A265185 -id:A265185 - cp's OEIS frontend

A025192 a(0)=1; a(n) = 2*3^(n-1) for n >= 1.

1, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974

Offset: 0

Clark Kimberling

Warning: there is considerable overlap between this entry and the essentially identical A008776.
Shifts one place left when plus-convolved (PLUSCONV) with itself. a(n) = 2*Sum_{i=0..n-1} a(i). - Antti Karttunen, May 15 2001
Let M = { 0, 1, ..., 2^n-1 } be the set of all n-bit numbers. Consider two operations on this set: "sum modulo 2^n" (+) and "bitwise exclusive or" (XOR). The results of these operations are correlated.
To give a numerical measure, consider the equations over M: u = x + y, v = x XOR y and ask for how many pairs (u,v) is there a solution? The answer is exactly a(n) = 2*3^(n-1) for n >= 1. The fraction a(n)/4^n of such pairs vanishes as n goes to infinity. - Max Alekseyev, Feb 26 2003
Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+2, s(0) = 3, s(2n+2) = 3. - Herbert Kociemba, Jun 10 2004
Number of compositions of n into parts of two kinds. For a string of n objects, before the first, choose first kind or second kind; before each subsequent object, choose continue, first kind, or second kind. For example, compositions of 3 are 3; 2,1; 1,2; and 1,1,1. Using parts of two kinds, these produce respectively 2, 4, 4 and 8 compositions, 2+4+4+8 = 18. - Franklin T. Adams-Watters, Aug 18 2006
In the compositions the kinds of parts are ordered inside a run of identical parts, see example.  Replacing "ordered" by "unordered" gives A052945. - Joerg Arndt, Apr 28 2013
Number of permutations of {1, 2, ..., n+1} such that no term is more than 2 larger than its predecessor. For example, a(3) = 18 because all permutations of {1, 2, 3, 4} are valid except 1423, 1432, 2143, 3142, 2314, 3214, in which 1 is followed by 4. Proof: removing (n + 1) gives a still-valid sequence. For n >= 2, can insert (n + 1) either at the beginning or immediately following n or immediately following (n - 1), but nowhere else. Thus the number of such permutations triples when we increase the sequence length by 1. - Joel B. Lewis, Nov 14 2006
Antidiagonal sums of square array A081277. - Philippe Deléham, Dec 04 2006
Equals row sums of triangle A160760. - Gary W. Adamson, May 25 2009
Let M = a triangle with (1, 2, 4, 8, ...) as the left border and all other columns = (0, 1, 2, 4, 8, ...). A025192 = lim_{n->oo} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 27 2010
Number of nonisomorphic graded posets with 0 and uniform hasse graph of rank n with no 3-element antichain. ("Uniform" used in the sense of Retakh, Serconek and Wilson. By "graded" we mean that all maximal chains have the same length n.) - David Nacin, Feb 13 2012
Equals partial sums of A003946 prefaced with a 1: (1, 1, 4, 12, 36, 108, ...). - Gary W. Adamson, Feb 15 2012
Number of vertices (or sides) of the (n-1)-th iteration of a Gosper island. - Arkadiusz Wesolowski, Feb 07 2013
Row sums of triangle in A035002. - Jon Perry, May 30 2013
a(n) counts walks (closed) on the graph G(1-vertex; 1-loop, 1-loop, 2-loop, 2-loop, 3-loop, 3-loop, ...). - David Neil McGrath, Jan 01 2015
From Tom Copeland, Dec 03 2015: (Start)
For n > 0, a(n) are the traces of the even powers of the adjacency matrix M of the simple Lie algebra B_3, tr(M^(2n)) where M = Matrix(row 1; row 2; row 3) = Matrix[0,1,0; 1,0,2; 0,1,0], same as the traces of Matrix[0,2,0; 1,0,1; 0,1,0] (cf. Damianou). The traces of the odd powers vanish.
The characteristic polynomial of M equals determinant(x*I - M) = x^3 - 3x = A127672(3,x), so 1 - 3*x^2 = det(I - x M) = exp(-Sum_{n>=1} tr(M^n) x^n / n), implying Sum_{n>=1} a(n+1) x^(2n) / (2n) = -log(1 - 3*x^2), giving a logarithmic generating function for the aerated sequence, excluding a(0) and a(1).
a(n+1) = tr(M^(2n)), where tr(M^n) = 3^(n/2) + (-1)^n * 3^(n/2) = 2^n*(cos(Pi/6)^n + cos(5*Pi/6)^n) = n-th power sum of the eigenvalues of M = n-th power sum of the zeros of the characteristic polynomial.
The relation det(I - x M) = exp(-Sum_{n>=1} tr(M^n) x^n / n) = Sum_{n>=0} P_n(-tr(M), -tr(M^2), ..., -tr(M^n)) x^n/n! = exp(P.(-tr(M), -tr(M^2), ...)x), where P_n(x(1), ..., x(n)) are the partition polynomials of A036039 implies that with x(2n) = -tr(M^(2n)) = -a(n+1) for n > 0 and x(n) = 0 otherwise, the partition polynomials evaluate to zero except for P_2(x(1), x(2)) = P_2(0,-6) = -6.
Because of the inverse relation between the partition polynomials of A036039 and the Faber polynomials F_k(b1,b2,...,bk) of A263916, F_k(0,-3,0,0,...) = tr(M^k) gives aerated a(n), excluding n=0,1. E.g., F_2(0,-3) = -2(-3) = 6, F_4(0,-3,0,0) = 2 (-3)^2 = 18, and F_6(0,-3,0,0,0,0) = -2(-3)^3 = 54. (Cf. A265185.)
(End)
Number of permutations of length n > 0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is the largest. - Sergey Kitaev, Dec 08 2020
For n > 0, a(n) is the number of 3-colorings of the grid graph P_2 X P_(n-1). More generally, for q > 1, the number of q-colorings of the grid graph P_2 X P_n is given by q*(q - 1)*((q - 1)*(q - 2) + 1)^(n - 1). - Sela Fried, Sep 25 2023
For n > 1, a(n) is the largest solution to the equation phi(x) = a(n-1). - M. Farrokhi D. G., Oct 25 2023
Number of dotted compositions of degree n. - Diego Arcis, Feb 01 2024

			There are a(3)=18 compositions of 3 into 2 kinds of parts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:1  1:0  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:0  2:0  ]
06:  [ 1:0  2:1  ]
07:  [ 1:1  1:0  1:0  ]
08:  [ 1:1  1:0  1:1  ]
09:  [ 1:1  1:1  1:0  ]
10:  [ 1:1  1:1  1:1  ]
11:  [ 1:1  2:0  ]
12:  [ 1:1  2:1  ]
13:  [ 2:0  1:0  ]
14:  [ 2:0  1:1  ]
15:  [ 2:1  1:0  ]
16:  [ 2:1  1:1  ]
17:  [ 3:0  ]
18:  [ 3:1  ]
- _Joerg Arndt_, Apr 28 2013
G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 54*x^4 + 162*x^5 + 486*x^6 + 1458*x^7 + ...

Richard P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

T. D. Noe, Table of n, a(n) for n = 0..200
George E. Andrews and Greg Simay, Parity palindrome compositions, Integers 21, Paper No A85, (2021), 13 pp.
Diego Arcis, Camilo González, and Sebastián Márquez On the Hopf algebra of noncommutative symmetric functions in superspace, arXiv:2205.11813 [math.CO], 2022.
D. Bevan, D. Levin, P. Nugent, J. Pantone and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510.08036 [math.CO], 2015.
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 13.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Richard P. Stanley, An Equivalence Relation on the Symmetric Group and Multiplicity-free Flag h-Vectors.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Vincent Vatter, Counting parity palindrome compositions, arXiv:2109.13155 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3).

First differences of 3^n (A000244). Other self-convolved sequences: A000108, A007460, A007461, A007462, A007463, A007464, A061922.
Apart from initial term, same as A008776.
Cf. A132019, A160760, A003946, A035002.
Cf. A036039, A263916, A265185, A127672, A328778.

a025192 0 = 1
a025192 n = 2 * 3 ^ (n -1)
a025192_list = 1 : iterate (* 3) 2  -- Reinhard Zumkeller, Nov 27 2012

A025192 := proc(n): if n=0 then 1 else 2*3^(n-1) fi: end: seq(A025192(n),n=0..26);

Join[{1},2*3^(Range[30]-1)]  (* Harvey P. Dale, Mar 22 2011 *)

a(n)=max(1,2*3^(n-1)) \\ Charles R Greathouse IV, Jul 25 2011

Vec((1-x)/(1-3*x) + O(x^100)) \\ Altug Alkan, Dec 05 2015

[1]+[2*3**(n-1) for n in range(1,30)] # David Nacin, Mar 04 2012

G.f.: (1-x)/(1-3*x).
E.g.f.: (2*exp(3*x) + exp(0))/3. - Paul Barry, Apr 20 2003
a(n) = phi(3^n) = A000010(A000244(n)). - Labos Elemer, Apr 14 2003
a(0) = 1, a(n) = Sum_{k=0..n-1} (a(k) + a(n-k-1)). - Benoit Cloitre, Jun 24 2003
a(n) = A002326((3^n-1)/2). - Vladimir Shevelev, May 26 2008
a(1) = 2, a(n) = 3*a(n-1). - Vincenzo Librandi, Jan 01 2011
a(n) = lcm(a(n-1), Sum_{k=1..n-1} a(k)) for n >= 3. - David W. Wilson, Sep 27 2011
a(n) = ((2*n-1)*a(n-1) + (3*n-6)*a(n-2))/(n-1); a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 16 2012
From Sergei N. Gladkovskii, Jul 17 2012: (Start)
For the e.g.f. E(x) = (2/3)*exp(3*x) + exp(0)/3 we have
E(x) = 2*G(0)/3 where G(k) = 1 + k!/(3*(9*x)^k - 3*(9*x)^(2*k+1)/((9*x)^(k+1) + (k+1)!/G(k+1))); (continued fraction, 3rd kind, 3-step).
E(x) = 1+2*x/(G(0)-3*x) where G(k) = 3*x + 1 + k - 3*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). (End)
a(n) = A114283(0,0). - Reinhard Zumkeller, Nov 27 2012
G.f.: 1 + ((1/2)/G(0) - 1)/x where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: 1 + x*W(0), where W(k) = 1 + 1/(1 - x*(2*k+3)/(x*(2*k+4) + 1/W(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
G.f.: 1 / (1 - 2*x / (1 - x)). - Michael Somos, Apr 03 2014
Construct the power matrix T(n,j) = [A(n)^*j]*[S(n)^*(j-1)] where A(n)=(2,2,2,...) and S(n)=(0,1,0,0,...). (* is convolution operation.) Then a(n) = Sum_{j=1..n} T(n,j). - David Neil McGrath, Jan 01 2015
G.f.: 1 + 2*x/(1 + 2*x)*( 1 + 5*x/(1 + 5*x)*( 1 + 8*x/(1 + 8*x)*( 1 + 11*x/(1 + 11*x)*( 1 + .... - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 7/4. - Bernard Schott, Oct 02 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} (-1)^n/a(n) = 5/8.
Product_{n>=1} (1 - 1/a(n)) = A132019. (End)

Additional comments from Barry E. Williams, May 27 2000
a(22) corrected by T. D. Noe, Feb 08 2008
Maple programs simplified by Johannes W. Meijer, Jun 02 2011

A007070 a(n) = 4a(n-1) - 2a(n-2) with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 14, 48, 164, 560, 1912, 6528, 22288, 76096, 259808, 887040, 3028544, 10340096, 35303296, 120532992, 411525376, 1405035520, 4797091328, 16378294272, 55918994432, 190919389184, 651839567872, 2225519493120, 7598398836736, 25942556360704, 88573427769344, 302408598355968

Offset: 0

N. J. A. Sloane, Mira Bernstein, Simon Plouffe

Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 4) is "size of raises in pot-limit poker, one blind, maximum raising."
It appears that this sequence is the BinomialMean transform of A002315 - see A075271. - John W. Layman, Oct 02 2002
Number of (s(0), s(1), ..., s(2n+3)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+3, s(0) = 1, s(2n+3) = 4. - Herbert Kociemba, Jun 11 2004
a(n) = number of distinct matrix products in (A+B+C+D)^n where commutators [A,B]=[C,D]=0 but neither A nor B commutes with C or D. - Paul D. Hanna and Joshua Zucker, Feb 01 2006
The n-th term of the sequence is the entry (1,2) in the n-th power of the matrix M=[1,-1;-1,3]. - Simone Severini, Feb 15 2006
Hankel transform of this sequence is [1,-2,0,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
A204089 convolved with A000225, e.g., a(4) = 164 = (1*31 + 1*15 + 4*7 + 14*3 + 48*1) = (31 + 15 + 28 + 42 + 48). - Gary W. Adamson, Dec 23 2008
Equals INVERT transform of A000225: (1, 3, 7, 15, 31, ...). - Gary W. Adamson, May 03 2009
For n>=1, a(n-1) is the number of generalized compositions of n when there are 2^i-1 different types of the part i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Binomial transform of A078057. - R. J. Mathar, Mar 28 2011
Pisano period lengths:  1, 1, 8, 1, 24, 8, 6, 1, 24, 24, 120, 8, 168, 6, 24, 1, 8, 24, 360, 24, ... . - R. J. Mathar, Aug 10 2012
a(n) is the diagonal of array A228405. - Richard R. Forberg, Sep 02 2013
From Wolfdieter Lang, Oct 01 2013: (Start)
a(n) appears together with A106731, both interspersed with zeros, in the representation of nonnegative powers of the algebraic number rho(8) = 2*cos(Pi/8) = A179260 of degree 4, which is the length ratio of the smallest diagonal and the side in the regular octagon.
The minimal polynomial for rho(8) is C(8,x) = x^4 - 4*x^2 + 2, hence rho(8)^n = A(n+1)*1 + A(n)*rho(8) + B(n+1)*rho(8)^2 + B(n)*rho(8)^3, n >= 0, with A(2*k) = 0, k >= 0, A(1) = 1, A(2*k+1) = A106731(k-1), k >= 1, and  B(2*k) = 0, k >= 0, B(1) = 0, B(2*k+1) = a(k-1), k >= 1. See also the P. Steinbach reference given under A049310. (End)
The ratio a(n)/A006012(n) converges to 1+sqrt(2). - Karl V. Keller, Jr., May 16 2015
From Tom Copeland, Dec 04 2015: (Start)
An aerated version of this sequence is given by the o.g.f. = 1 / (1 - 4 x^2 + 2 x^4) =  1 / [x^4 a_4(1/x)] = 1 / determinant(I - x M) = exp[-log(1 -4 x + 2 x^4)], where M is the adjacency matrix for the simple Lie algebra B_4 given in A265185 with the characteristic polynomial a_4(x) = x^4 - 4 x^2 + 2 = 2 T_4(x/2) = A127672(4,x), where T denotes a Chebyshev polynomial of the first kind.
A133314 relates a(n) to the reciprocal of the e.g.f. 1 - 4 x + 4 x^2/2!. (End)
a(n) is the number of vertices of the Minkowski sum of n simplices with vertices e_(2*i+1), e_(2*i+2), e_(2*i+3), e_(2*i+4) for i=0,...,n-1, where e_i is a standard basis vector. - Alejandro H. Morales, Oct 03 2022

			a(3) = 48 = 3 * 4 + 4 + 1 + 1 = 3*a(2) + a(1) + a(0) + 1.
Example for the octagon rho(8) powers: rho(8)^4  = 2 + sqrt(2) = -2*1 + 4*rho(8)^2  = A(5)*1 + A(4)*rho(8) + B(5)*rho(8)^2 + B(4)*rho(8)^3, with a(5) = A106731(1) = -2, B(5) = a(1) = 4, A(4) = 0, B(4) = 0. - _Wolfdieter Lang_, Oct 01 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
A. Bernini, F. Disanto, R. Pinzani, and S. Rinaldi, Permutations defining convex permutominoes, J. Int. Seq. 10 (2007) # 07.9.7.
A. Burstein, S. Kitaev, and T. Mansour, Independent sets in certain classes of (almost) regular graphs, arXiv:math/0310379 [math.CO], 2003.
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. (See Cor. 3.7(e).)
Tomislav Doslic and I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276.
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022. (See Ex. 5.1)
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
A. S. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
Sela Fried, Toufik Mansour, and Mark Shattuck, Counting k-ary words by number of adjacency differences of a prescribed size, arXiv:2504.03013 [math.CO], 2025. See p. 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 440
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Index entries for sequences related to poker
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Row sums of A059474. - David W. Wilson, Aug 14 2006
Cf. A000129, A000225, A002307, A002315, A006012 (same recurrence), A007052, A075271, A078057, A106731, A179260, A204089, A228405.
Equals 2 * A003480, n>0.
Row sums of A140071.
Cf. A127672, A265185, A133314.

a007070 n = a007070_list !! n
a007070_list = 1 : 4 : (map (* 2) $ zipWith (-)
   (tail $ map (* 2) a007070_list) a007070_list)
-- Reinhard Zumkeller, Jan 16 2012

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((4+r)^(1+n)-(4-r)^(1+n))/((2^(1+n))*r): n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Vincenzo Librandi, Mar 27 2011

[n le 2 select 3*n-2 else 4*Self(n-1)-2*Self(n-2): n in [1..23]];  // Bruno Berselli, Mar 28 2011

A007070 :=proc(n) option remember; if n=0 then 1 elif n=1 then 4 else 4*procname(n-1)-2*procname(n-2); fi; end:
seq(A007070(n), n=0..30); # Wesley Ivan Hurt, Dec 06 2015

LinearRecurrence[{4,-2}, {1,4}, 30] (* Harvey P. Dale, Sep 16 2014 *)

a(n)=polcoeff(1/(1-4*x+2*x^2)+x*O(x^n),n)

a(n)=if(n<1,1,ceil((2+sqrt(2))*a(n-1)))

[lucas_number1(n,4,2) for n in range(1, 24)]# Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1 - 4*x + 2*x^2).
Preceded by 0, this is the binomial transform of the Pell numbers A000129. Its e.g.f. is then exp(2*x)*sinh(sqrt(2)*x)/sqrt(2). - Paul Barry, May 09 2003
a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/sqrt(8). - Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008, corrected Mar 28 2011
a(n) = (2 - sqrt(2))^n*(1/2 - sqrt(2)/2) + (2 + sqrt(2))^n*(1/2 + sqrt(2)/2). - Paul Barry, May 09 2003
a(n) = ceiling((2 + sqrt(2))*a(n-1)). - Benoit Cloitre, Aug 15 2003
a(n) = U(n, sqrt(2))*sqrt(2)^n. - Paul Barry, Nov 19 2003
a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)*sin(r*Pi/2)*(2*cos(r*Pi/8))^(2*n+3). - Herbert Kociemba, Jun 11 2004
a(n) = center term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [A007052(n) a(n) A007052(n)]. E.g., a(3) = 48 since M^3 * [1 1 1] = [34 48 34], where 34 = A007052(3). - Gary W. Adamson, Dec 18 2004
This is the binomial mean transform of A002307. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
a(2n) = Sum_{r=0..n} 2^(2n-1-r)*(4*binomial(2n-1,2r) + 3*binomial(2n-1,2r+1)) a(2n-1) = Sum_{r=0..n} 2^(2n-2-r)*(4*binomial(2n-2,2r) + 3*binomial(2n-2,2r+1)). - Jeffrey Liese, Oct 12 2006
a(n) = 3*a(n - 1) + a(n - 2) + a(n - 3) + ... + a(0) + 1. - Gary W. Adamson, Feb 18 2011
G.f.: 1/(1 - 4*x + 2*x^2) = 1/( x*(1 + U(0)) ) - 1/x  where U(k)= 1 - 2^k/(1 - x/(x - 2^k/U(k+1) )); (continued fraction 3rd kind, 3-step). - Sergei N. Gladkovskii, Dec 05 2012
G.f.: A(x) = G(0)/(1-2*x) where G(k) = 1 + 2*x/(1 - 2*x - x*(1-2*x)/(x + (1-2*x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 04 2013
G.f.: G(0)/(2*x) - 1/x, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - (1-x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n-1) = Sum_{k=0..n} binomial(2*n, n+k)*(k|8) where (k|8) is the Kronecker symbol. - Greg Dresden, Oct 11 2022
E.g.f.: exp(2*x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, May 20 2024

A006012 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - 2a(n-2), n >= 2.

Original entry on oeis.org

1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, 367424, 1254464, 4283008, 14623104, 49926400, 170459392, 581984768, 1987020288, 6784111616, 23162405888, 79081400320, 270000789504, 921840357376, 3147359850496

Offset: 0

N. J. A. Sloane

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 4, s(2n) = 4. - Herbert Kociemba, Jun 12 2004
a(n-1) counts permutations pi on [n] for which the pairs {i, pi(i)} with i < pi(i), considered as closed intervals [i+1,pi(i)], do not overlap; equivalently, for each i in [n] there is at most one j <= i with pi(j) > i. Counting these permutations by the position of n yields the recurrence relation. - David Callan, Sep 02 2003
a(n) is the sum of (n+1)-th row terms of triangle A140070. - Gary W. Adamson, May 04 2008
The binomial transform is in A083878, the Catalan transform in A084868. - R. J. Mathar, Nov 23 2008
Equals row sums of triangle A152252. - Gary W. Adamson, Nov 30 2008
Counts all paths of length (2*n), n >= 0, starting at the initial node on the path graph P_7, see the second Maple program. - Johannes W. Meijer, May 29 2010
From L. Edson Jeffery, Apr 04 2011: (Start)
Let U_1 and U_3 be the unit-primitive matrices (see [Jeffery])
U_1 = U_(8,1) = [(0,1,0,0); (1,0,1,0); (0,1,0,1); (0,0,2,0)] and
U_3 = U_(8,3) = [(0,0,0,1); (0,0,2,0); (0,2,0,1); (2,0,2,0)]. Then a(n) = (1/4) * Trace(U_1^(2*n)) = (1/2^(n+2)) * Trace(U_3^(2*n)). (See also A084130, A001333.) (End)
Pisano period lengths: 1, 1, 8, 1, 24, 8, 6, 1, 24, 24, 120, 8, 168, 6, 24, 1, 8, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012
a(n) is the first superdiagonal of array A228405. - Richard R. Forberg, Sep 02 2013
Conjecture: With offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). For example, the 4 permutations of [4] not counted by a(4) are 1324, 1423, 2314, 2413. - David Callan, Aug 27 2014
The conjecture of David Callan above is correct - with offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). - Yonah Biers-Ariel, Jun 27 2017
From Gary W. Adamson, Jul 22 2016: (Start)
A production matrix for the sequence is M =
  1, 1, 0, 0, 0, 0, ...
  1, 0, 3, 0, 0, 0, ...
  1, 0, 0, 3, 0, 0, ...
  1, 0, 0, 0, 3, 0, ...
  1, 0, 0, 0, 0, 3, ...
  ...
Take powers of M, extracting the upper left terms; getting the sequence starting: (1, 1, 2, 6, 20, 68, ...). (End)
From Gary W. Adamson, Jul 24 2016: (Start)
The sequence is the INVERT transform of the powers of 3 prefaced with a "1": (1, 1, 3, 9, 27, ...) and is N=3 in an infinite of analogous sequences starting:
   N=1 (A000079):  1, 2, 4,  8,  16,   32, ...
   N=2 (A001519):  1, 2, 5, 13,  34,   89, ...
   N=3 (A006012):  1, 2, 6, 20,  68,  232, ...
   N=4 (A052961):  1, 2, 7, 29, 124,  533, ...
   N=5 (A154626):  1, 2, 8, 40, 208, 1088, ...
   N=6:            1, 2, 9, 53, 326, 2017, ...
   ... (End)
Number of permutations of length n > 0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>2, 4>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first and fourth elements are larger than the second and third elements. - Sergey Kitaev, Dec 08 2020
a(n-1) is the number of permutations of [n] that can be obtained by placing n points on an X-shape (two crossing lines with slopes 1 and -1), labeling them 1,2,...,n by increasing y-coordinate, and then reading the labels by increasing x-coordinate. - Sergi Elizalde, Sep 27 2021
Consider a stack of pancakes of height n, where the only allowed operation is reversing the top portion of the stack. First, perform a series of reversals of decreasing sizes, followed by a series of reversals of increasing sizes. The number of distinct permutations of the initial stack that can be reached through these operations is a(n). - Thomas Baruchel, May 12 2025
Number of permutations of [n] that are correctly sorted after performing one left-to-right pass and one right-to-left pass of the cocktail sort. - Thomas Baruchel, May 16 2025

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. Birkhäuser, Boston, 3rd edition, 1990, p. 86.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.4.8 Answer to Exer. 8.
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
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Andrei Asinowski and Toufik Mansour, Separable d-Permutations and Guillotine Partitions, arXiv 0803.3414 [math.CO], 2008. Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; Abstract
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 155
George Balla, Ghislain Fourier, and Kunda Kambaso, PBW filtration and monomial bases for Demazure modules in types A and C, arXiv:2205.01747 [math.RT], 2022.
M. Barnabei, F. Bonetti, and M. Silimbani, Two permutation classes related to the Bubble Sort operator, Electronic Journal of Combinatorics 19(3) (2012), #P25. - From _N. J. A. Sloane_, Dec 25 2012
Yonah Biers-Ariel, A New Quantity Counted by OEIS Sequence A006012, arXiv:1706.07064 [math.CO], 2017.
Yonah Biers-Ariel, The Number of Permutations Avoiding a Set of Generalized Permutation Patterns, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.3.
Rocco Chirivì, Xin Fang, and Ghislain Fourier, Degenerate Schubert varieties in type A, Transformation Groups (2020).
CombOS - Combinatorial Object Server, Generate pattern-avoiding permutations
Sergi Elizalde, The X-class and almost-increasing permutations, arXiv:0710.5168 [math:CO], 2007. Ann. Comb. 15 (2011), 51-68.
S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Luca Ferrari, Enhancing the connections between patterns in permutations and forbidden configurations in restricted elections, arXiv:1906.10553 [math.CO], 2019.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
L. E. Jeffery, Unit-primitive matrices
Donghyun Kim and Lauren Williams, Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations, arXiv:2106.13378 [math.CO], 2021.
Sergey Kitaev and Artem Pyatkin, On permutations avoiding partially ordered patterns defined by bipartite graphs, arXiv:2204.08936 [math.CO], 2022.
Joshua Marsh and Nathan Williams, Nesting Nonpartitions, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
Markus Saers, Dekai Wu, and Chris Quirk, On the Expressivity of Linear Transductions.
Yi-dong Sun and Cang-zhi Jia, Counting Dyck paths with strictly increasing peak sequences, J. Math. Res. Expos. 27 (2) (2007) 253, Theorem 3.11.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A000079, A000129, A001333, A001519, A002203, A003480, A007052, A007070, A024175, A030436, A052961, A056236, A083978, A084130, A084868, A094803, A098158, A140070, A147703, A152252, A154626, A201730, A228405, A265185.

a006012 n = a006012_list !! n
a006012_list = 1 : 2 : zipWith (-) (tail $ map (* 4) a006012_list)
(map (* 2) a006012_list)
-- Reinhard Zumkeller, Oct 03 2011

[n le 2 select n else 4*Self(n-1)- 2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 05 2011

A006012:=-(-1+2*z)/(1-4*z+2*z**2); # Simon Plouffe in his 1992 dissertation
with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=24; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..7); od: seq(a(2*n),n=0..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{4,-2},{1,2},50] (* or *) With[{c=Sqrt[2]}, Simplify[ Table[((2+c)^n+(3+2c)(2-c)^n)/(2(2+c)),{n,50}]]] (* Harvey P. Dale, Aug 29 2011 *)

{a(n) = real(((2 + quadgen(8))^n))}; /* Michael Somos, Feb 12 2004 */

{a(n) = if( n<0, 2^n, 1) * polsym(x^2 - 4*x + 2, abs(n))[abs(n)+1] / 2}; /* Michael Somos, Feb 12 2004 */

Vec((1-2*x)/(1-4*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 05 2015

l = [1, 2]
for n in range(2, 101): l.append(4 * l[n - 1] - 2 * l[n - 2])
print(l)  # Indranil Ghosh, Jul 02 2017

A006012=BinaryRecurrenceSequence(4,-2,1,2)
print([A006012(n) for n in range(41)]) # G. C. Greubel, Aug 27 2025

G.f.: (1-2*x)/(1 - 4*x + 2*x^2).
a(n) = 2*A007052(n-1) = A056236(n)/2.
Limit_{n -> oo} a(n)/a(n-1) = 2 + sqrt(2). - Zak Seidov, Oct 12 2002
From Paul Barry, May 08 2003: (Start)
Binomial transform of A001333.
E.g.f.: exp(2*x)*cosh(sqrt(2)*x). (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*2^(n-k) = Sum_{k=0..n} binomial(n, k)*2^(n-k/2)(1+(-1)^k)/2. - Paul Barry, Nov 22 2003 (typo corrected by Manfred Scheucher, Jan 17 2023)
a(n) = ((2+sqrt(2))^n + (2-sqrt(2))^n)/2.
a(n) = [M^n]{2,2}, where M is the 3 X 3 matrix: M = [1,1,1;1,2,1;1,1,1]. - _Simone Severini, Jun 11 2006
a(n) = Sum_{k=0..n} 2^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
a(n) = A007070(n) - 2*A007070(n-1). - R. J. Mathar, Nov 16 2007
a(n) = Sum_{k=0..n} A147703(n,k). - Philippe Deléham, Nov 29 2008
a(n) = Sum_{k=0..n} A201730(n,k). - Philippe Deléham, Dec 05 2011
G.f.: G(0) where G(k)= 1 + 2*x/((1-2*x) - 2*x*(1-2*x)/(2*x + (1-2*x)*2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
G.f.: G(0)*(1-2*x)/2, where G(k) = 1 + 1/(1 - 2*x*(4*k+2-x)/( 2*x*(4*k+4-x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 27 2014
a(-n) = a(n) / 2^n for all n in Z. - Michael Somos, Aug 24 2014
a(n) = A265185(n) / 4, connecting this sequence to the simple Lie algebra B_4. - Tom Copeland, Dec 04 2015
From G. C. Greubel, Aug 27 2025: (Start)
a(n) = 2^((n-2)/2)*( (n+1 mod 2)*A002203(n) + 2*sqrt(2)*(n mod 2)*A000129(n) ).
a(n) = 2^(n/2)*ChebyshevT(n, sqrt(2)). (End)

A189315 Expansion of g.f. 5(1-3x+x^2)/(1-5x+5x^2).

Original entry on oeis.org

5, 10, 30, 100, 350, 1250, 4500, 16250, 58750, 212500, 768750, 2781250, 10062500, 36406250, 131718750, 476562500, 1724218750, 6238281250, 22570312500, 81660156250, 295449218750, 1068945312500, 3867480468750, 13992675781250, 50625976562500, 183166503906250, 662702636718750

Offset: 0

L. Edson Jeffery, Apr 20 2011

Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,1)=
(0 1 0 0 0)
(1 0 1 0 0)
(0 1 0 1 0)
(0 0 1 0 1)
(0 0 0 2 0).
Then a(n) = Trace(A^(2*n)).
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0From Tom Copeland, Dec 08 2015: (Start)
These are also the non-vanishing traces for the adjacency matrices of the simple Lie algebras B_5 and C_5. See links for B_4, A265185, and B_3, A025192.
a(n+1) = 10 * A081567(n), and, ignoring a(0), a G.F. is 10 *(1-2*x)/(1-5*x+5*x^2) whose denominator is y^5 * A127672(5,1/y) with y = sqrt(x).
-log(1 - 5x^2 + 5x^4) = 10 x^2/2 + 30 x^4/4 + ... provides a logarithmic series for the traces of both the odd and even powers of the matrix beginning  with the first power. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A147748, A189316, A189317, A189318.

Cf. A025192, A081567, A127672, A265185.

I:=[5,10,30]; [n le 3 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015

CoefficientList[Series[5(1-3x+x^2)/(1-5x+5x^2),{x,0,40}],x] (* or *)
Join[{5},LinearRecurrence[{5,-5},{10,30},40]]  (* Harvey P. Dale, Apr 25 2011 *)

Vec(5*(1-3*x+x^2)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

a(n) = 5*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=10, a(2)=30.
a(n) = Sum_{k=1..5} (w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
a(n) = 2^(1-n)*((5-Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5.
a(n) = 5*A147748(n).
E.g.f.: 1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2). - Stefano Spezia, Jul 09 2024

Showing 1-4 of 4 results.

cp's OEIS Frontend

A025192 a(0)=1; a(n) = 2*3^(n-1) for n >= 1.

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A007070 a(n) = 4*a(n-1) - 2*a(n-2) with a(0) = 1, a(1) = 4.

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A006012 a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - 2*a(n-2), n >= 2.

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A189315 Expansion of g.f. 5*(1-3*x+x^2)/(1-5*x+5*x^2).

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A007070 a(n) = 4a(n-1) - 2a(n-2) with a(0) = 1, a(1) = 4.

A006012 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - 2a(n-2), n >= 2.

A189315 Expansion of g.f. 5(1-3x+x^2)/(1-5x+5x^2).