A039301 -id:A039301 - cp's OEIS frontend

A007583 a(n) = (2^(2*n + 1) + 1)/3.

1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203, 11184811, 44739243, 178956971, 715827883, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 2932031007403, 11728124029611, 46912496118443, 187649984473771, 750599937895083

Offset: 0

Simon Plouffe

Let u(k), v(k), w(k) be the 3 sequences defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)-w(k), v(k+1)=u(k)-v(k)+w(k), w(k+1)=-u(k)+v(k)+w(k); let M(k)=Max(u(k),v(k),w(k)); then a(n)=M(2n)=M(2n-1). - Benoit Cloitre, Mar 25 2002
Also the number of words of length 2n generated by the two letters s and t that reduce to the identity 1 by using the relations ssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the dihedral group D6=C2xD3. - Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com) and John W. Layman, Jul 08 2002
Binomial transform of A025192. - Paul Barry, Apr 11 2003
Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_6. Example: a(1)=3 because in the cycle ABCDEF we have three walks of length 3 between A and B: ABAB, ABCB and AFAB. - Emeric Deutsch, Apr 01 2004
Numbers of the form 1 + Sum_{i=1..m} 2^(2*i-1). - Artur Jasinski, Feb 09 2007
Prime numbers of the form 1+Sum[2^(2n-1)] are in A000979. Numbers x such that 1+Sum[2^(2n-1)] is prime for n=1,2,...,x is A127936. - Artur Jasinski, Feb 09 2007
Related to A024493(6n+1), A131708(6n+3), A024495(6n+5). - Paul Curtz, Mar 27 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - Milan Janjic, Feb 21 2010
Number of toothpicks in the toothpick structure of A139250 after 2^n stages. - Omar E. Pol, Feb 28 2011
Numbers whose binary representation is "10" repeated (n-1) times with "11" appended on the end, n >= 1. For example 171 = 10101011 (2). - Omar E. Pol, Nov 22 2012
a(n) is the smallest number for which A072219(a(n)) = 2*n+1. - Ramasamy Chandramouli, Dec 22 2012
An Engel expansion of 2 to the base b := 4/3 as defined in A181565, with the associated series expansion 2 = b + b^2/3 + b^3/(3*11) + b^4/(3*11*43) + .... Cf. A007051. - Peter Bala, Oct 29 2013
The positive integer solution (x,y) of 3*x - 2^n*y = 1, n>=0, with smallest x is (a(n/2), 2) if n is even and (a((n-1)/2), 1) if n is odd. - Wolfdieter Lang, Feb 15 2014
The smallest positive number that requires at least n additions and subtractions of powers of 2 to be formed. See Puzzling StackExchange link. - Alexander Cooke Jul 16 2023

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 10.
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..170
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Cecilia Bebeacua, Toufik Mansour, Alexander Postnikov, and Simone Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Greg Bell, Austin Lawson, Neil Pritchard, and Dan Yasaki, On locally infinite Cayley graphs of the integers, arXiv preprint arXiv:1711.00809 [math.GT], 2017. See lambda_2.
Phillip G. Bradford, Efficient Exact Paths For Dyck and semi-Dyck Labeled Path Reachability, arXiv:1802.05239 [cs.DS], 2018.
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv:1507.04803 [math.CO], 2015.
Ernesto Estrada and José A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176 [math.CO], 2013.
Ernesto Estrada and José A. de la Pena, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 35.
Sungpyo Hong and Jin Ho Kwak, Regular fourfold coverings with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 893
Dmitry Kamenetsky, A magic grasshopper, Puzzling StackExchange, 2023.
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.11.7.
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Nathan Mihm, Optimal Addition-Subtraction Chains, Bachelor Honors Thesis, Univ. Minnesota-Twin Cities (2025). See p. 5.
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1.
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. also A006054, A006356, A005578, A075680.
Partial sums of A081294.
Cf. A002450, A004171, A007051, A083065, A083066, A083884.
Cf. A000978, A000979, A124400, A124401, A127936, A127955, A127956, A127957, A127958.
Cf. location of records in A007302.

List([0..25], n-> (2^(2*n+1) + 1)/3); # G. C. Greubel, Dec 25 2019

a007583 = (`div` 3) . (+ 1) . a004171
-- Reinhard Zumkeller, Jan 09 2013

[(2^(2*n+1) + 1)/3: n in [0..30] ]; // Vincenzo Librandi, Apr 28 2011

a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-1 od: seq(a[n], n=0..23); # Zerinvary Lajos, Feb 22 2008, with correction by K. Spage, Aug 20 2014
A007583 := proc(n)
    (2^(2*n+1)+1)/3 ;
end proc: # R. J. Mathar, Feb 19 2015

(* From Michael De Vlieger, Aug 22 2016 *)
Table[(2^(2n+1) + 1)/3, {n, 0, 23}]
Table[1 + 2Sum[4^k, {k, 0, n-1}], {n, 0, 23}]
NestList[4# -1 &, 1, 23]
Table[Sum[Binomial[n+k, 2k]/2^(k-n), {k, 0, n}], {n, 0, 23}]
CoefficientList[Series[(1-2x)/(1-5x+4x^2), {x, 0, 23}], x] (* End *)

a(n)=sum(k=-n\3,n\3,binomial(2*n+1,n+1+3*k))

a=1; for(n=1,23, print1(a,", "); a=bitor(a,3*a)) \\ K. Spage, Aug 20 2014

Vec((1-2*x)/(1-5*x+4*x^2) + O(x^30)) \\ Altug Alkan, Dec 08 2015

apply( {A007583(n)=2<<(2*n)\/3}, [0..25]) \\ M. F. Hasler, Nov 30 2021

[(2^(2*n+1) + 1)/3 for n in (0..25)] # G. C. Greubel, Dec 25 2019

a(n) = 2*A002450(n) + 1.
From Wolfdieter Lang, Apr 24 2001: (Start)
a(n) = Sum_{m = 0..n} A060920(n, m) = A002450(n+1) - 2*A002450(n).
G.f.: (1-2*x)/(1-5*x+4*x^2). (End)
a(n) = Sum_{k = 0..n} binomial(n+k, 2*k)/2^(k - n).
a(n) = 4*a(n-1) - 1, n > 0.
From Paul Barry, Mar 17 2003: (Start)
a(n) = 1 + 2*Sum_{k = 0..n-1} 4^k;
a(n) = A001045(2n+1). (End)
a(n) = A020988(n-1) + 1 = A039301(n+1) - 1 = A083584(n-1) + 2. - Ralf Stephan, Jun 14 2003
a(0) = 1; a(n+1) = a(n) * 4 - 1. - Regis Decamps (decamps(AT)users.sf.net), Feb 04 2004 (correction to lead index by K. Spage, Aug 20 2014)
a(n) = Sum_{i + j + k = n; 0 <= i, j, k <= n} (n+k)!/i!/j!/(2*k)!. - Benoit Cloitre, Mar 25 2004
a(n) = 5*a(n-1) - 4*a(n-2). - Emeric Deutsch, Apr 01 2004
a(n) = 4^n - A001045(2*n). - Paul Barry, Apr 17 2004
a(n) = 2*(A001045(n))^2 + (A001045(n+1))^2. - Paul Barry, Jul 15 2004
a(n) = left and right terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A002450(n+1) a(n)] E.g. a(3) = 43 since M^n * [1 1 1] = [43 85 43] = [a(3) A002450(4) a(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = A072197(n) - A020988(n). - Creighton Dement, Dec 31 2004
a(n) = A139250(2^n). - Omar E. Pol, Feb 28 2011
a(n) = A193652(2*n+1). - Reinhard Zumkeller, Aug 08 2011
a(n) = Sum_{k = -floor(n/3)..floor(n/3)} binomial(2*n, n+3*k)/2. - Mircea Merca, Jan 28 2012
a(n) = 2^(2*(n+1)) - A072197(n). - Vladimir Pletser, Apr 12 2014
a(n) == 2*n + 1 (mod 3). Indeed, from Regis Decamps' formula (Feb 04 2004) we have a(i+1) - a(i) == -1 (mod 3), i= 0, 1, ..., n - 1. Summing, we have a(n) - 1 == -n (mod 3), and the formula follows. - Vladimir Shevelev, May 20 2015
For n > 0 a(n) = A133494(0) + 2 * (A133494(n) + Sum_{x = 1..n - 1}Sum_{k = 0..x - 1}(binomial(x - 1, k)*(A133494(k+1) + A133494(n-x+k)))). - J. Conrad, Dec 06 2015
a(n) = Sum_{k = 0..2n} (-2)^k == 1 + Sum_{k = 1..n} 2^(2k-1). - Bob Selcoe, Aug 21 2016
E.g.f.: (1 + 2*exp(3*x))*exp(x)/3. - Ilya Gutkovskiy, Aug 21 2016
A075680(a(n)) = 1, for n > 0. - Ralf Stephan, Jun 17 2025

A020988 a(n) = (2/3)*(4^n-1).

Original entry on oeis.org

0, 2, 10, 42, 170, 682, 2730, 10922, 43690, 174762, 699050, 2796202, 11184810, 44739242, 178956970, 715827882, 2863311530, 11453246122, 45812984490, 183251937962, 733007751850, 2932031007402, 11728124029610, 46912496118442, 187649984473770, 750599937895082

Offset: 0

N. J. A. Sloane

Numbers whose binary representation is 10, n times (see A163662(n) for n >= 1). - Alexandre Wajnberg, May 31 2005
Numbers whose base-4 representation consists entirely of 2's; twice base-4 repunits. - Franklin T. Adams-Watters, Mar 29 2006
Expected time to finish a random Tower of Hanoi problem with 2n disks using optimal moves, so (since 2n is even and A010684(2n) = 1) a(n) = A060590(2n). - Henry Bottomley, Apr 05 2001
a(n) is the number of derangements of [2n + 3] with runs consisting of consecutive integers. E.g., a(1) = 10 because the derangements of {1, 2, 3, 4, 5} with runs consisting of consecutive integers are 5|1234, 45|123, 345|12, 2345|1, 5|4|123, 5|34|12, 45|23|1, 345|2|1, 5|4|23|1, 5|34|2|1 (the bars delimit the runs). - Emeric Deutsch, May 26 2003
For n > 0, also smallest numbers having in binary representation exactly n + 1 maximal groups of consecutive zeros: A087120(n) = a(n-1), see A087116. - Reinhard Zumkeller, Aug 14 2003
Number of walks of length 2n + 3 between any two diametrically opposite vertices of the cycle graph C_6. Example: a(0) = 2 because in the cycle ABCDEF we have two walks of length 3 between A and D: ABCD and AFED. - Emeric Deutsch, Apr 01 2004
From Paul Barry, May 18 2003: (Start)
Row sums of triangle using cumulative sums of odd-indexed rows of Pascal's triangle (start with zeros for completeness):
            0  0
            1  1
         1  4  4  1
      1  6 14 14  6  1
   1  8 27 49 49 27  8  1 (End)
a(n) gives the position of the n-th zero in A173732, i.e., A173732(a(n)) = 0 for all n and this gives all the zeros in A173732. - Howard A. Landman, Mar 14 2010
Smallest number having alternating bit sum -n. Cf. A065359. For n = 0, 1, ..., the last digit of a(n) is 0, 2, 0, 2, ... . - Washington Bomfim, Jan 22 2011
Number of toothpicks minus 1 in the toothpick structure of A139250 after 2^n stages. - Omar E. Pol, Mar 15 2012
For n > 0 also partial sums of the odd powers of 2 (A004171). - K. G. Stier, Nov 04 2013
Values of m such that binomial(4*m + 2, m) is odd. Cf. A002450. - Peter Bala, Oct 06 2015
For a(n) > 2, values of m such that m is two steps away from a power of 2 under the Collatz iteration. - Roderick MacPhee, Nov 10 2016
a(n) is the position of the first occurrence of 2^(n+1)-1 in A020986. See the Brillhart and Morton link, pp. 856-857. - John Keith, Jan 12 2021
a(n) is the number of monotone paths in the n-dimensional cross-polytope for a generic linear orientation. See the Black and De Loera link. - Alexander E. Black, Feb 15 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..170 from Vincenzo Librandi)
Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, On extremal cases of pop-stack sorting, Permutation Patterns (Zürich, Switzerland, 2019) [link is not very stable].
Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, Flip-sort and combinatorial aspects of pop-stack sorting, arXiv:2003.04912 [math.CO], 2020.
Peter Bala, A characterization of A002450, A020988 and A080674.
Alexander E. Black, Monotone Paths on Polytopes: Combinatorics and Optimization, Ph. D. Dissertation, Univ. Calif. Davis (2024). See p. 59.
Alexander Black and Jesús De Loera, Monotone paths on cross-polytopes, arXiv:2102.01237 [math.CO], Feb 2021
John Brillhart and Peter Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
Nobushige Kurokawa, Zeta functions over F_1, Proc. Japan Acad., 81, Ser. A (2005), 180-184. See Theorem 3 (3).
Jonathan L. Merzel, Ján Minač, Tung T. Nguyen, and Nguyên Duy Tân, On divisibility relation graphs, 2025. See p. 14.
Andrei K. Svinin, Tuenter polynomials and a Catalan triangle, arXiv:1603.05748 [math.CO], 2016. See p.3.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A020989, A047849, A108019, A295521, A020522.

[(2/3)*(4^n-1): n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

A020988 := proc(n)
    2*(4^n-1)/3 ;
end proc: # R. J. Mathar, Feb 19 2015

(2(4^Range[0, 30] - 1))/3 (* or *) LinearRecurrence[{5, -4}, {0, 2}, 30] (* Harvey P. Dale, Sep 25 2013 *)

vector(100, n, n--; (2/3)*(4^n-1)) \\ Altug Alkan, Oct 06 2015

Vec(2*z/((1-z)*(1-4*z)) + O(z^30)) \\ Altug Alkan, Oct 11 2015

def A020988(n): return (2 * ((1 << (2 * n)) - 1)) // 3 # John Reimer Morales, Aug 05 2025

(((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * )).map(2 * )).scanLeft(0: BigInt)( + ) // _Alonso del Arte, Sep 12 2019

a(n) = 4*a(n-1) + 2, a(0) = 0.
a(n) = A026644(2*n).
a(n) = A007583(n) - 1 = A039301(n+1) - 2 = A083584(n-1) + 1.
E.g.f. : (2/3)*(exp(4*x)-exp(x)). - Paul Barry, May 18 2003
a(n) = A007583(n+1) - 1 = A039301(n+2) - 2 = A083584(n) + 1. - Ralf Stephan, Jun 14 2003
G.f.: 2*x/((1-x)*(1-4*x)). - R. J. Mathar, Sep 17 2008
a(n) = a(n-1) + 2^(2n-1), a(0) = 0. - Washington Bomfim, Jan 22 2011
a(n) = A193652(2*n). - Reinhard Zumkeller, Aug 08 2011
a(n) = 5*a(n-1) - 4*a(n-2) (n > 1), a(0) = 0, a(1) = 2. - L. Edson Jeffery, Mar 02 2012
a(n) = (2/3)*A024036(n). - Omar E. Pol, Mar 15 2012
a(n) = 2*A002450(n). - Yosu Yurramendi, Jan 24 2017
From Seiichi Manyama, Nov 24 2017: (Start)
Zeta_{GL(2)/F_1}(s) = Product_{k = 1..4} (s-k)^(-b(2,k)), where Sum b(2,k)*t^k = t*(t-1)*(t^2-1). That is Zeta_{GL(2)/F_1}(s) = (s-3)*(s-2)/((s-4)*(s-1)).
Zeta_{GL(2)/F_1}(s) = Product_{n > 0} (1 - (1/s)^n)^(-A295521(n)) = Product_{n > 0} (1 - x^n)^(-A295521(n)) = (1-3*x)*(1-2*x)/((1-4*x)*(1-x)) = 1 + Sum_{k > 0} a(k-1)*x^k (x=1/s). (End)
From Oboifeng Dira, May 29 2020: (Start)
a(n) = A078008(2n+1) (second bisection).
a(n) = Sum_{k=0..n} binomial(2n+1, ((n+2) mod 3)+3k). (End)
From John Reimer Morales, Aug 04 2025: (Start)
a(n) = A000302(n) - A047849(n).
a(n) = A020522(n) + A000079(n) - A047849(n). (End)

Edited by N. J. A. Sloane, Sep 06 2006

A047849 a(n) = (4^n + 2)/3.

Original entry on oeis.org

1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, 89478486, 357913942, 1431655766, 5726623062, 22906492246, 91625968982, 366503875926, 1466015503702, 5864062014806, 23456248059222, 93824992236886, 375299968947542

Offset: 0

Clark Kimberling

Counts closed walks of length 2n at a vertex of the cyclic graph on 6 nodes C_6. - Paul Barry, Mar 10 2004
The number of closed walks of odd length of the cyclic graph is zero. See the array w(N,L) and triangle a(K,N) given in A199571 for the general case. - Wolfdieter Lang, Nov 08 2011
A. A. Ivanov conjectures that the dimension of the universal embedding of the unitary dual polar space DSU(2n,4) is a(n). - J. Taylor (jt_cpp(AT)yahoo.com), Apr 02 2004
Permutations with two fixed points avoiding 123 and 132.
Related to A024495(6n), A131708(6n+2), A024493(6n+4). First differences give A000302. - Paul Curtz, Mar 25 2008
Also the number of permutations of length n which avoid 4321 and 4123 (or 4321 and 3412, or 4123 and 3214, or 4123 and 2143). - Vincent Vatter, Aug 17 2009; minor correction by Henning Ulfarsson, May 14 2017
This sequence is related to A014916 by A014916(n) = n*a(n)-Sum_{i=0..n-1} a(i). - Bruno Berselli, Jul 27 2010, Mar 02 2012
For n >= 2, a(n) equals 2^n times the permanent of the (2n-2) X (2n-2) tridiagonal matrix with 1/sqrt(2)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
For n > 0, counts closed walks of length (n) at a vertex of a triangle with two (x2) loops at each vertex. - David Neil McGrath, Sep 11 2014
From Michel Lagneau, Feb 26 2015: (Start)
a(n) is also the sum of the largest odd divisors of the integers 1,2,3, ..., 2^n.
Proof:
All integers of the set {2^(n-1)+1, 2^(n-1)+2, ..., 2^n} are of the form 2^k(2m+1) where k and m integers. The greatest odd divisor of a such integer is 2m+1. Reciprocally, if 2m+1 is an odd integer <= 2^n, there exists a unique integer in the set {2^(n-1)+1, 2^(n-1)+2, ..., 2^n} where 2m+1 is the greatest odd divisor. Hence the recurrence relation:
   a(n) = a(n+1) + (1 + 3 + ... + 2*2^(n-1) - 1) = a(n-1) + 4^(n-1) for n >= 2.
We obtain immediately: a(n) = a(1) + 4 + ... + 4^n = (4^n+2)/3. (End)
The number of Riordan graphs of order n+1. See Cheon et al., Proposition 2.8. - Peter Bala, Aug 12 2021
Let q = 2^(2n+1) and Omega_n be the Suzuki ovoid with q^2 + 1 points. Then a(n) is the number of orbits of the finite Suzuki group Sz(q) on 3-subsets of Omega_n. Link to result in References. - Paul M. Bradley, Jun 04 2023
Also the cogrowth sequence for the 8-element dihedral group D8 = . - Sean A. Irvine, Nov 04 2024

			a(2) = 6 for the number of round trips in C_6 from the six round trips from, say, vertex no. 1: 12121, 16161, 12161, 16121, 12321 and 16561. - _Wolfdieter Lang_, Nov 08 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Jeffrey M. Barnes, Georgia Benkart, and Tom Halverson, McKay centralizer algebras. Proc. Lond. Math. Soc. (3) 112, No. 2, 375-414 (2016).
Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
Georgia Benkart and Tom Halverson, McKay Centralizer Algebras, hal-02173744 [math.CO], 2020.
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 37-51.
Pascal Caron, Jean-Gabriel Luque, and Bruno Patrou, A combinatorial approach for the state complexity of the Shuffle product, arXiv:1905.08120 [cs.FL], 2019.
Gi-Sang Cheon, Ji-Hwan Jung, Sergey Kitaev, and Seyed Ahmad Mojallal, Riordan graphs I: structural properties, Linear Algebra and its Applications, 579. pp. 89-135, Prop. 2.8. (2019).
B. N. Cooperstein and E. E. Shult, A note on embedding and generating dual polar spaces. Adv. Geom. 1 (2001), 37-48. See Conjecture 5.5.
Kittitat Iamthong, Ji-Hwan Jung, and Sergey Kitaev, Encoding labelled p-Riordan graphs by words and pattern-avoiding permutations, arXiv:2009.01410 [math.CO], 2020.
D. Kremer and W. C. Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Mathematics 268 (2003), 171-183.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002.
Wikipedia, Permutation classes avoiding two patterns of length 4.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000302, A001045, A002450, A007583, A024493, A047848, A078008, A121314, A131708, A178789, A199571.

Cf. A014916, A073724, A020988, A039301.

[(4^n+2)/3: n in [0..30]]; // Vincenzo Librandi, Dec 07 2015

(4^Range[0,30] +2)/3 (* or *) LinearRecurrence[{5,-4},{1,2},30] (* Harvey P. Dale, Nov 27 2015 *)

PARI
```
a(n)=(4^n+2)/3;
```

def A047849(n): return (pow(4, n) +2)//3
print([A047849(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

def A047849(n): return ((1 << (2 * n)) + 2) // 3 # John Reimer Morales, Aug 05 2025

n-th difference of a(n), a(n-1), ..., a(0) is 3^(n-1) for n >= 1.
From Henry Bottomley, Aug 29 2000: (Start)
a(n) = (4^n + 2)/3.
a(n) = 4*a(n-1) - 2.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = 2*A007583(n-1) = A002450(n) + 1. (End)
a(n) = A047848(1,n).
With interpolated zeros, this is (-2)^n/6 + 2^n/6 + (-1)^n/3 + 1/3. - Paul Barry, Aug 26 2003
a(n) = A007583(n) - A002450(n) = A001045(2n+1) - A001045(2n) . - Philippe Deléham, Feb 25 2004
Second binomial transform of A078008. Binomial transform of 1, 1, 3, 9, 81, ... (3^n/3 + 2*0^n/3). a(n) = A078008(2n). - Paul Barry, Mar 14 2004
G.f.: (1-3*x)/((1-x)*(1-4*x)). - Herbert Kociemba, Jun 06 2004
a(n) = Sum_{k=0..n} 2^k*A121314(n,k). - Philippe Deléham, Sep 15 2006
a(n) = (A001045(2*n+1) + 1)/2. - Paul Barry, Dec 05 2007
From Bruno Berselli, Jul 27 2010: (Start)
a(n) = (A020988(n) + 2)/2 = A039301(n+1)/2.
Sum_{i=0..n} a(i) = A073724(n). (End)
For n >= 3, a(n) equals [2, 1, 1; 1, 2, 1; 1, 1, 2]^(n - 2)*{1, 1, 2}*{1, 1, 2}. - John M. Campbell, Jul 09 2011
a(n) = Sum_{k=0..n} binomial(2*n, mod(n,3) + 3*k). - Oboifeng Dira, May 29 2020
From Elmo R. Oliveira, Dec 21 2023: (Start)
E.g.f.: (exp(x)*(exp(3*x) + 2))/3.
a(n) = A178789(n+1)/3. (End)
a(n) = A000302(n) - A020988(n). - John Reimer Morales, Aug 03 2025

New name from Charles R Greathouse IV, Dec 22 2011

A083584 a(n) = (8*4^n - 5)/3.

Original entry on oeis.org

1, 9, 41, 169, 681, 2729, 10921, 43689, 174761, 699049, 2796201, 11184809, 44739241, 178956969, 715827881, 2863311529, 11453246121, 45812984489, 183251937961, 733007751849, 2932031007401, 11728124029609, 46912496118441

Offset: 0

Paul Barry, May 01 2003

a(n) = A007583(n+1) - 2 = A020988(n) - 1 = A039301(n+2) - 3. - Ralf Stephan, Jun 14 2003

Sum of n-th row of triangle of powers of 4: 1; 4 1 4; 16 4 1 4 16; 64 16 4 1 4 16 64; .... - Philippe Deléham, Feb 24 2014

			a(0) = 1;
a(1) = 4 + 1 + 4 = 9;
a(2) = 16 + 4 + 1 + 4 + 16 = 41;
a(3) = 64 + 16 + 4 + 1 + 4 + 16 + 64 = 169; etc. - _Philippe Deléham_, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..170
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A083855.

[(8*4^n-5)/3: n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

(8 4^Range[0,30]-5)/3 (* or *) LinearRecurrence[{5,-4},{1,9},30] (* Harvey P. Dale, Oct 23 2011 *)

a(n)=(8*4^n-5)/3 \\ Charles R Greathouse IV, Oct 07 2015

print([8*4**n//3 - 1 for n in range(50)]) # Karl V. Keller, Jr., May 21 2022

a(n) = (8*4^n - 5)/3.
G.f.: (1+4*x)/((1-x)*(1-4*x)).
E.g.f.: (8*exp(4*x) - exp(x))/3.
a(0)=1, a(1)=9, a(n) = 5*a(n-1) - 4*a(n-2). - Harvey P. Dale, Oct 23 2011
a(n) = 4*a(n-1) + 5, a(0) = 1. - Philippe Deléham, Feb 24 2014
a(n+1) = 2^(2^n+1) + a(n), a(1)=1. - Ben Paul Thurston, Dec 27 2015

A106285 Number of orbits of the 3-step recursion mod n.

Original entry on oeis.org

1, 4, 3, 12, 5, 12, 9, 44, 21, 20, 25, 36, 15, 66, 15, 172, 53, 84, 21, 60, 27, 144, 23, 132, 105, 116, 183, 482, 177, 60, 91, 684, 75, 420, 45, 252, 109, 162, 45, 220, 125, 198, 265, 520, 105, 92, 2259, 516, 359, 420, 159, 884, 2867, 732, 125, 3714, 63, 1408, 59, 180

Offset: 1

T. D. Noe, May 02 2005

nonn

Consider the 3-step recursion x(k)=x(k-1)+x(k-2)+x(k-3) mod n. For any of the n^3 initial conditions x(1), x(2) and x(3) in Zn, the recursion has a finite period. Each of these n^3 vectors belongs to exactly one orbit. In general, there are only a few different orbit lengths (A106288) for each n. For instance, the orbits mod 8 have lengths of 1, 2, 4, 8, 16. Interestingly, for n=2^k and n=3^k, the number of orbits appear to be A039301 and A054879, respectively.

			Orbits for n=2: {(0,0,0)}, {(1,1,1)}, {(0,1,0), (1,0,1)} and {(0,0,1), (0,1,1), (1,1,0), (1,0,0)}

D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Cf. A015134 (orbits of Fibonacci sequences), A106286 (orbits of 4-step sequences), A106287 (orbits of 5-step sequences), A106288 (number of different orbit lengths), A106307 (n producing a simple orbit structure).

A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 5, 8, 9, 10, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 2, 5, 8, 9, 10, 17, 20, 21, 26, 29, 32, 33, 34, 33, 36, 37, 34, 37, 40, 41, 42, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 34, 37, 40, 41, 42, 49, 52, 53, 58, 61, 64, 65, 66, 65, 68

Offset: 0

Alex Ratushnyak, Jun 13 2012

Indices of 1's: A007583(n),
indices of 2's: A047849(n+1),
indices of 4's: A039301(n+2),
indices of 5's: A153643(n+3),
indices of 8's: A155701(n+2),
indices of 9's: A155701(n+2)+1 = A163868(n+2),
indices of 10's: A153643(n+4)+3^((n+1) mod 2),
indices of 13's: A039301(n+3)+3,
indices of 16's: A039301(n+3)+4,
indices of 17's: 17, 19, 27, 49, 51, 91, 177, 179, 347, 689, 691, 1371, 2737, 2739, 5467, 10929, 10931, 21851, 43697, 43699, 87387, 174769, 174771, 349531, 699057, 699059, 1398107, 2796209, 2796211, 5592411, 11184817, 11184819, 22369627, 44739249, 44739251, 89478491, ...
indices of 18's: A039301(n+3)+6,
n's such that a(n)<3: A005578, except the first term.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a:= proc(n) local i, k, m, r;
      k, m, r:= n, 3*n, 0;
      for i from 0 while (m>0 or k>0) do
        r:= r +2^i* irem(m, 2, 'm') *irem(k, 2, 'k')
      od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 22 2012

Table[BitAnd[n, 3*n], {n, 0, 68}] (* Arkadiusz Wesolowski, Jun 23 2012 *)

a(n)=bitand(n,3*n) \\ Charles R Greathouse IV, Feb 05 2013

for n in range(99):
    print(3*n & n, end=',')

A039306 Number of distinct quadratic residues mod 9^n.

Original entry on oeis.org

1, 4, 31, 274, 2461, 22144, 199291, 1793614, 16142521, 145282684, 1307544151, 11767897354, 105911076181, 953199685624, 8578797170611, 77209174535494, 694882570819441, 6253943137374964, 56285488236374671, 506569394127372034

Offset: 0

David W. Wilson

Number of distinct n-digit suffixes of base 9 squares.
From Danny Rorabaugh, Dec 15 2015: (Start)
Construct the word y_n as follows: y_0 = a; y_{n+1} is three concatenated copies of y_n, except that the middle copy is written with letters not used in y_n. For example:
y_0 = a;
y_1 = aba;
y_2 = abacdcaba;
y_3 = abacdcabaefeghgefeabacdcaba.
a(n) is the number of nonempty subwords of y_n that occur as a subword exactly once.
Let s(n, k) be the number of subwords of y_n that occur exactly 2^k times. One can show that s(n, 0) = a(n) using s(n+1, k+1) = s(n, k) + s(n, k+1), binomial(3^n+1, 2) = Sum_{k=0..n) s(n, k)*2^k, and the formulas for a(n) below.
(End)

			From _Danny Rorabaugh_, Dec 15 2015: (Start)
The squares of the numbers 0..8 are [0, 1, 4, 9, 16, 25, 36, 49, 64]. Modulo 9, these are [0, 1, 4, 0, 7, 7, 0, 4, 1]. Thus there are a(1) = 4 distinct quadratic residues module 9^1 = 9: 0, 1, 4, and 7.
There are a(2) = 31 subwords of y_2 = abacdcaba which occur in y_2 exactly once: [abac, abacd, abacdc, abacdca, abacdcab, abacdcaba, bac, bacd, bacdc, bacdca, bacdcab, bacdcaba, ac, acd, acdc, acdca, acdcab, acdcaba, cd, cdc, cdca, cdcab, cdcaba, d, dc, dca, dcab, dcaba, ca, cab, caba].
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Quadratic residues modulo k^n: A023105 (k=2), A039300 (k=3), A039301 (k=4), A039302 (k=5), A039303 (k=6), A039304 (k=7), A039305 (k=8), this sequence (k=9), A000993 (k=10).

I:=[1, 4, 31]; [n le 3 select I[n] else 9*Self(n-1)+Self(n-2)-9*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 22 2012

CoefficientList[Series[(1-6*x)/((1-x)*(1-9*x)),{x,0,30}],x] (* Vincenzo Librandi, Apr 22 2012 *)

a(n) = floor((9^n+3)*3/8).
G.f.: (1-6*x)/((1-x)*(1-9*x)). - _Colin Barker, Mar 14 2012
a(n) = 9*a(n-1) +a(n-2) -9*a(n-3). - Vincenzo Librandi, Apr 22 2012
a(n) = (5+3^(2n+1))/8 = a(n-1) + 3^(2n-1). - Danny Rorabaugh, Dec 15 2015

cp's OEIS Frontend

A007583 a(n) = (2^(2*n + 1) + 1)/3.

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A020988 a(n) = (2/3)*(4^n-1).

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A047849 a(n) = (4^n + 2)/3.

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A083584 a(n) = (8*4^n - 5)/3.

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A106285 Number of orbits of the 3-step recursion mod n.

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A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

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