A084247 -id:A084247 - cp's OEIS frontend

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

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

A112555 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies log(T) = T - I, where I is the identity matrix.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -2, 0, 1, 1, 3, 4, 2, 1, 1, -1, -4, -7, -6, -3, 0, 1, 1, 5, 11, 13, 9, 3, 1, 1, -1, -6, -16, -24, -22, -12, -4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, -1, -8, -29, -62, -86, -80, -50, -20, -5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, -1, -10, -46, -128, -239, -314, -296, -200, -95, -30, -6, 0

Offset: 0

Paul D. Hanna, Sep 21 2005

Signed version of A108561. Row sums equal A084247. The n-th unsigned row sum = A001045(n) + 1 (Jacobsthal numbers). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms.
Equals row reversal of triangle A112468 up to sign, where A112468 is the Riordan array (1/(1-x),x/(1+x)). - Paul D. Hanna, Jan 20 2006
The elements here match A108561 in absolute value, but the signs are crucial to the properties that the matrix A112555 exhibits; the main property being T^m = I + m*(T - I). This property is not satisfied by A108561. - Paul D. Hanna, Nov 10 2009
Eigensequence of the triangle = A140165. - Gary W. Adamson, Jan 30 2009
Triangle T(n,k), read by rows, given by [1,-2,0,0,0,0,0,0,0,...] DELTA [1,0,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 17 2009

			Triangle T begins:
   1;
   1,   1;
  -1,   0,   1;
   1,   1,   1,   1;
  -1,  -2,  -2,   0,   1;
   1,   3,   4,   2,   1,   1;
  -1,  -4,  -7,  -6,  -3,   0,   1;
   1,   5,  11,  13,   9,   3,   1,   1;
  -1,  -6, -16, -24, -22, -12,  -4,   0,   1;
   1,   7,  22,  40,  46,  34,  16,   4,   1,   1;
  -1,  -8, -29, -62, -86, -80, -50, -20,  -5,   0,   1;
  ...
Matrix log, log(T) = T - I, begins:
   0;
   1,  0;
  -1,  0,  0;
   1,  1,  1,  0;
  -1, -2, -2,  0,  0;
   1,  3,  4,  2,  1,  0;
  -1, -4, -7, -6, -3,  0,  0;
  ...
Matrix inverse, T^-1 = 2*I - T, begins:
   1;
  -1,  1;
   1,  0,  1;
  -1, -1, -1,  1;
   1,  2,  2,  0,  1;
  -1, -3, -4, -2, -1,  1;
  ...
where adjacent sums in row n of T^-1 gives row n+1 of T.

Paul D. Hanna, Table of n, a(n) for n = 0..1080

Cf. A112468 (reversed rows), A108561, A084247, A001045, A072547, A112556, A279006, A140165.
From Philippe Deléham, Oct 07 2009: (Start)
Sum_{k=0..n} T(n, k)*x^(n-k) = A165760(n), A165759(n), A165758(n), A165755(n), A165752(n), A165746(n), A165751(n), A165747(n), A000007(n), A000012(n), A084247(n), A165553(n), A165622(n), A165625(n), A165638(n), A165639(n), A165748(n), A165749(n), A165750(n) for x= -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
Sum_{k=0..n} T(n, k)*x^k = A166157(n), A166153(n), A166152(n), A166149(n), A166036(n), A166035(n), A091004(n+1), A077925(n), A000007(n), A165326(n), A084247(n), A165405(n), A165458(n), A165470(n), A165491(n), A165505(n), A165506(n), A165510(n), A165511(n) for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.  (End)

Clear[t]; t[0, 0] = 1; t[n_, 0] = (-1)^(Mod[n, 2]+1); t[n_, n_] = 1; t[n_, k_] /; k == n-1 := t[n, k] = Mod[n, 2]; t[n_, k_] /; 0 < k < n-1 := t[n, k] = -t[n-1, k] - t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff( polcoeff( (1+2*x+x*y)/((1-x*y)*(1+x+x*y)),n,X),k,Y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

{T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff(1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)),n,X),k,Y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

def A112555_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k+1)*prec(n+1, k) for k in (1..n+1)]
for n in (0..12): print(A112555_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 1/(1-x*y) + x/((1-x*y)*(1+x+x*y)).
The m-th matrix power T^m has the g.f.: 1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)).
Recurrence: T(n, k) = [T^-1](n-1, k) + [T^-1](n-1, k-1), where T^-1 is the matrix inverse of T.
From Peter Bala, Jun 23 2025: (Start)
T^z = exp(z*log(T)) = I + z*(T - I) for arbitrary complex z, where I is the identity array.
exp(T) = e*T. More generally, exp(z * T^u) = exp(z)*T^(u*z) = exp(z)*I + u*z*exp(z)*(T - I).
sin(z * T^u) =  sin(z)*I + u*z*cos(z)*(T - I).
cos(z * T^u) =  cos(z)*I - u*z*sin(z)*(T - I).
tan(z * T^u) =  tan(z)*I + u*z*sec(z)^2*(T - I).
Chebyshev_T(n, T^u) = I + (n^2)*u*(T - I) and
Legendre_P(n, T^u) = I + (n*(n+1)/2)*u*(T - I).
More generally, for n >= 1,
Chebyshev_T(n, z*T^u) = Chebyshev_T(n, z)*I + n*u*z*Chebyshev_U(n-1, z)*(T - I) and
Legendre_P(n, z*T^u) = Legendre_P(n, z)*I + u*Q(n, z)*(T - I), where Q(1, z) = z and Q(n, z) = n*Legendre_P(n, z) + Q(n-1, z)/z for n > 1.
All the above properties may also hold for the triangle A279006. (End)

A151575 G.f.: (1+x)/(1+x-2*x^2).

Original entry on oeis.org

1, 0, 2, -2, 6, -10, 22, -42, 86, -170, 342, -682, 1366, -2730, 5462, -10922, 21846, -43690, 87382, -174762, 349526, -699050, 1398102, -2796202, 5592406, -11184810, 22369622, -44739242, 89478486, -178956970, 357913942, -715827882, 1431655766, -2863311530, 5726623062

Offset: 0

N. J. A. Sloane, May 25 2009, based on a suggestion from Gary W. Adamson

Or, g.f. = (1+x)/((1-x)*(1-2*x)).
A signed version of A078008, which is the main entry.
[1, 0, 2, -2, 6, -10, 22, -42, 86, ...] = an operator for toothpick sequences. The sequence convolved with A151548 = toothpick sequence A139250. The sequence convolved with A151555 = toothpick sequence A153006. - Gary W. Adamson, May 25 2009

Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A078008, A077925, A084247.

Cf. A139250, A151548, A151555.

CoefficientList[Series[(1+x)/(1+x-2x^2),{x,0,40}],x] (* or *) LinearRecurrence[{-1,2},{1,0},40] (* Harvey P. Dale, May 31 2023 *)

From R. J. Mathar, Jul 08 2009: (Start)
a(n) = (2 + (-2)^n)/3 = (-1)^n*A078008(n), n>=0.
a(n) = 2*A077925(n-2), n>1. (End)
a(n) = A084247(n+1)/2. - Philippe Deléham, Sep 21 2009
G.f.: 1 + x - x*Q(0), where Q(k) = 1 + 2*x^2 - (2*k+3)*x + x*(2*k+1 - 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013

A154879 Third differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

3, -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, 5726623064

Offset: 0

Paul Curtz, Jan 16 2009

Second differences of A078008. First differences of the sequence (-1)^(n+1)*A084247(n).

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

Cf. A115341.

[(1/3)*(8*(-1)^n+2^n): n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

Differences[LinearRecurrence[{1,2},{0,1},40],3] (* or *) LinearRecurrence[ {1,2},{3,-2},40] (* Harvey P. Dale, Apr 20 2018 *)

def A154879(n): return ((1<2 else (3,-2,4)[n] # Chai Wah Wu, Apr 18 2025

a(n) + a(n+1) = A000079(n), n > 1.
a(n+3) = 8*A001045(n) = 4*A078008(n+1) = 2*A097073(n+1).
From R. J. Mathar, Jan 23 2009: (Start)
a(n) = a(n-1) + 2*a(n-2).
G.f.: (3-5*x)/((1+x)*(1-2*x)). (End)

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

Typo in A-number in formula corrected by R. J. Mathar, Feb 23 2009

A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

Original entry on oeis.org

0, 2, 0, 3, 2, 7, 10, 23, 42, 87, 170, 343, 682, 1367, 2730, 5463, 10922, 21847, 43690, 87383, 174762, 349527, 699050, 1398103, 2796202, 5592407, 11184810, 22369623, 44739242, 89478487, 178956970, 357913943, 715827882, 1431655767, 2863311530, 5726623063, 11453246122, 22906492247, 45812984490

Offset: 0

Miklos Kristof, Dec 07 2007

Partial sums of A155980 for n > 2. - Klaus Purath, Jan 30 2021

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

Cf. A005578, A099754.
Cf. A001045, A327767, A328284.
Cf. A007583, A062092, A087289, A020988 (even bisection), A163834 (odd bisection), A078008, A084247, A181565.

List([0..40], n-> (2^n+3-7*(-1)^n+3*0^n)/6); # G. C. Greubel, Sep 02 2019

a135351:=func< n | (2^n+3-7*(-1)^n+3*0^n)/6 >; [ a135351(n): n in [0..32] ]; // Klaus Brockhaus, Dec 05 2009

G(x):=x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)): f[0]:=G(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/n!,n=0..30);

Join[{0}, Table[(2^n +3 -7*(-1)^n)/6, {n,40}]] (* G. C. Greubel, Oct 11 2016 *)
LinearRecurrence[{2,1,-2},{0,2,0,3},40] (* Harvey P. Dale, Feb 13 2024 *)

a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; \\ Michel Marcus, Oct 11 2016

[(2^n+3-7*(-1)^n+3*0^n)/6 for n in (0..40)] # G. C. Greubel, Sep 02 2019

G.f.: x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)).
E.g.f.: (exp(2*x) + 3*exp(x) - 7*exp(-x) + 3)/6.
From Paul Curtz, Dec 20 2020: (Start)
a(n) + (period 2 sequence: repeat [1, -2]) = A328284(n+2).
a(n+1) + (period 2 sequence: repeat [-2, 1]) = A001045(n).
a(n+1) + (period 2 sequence: repeat [-1, 0]) = A078008(n).
a(n+1) + (period 2 sequence : repeat [-3, 2]) = -(-1)^n*A084247(n).
a(n+4) = a(n+1) + 7*A001045(n).
a(n+4) + a(n+1) = A181565(n).
a(2*n+2) + a(2*n+3) = A087289(n) = 3*A007583(n).
a(2*n+1) = A163834(n), a(2*n+2) = A020988(n). (End)

First part of definition corrected by Klaus Brockhaus, Dec 05 2009

A140642 Triangle of sorted absolute values of Jacobsthal successive differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 16, 20, 21, 22, 24, 32, 40, 42, 43, 44, 48, 64, 80, 84, 85, 86, 88, 96, 128, 160, 168, 170, 171, 172, 176, 192, 256, 320, 336, 340, 341, 342, 344, 352, 384, 512, 640, 672, 680, 682, 683, 684, 688, 704, 768, 1024, 1280, 1344, 1360

Offset: 0

Paul Curtz, Jul 08 2008

The triangle is generated from the set of Jacobsthal numbers A001045 and all the iterated differences (see A078008, A084247), taking the absolute values and sorting into natural order.

The first differences generated individually along any row of this triangle here are all in A000079.

			The triangle starts
   1;
   2,  3;
   4,  5,  6;
   8, 10, 11, 12;
  16, 20, 21, 22, 24;
The Jacobsthal sequence and its differences in successive rows start:
    0,   1,   1,   3,   5,  11,  21,  43,  85, ...
    1,   0,   2,   2,   6,  10,  22,  42,  86, ...
   -1,   2,   0,   4,   4,  12,  20,  44,  84, ...
    3,  -2,   4,   0,   8,   8,  24,  40,  88, ...
   -5,   6,  -4,   8,   0,  16,  16,  48,  80, ...
   11, -10,  12,  -8,  16,   0,  32,  32,  96, ...
  -21,  22, -20,  24, -16,  32,   0,  64,  64, ...
   43, -42,  44, -40,  48, -32,  64,   0, 128, ...
The values +-7, +-9, +-13, for example, are missing there, so 7, 9 and 13 are not in the triangle.

Cf. A000079, A003945, A078008, A084247, A113861.

maxTerm = 384; FixedPoint[(nMax++; Print["nMax = ", nMax]; jj = Table[(2^n - (-1)^n)/3, {n, 0, nMax}]; Table[Differences[jj, n], {n, 0, nMax}] // Flatten // Abs // Union // Select[#, 0 < # <= maxTerm &] &) &, nMax = 5 ] (* Jean-François Alcover, Dec 16 2014 *)

Row sums: A113861(n+2).

Edited by R. J. Mathar, Dec 05 2008

a(45)-a(58) from Stefano Spezia, Mar 12 2024

A166114 a(n) = (6-(-4)^n)/5.

Original entry on oeis.org

1, 2, -2, 14, -50, 206, -818, 3278, -13106, 52430, -209714, 838862, -3355442, 13421774, -53687090, 214748366, -858993458, 3435973838, -13743895346, 54975581390, -219902325554, 879609302222, -3518437208882, 14073748835534, -56294995342130

Offset: 0

Philippe Deléham, Oct 06 2009

Georg Fischer, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,4).

Cf. A084222, A084247.

Table[(6 - (-4)^n)/5, {n, 0, 100}] (* G. C. Greubel, Apr 24 2016 *)

a(n)=(6-(-4)^n)/5 \\ Charles R Greathouse IV, Apr 28 2016

a(n) = 4*a(n-2) - 3*a(n-1), a(0)= 1, a(1)= 2, for n>1.
a(n) = 6 - 4*a(n-1), a(0)=1.
a(n) = a(n-1) + (-4)^(n-1), a(0)=1.
G.f.: (1+5x)/(1+3x-4x^2).
E.g.f.: (6*exp(x) - exp(-4x))/5.

a(12) onward changed by Georg Fischer, May 03 2019

A140505 Second differences of Jacobsthal sequence A001045, pairs with even and odd indices swapped.

Original entry on oeis.org

2, -1, 4, 0, 12, 4, 44, 20, 172, 84, 684, 340, 2732, 1364, 10924, 5460, 43692, 21844, 174764, 87380, 699052, 349524, 2796204, 1398100, 11184812, 5592404, 44739244, 22369620, 178956972, 89478484, 715827884, 357913940, 2863311532, 1431655764, 11453246124

Offset: 0

Paul Curtz, Jun 30 2008

The second differences are -1, 2, 0, 4, 4, 12, 20, 44, ... (-1)^(n+1)*A084247(n), essentially A097073, which are listed here with -1 <=> 2, 0 <=> 4 etc. swapped in pairs.

a(n) = 4*A092808(n-2), n>1.
a(n+1) - 2a(n) = (-1)^n*A140504(n).
O.g.f.: (2+x-5x^2)/[(1+x)(1-2x)(1+2x)]. - R. J. Mathar, Jul 08 2008

Edited by R. J. Mathar, Jul 08 2008

A165553 a(n) = (3/2)*(1+(-3)^(n-1)).

Original entry on oeis.org

1, 3, -3, 15, -39, 123, -363, 1095, -3279, 9843, -29523, 88575, -265719, 797163, -2391483, 7174455, -21523359, 64570083, -193710243, 581130735, -1743392199, 5230176603, -15690529803, 47071589415, -141214768239

Offset: 0

Philippe Deléham, Sep 21 2009

a(n)/a(n-1) tends to -3.

Index entries for linear recurrences with constant coefficients, signature (-2,3).

Cf. A046717, A084247, A112555.

3/2*(1 + (-3)^(Range[0, 29] - 1)) (* or *)
LinearRecurrence[{-2, 3}, {1, 3}, 30] (* Paolo Xausa, Apr 22 2024 *)

a(0)=1, a(1)=3, a(n)=3*a(n-2)-2*a(n-1).
G.f.: (1+5x)/(1+2x-3x^2).
a(n)= Sum_{k=0..n} A112555(n,k)*2^(n-k).

A166122 a(n) = (7-(-5)^n)/6.

Original entry on oeis.org

1, 2, -3, 22, -103, 522, -2603, 13022, -65103, 325522, -1627603, 8138022, -40690103, 203450522, -1017252603, 5086263022, -25431315103, 127156575522, -635782877603, 3178914388022, -15894571940103, 79472859700522

Offset: 0

Philippe Deléham, Oct 07 2009

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

Cf. A084222, A084247, A166114.

LinearRecurrence[{-4,5},{1,2},30] (* Harvey P. Dale, Mar 10 2016 *)
Table[(7 - (-5)^n)/6, {n, 24}] (* or *)
CoefficientList[Series[(1 + 6 x)/(1 + 4 x - 5 x^2), {x, 0, 24}], x] (* Michael De Vlieger, Apr 27 2016 *)

a(n) = 5*a(n-2) - 4*a(n-1), a(0)= 1, a(1)= 2, for n>1.
a(n) = 7-5*a(n-1), a(0)=1.
a(n) = a(n-1)+(-5)^(n-1), a(0)=1.
O.g.f.: (1+6*x)/(1+4*x-5*x^2).
E.g.f.: (7*exp(x)-exp(-5*x))/6.

cp's OEIS Frontend

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

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A112555 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies log(T) = T - I, where I is the identity matrix.

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A151575 G.f.: (1+x)/(1+x-2*x^2).

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A154879 Third differences of the Jacobsthal sequence A001045.

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A135351 a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

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A140642 Triangle of sorted absolute values of Jacobsthal successive differences.

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A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.