram Eldar, May 10 2021 *)

a(n+1) = 5*2^n - a(n) for n >= 0, with a(0) = 4.
a(n+2) = 5*2^n + a(n) for n >= 0, with a(0) = 4, a(1) = 1.
a(n+3) = 15*2^n - a(n) for n >= 0, with a(0) = 4, a(1) = 1, a(2) = 9.
a(n) = A001045(n+2) + A154879(n).
a(2*n+1) = A321421(n).
a(n) = a(n-1) + 2*a(n-2) for n >= 2. - Pontus von Brömssen, May 09 2021
G.f.: (4 - 3*x)/(1 - x - 2*x^2). - Stefano Spezia, May 10 2021
a(n) = 2*A014551(n) - A001045(n).
a(n) = abs(A156550(n)) - (-1)^n.
a(n+3) = a(n) + 7*A084214(n+1) for n >= 0, with a(0) = 4.
a(n) = 5*A001045(n+1) - A084214(n+1) for n >= 0.
a(n) = A084214(n+1) + 3*(-1)^n for n >= 0.

cp's OEIS Frontend

A115341 a(n) = abs(A154879(n+1)).

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A155734 Binomial transform of A154879.

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A007283 a(n) = 3*2^n.

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A062092 a(n) = 2*a(n-1) - (-1)^n for n > 0, a(0)=2.

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A155701 a(n) = (4^n + 8)/3.

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A156591 First differences of A154570.

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A156605 a(n) = (4^n + 20)/3.

A344109 a(n) = (52^n + 7(-1)^n)/3.