A010684 -id:A010684 - cp's OEIS frontend

A074323 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(1,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

1, 1, 3, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Instead of listing the coefficients of the highest power of q in each nu(n), if we list the coefficients of the smallest power of q (i.e., constant terms), we get a weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n>=2, f(n)=f(n-1)+2f(n-2).

The highest powers are given by the quarter-squares A002620(n-1). - Paul Barry, Mar 11 2007

			nu(0)=1;
nu(1)=1;
nu(2)=3;
nu(3)=5+2q;
nu(4)=11+8q+6q^2;
nu(5)=21+22q+20q^2+14q^3+4q^4;
nu(6)=43+60q+70q^2+64q^3+54q^4+28q^5+12q^6;
by listing the coefficients of the highest power in each nu(n), we get 1,1,3,2,6,4,12,...

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A001045.

Join[{1}, LinearRecurrence[{0, 2}, {1, 3}, 41]] (* Jean-François Alcover, Sep 22 2017 *)

For given b and lambda, the recurrence relation is given by; t(0)=1, t(1)=b, t(2)=b^2+lambda and for n>=3, t(n)=lambda*t(n-2).
G.f.: (1+x+x^2)/(1-2*x^2); a(n)=2^floor(n/2)+2^((n-2)/2)*(1+(-1)^n)/2-0^n/2. - Paul Barry, Mar 11 2007
a(0)=0, a(2n+1) = A000079, a(2n+2) = 3a(2n+1). a(2n)-a(2n+1) = A131577. - Paul Curtz, Mar 05 2008
a(2n+1) = 2^n = A000079(n), a(2n+2) = 3*A000079(n). Also a(2n)-a(2n+1) = A131577. a(2n+1)-a(2n)=2^n for n>0. - Paul Curtz, Apr 09 2008
a(n+1) = A010684(n)*A016116(n). - R. J. Mathar, Jul 08 2009

More terms from Paul Barry, Mar 11 2007

A126116 a(n) = a(n-1) + a(n-3) + a(n-4), with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 7, 11, 19, 31, 49, 79, 129, 209, 337, 545, 883, 1429, 2311, 3739, 6051, 9791, 15841, 25631, 41473, 67105, 108577, 175681, 284259, 459941, 744199, 1204139, 1948339, 3152479, 5100817, 8253295, 13354113, 21607409, 34961521

Offset: 0

Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007

nonn

This sequence has the same growth rate as the Fibonacci sequence, since x^4 - x^3 - x - 1 has the real roots phi and -1/phi.

The Ca1 sums, see A180662 for the definition of these sums, of triangle A035607 equal the terms of this sequence without the first term. - Johannes W. Meijer, Aug 05 2011

			G.f. = 1 + x + x^2 + x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 11*x^7 + 19*x^8 + 31*x^9 + ...

S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002

Seiichi Manyama, Table of n, a(n) for n = 0..4786
K. T. Atanassov, D. R. Deford, A. G. Shannon, Pulsated Fibonacci recurrences, Fibonacci Quarterly, Vol. 52, No. 5, Dec. 2014, pp. 22-27.
Kelley L. Ross, The Golden Ratio and The Fibonacci Numbers
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden Ratio
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. Fibonacci numbers A000045; Lucas numbers A000032; tribonacci numbers A000213; tetranacci numbers A000288; pentanacci numbers A000322; hexanacci numbers A000383; 7th-order Fibonacci numbers A060455; octanacci numbers A079262; 9th-order Fibonacci sequence A127193; 10th-order Fibonacci sequence A127194; 11th-order Fibonacci sequence A127624, A128429.

a:=[1,1,1,1];; for n in [5..50] do a[n]:=a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jul 15 2019

[n le 4 select 1 else Self(n-1) + Self(n-3) + Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2015

# From R. J. Mathar, Jul 22 2010: (Start)
A010684 := proc(n) 1+2*(n mod 2) ; end proc:
A000032 := proc(n) coeftayl((2-x)/(1-x-x^2),x=0,n) ; end proc:
A126116 := proc(n) ((-1)^floor(n/2)*A010684(n)+2*A000032(n))/5 ; end proc: seq(A126116(n),n=0..80) ; # (End)
with(combinat): A126116 := proc(n): fibonacci(n-1) + fibonacci(floor((n-4)/2)+1)* fibonacci(ceil((n-4)/2)+2) end: seq(A126116(n), n=0..38); # Johannes W. Meijer, Aug 05 2011

LinearRecurrence[{1,0,1,1},{1,1,1,1},50] (* Harvey P. Dale, Nov 08 2011 *)

Vec((x-1)*(1+x+x^2)/((x^2+x-1)*(x^2+1)) + O(x^50)) \\ Altug Alkan, Dec 25 2015

((1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 15 2019

From R. J. Mathar, Jul 22 2010: (Start)
G.f.: (1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2)).
a(n) = ( (-1)^floor(n/2) * A010684(n) + 2*A000032(n))/5.
a(2*n) = A061646(n). (End)
From Johannes W. Meijer, Aug 05 2011: (Start)
a(n) = F(n-1) + A070550(n-4) with F(n) = A000045(n).
a(n) = F(n-1) + F(floor((n-4)/2) + 1)*F(ceiling((n-4)/2) + 2). (End)
a(n) = (1/5)*((sqrt(5)-1)*(1/2*(1+sqrt(5)))^n - (1+sqrt(5))*(1/2*(1-sqrt(5)))^n + sin((Pi*n)/2) - 3*cos((Pi*n)/2)). - Harvey P. Dale, Nov 08 2011
(-1)^n * a(-n) = a(n) = F(n) - A070550(n - 6). - Michael Somos, Feb 05 2012
a(n)^2 + 3*a(n-2)^2 + 6*a(n-5)^2 + 3*a(n-7)^2 = a(n-8)^2 + 3*a(n-6)^2 + 6*a(n-3)^2 + 3*a(n-1)^2. - Greg Dresden, Jul 07 2021
a(n) = A293411(n)-A293411(n-1). - R. J. Mathar, Jul 20 2025

Edited by Don Reble, Mar 09 2007

A153643 Jacobsthal numbers A001045 incremented by 2.

Original entry on oeis.org

2, 3, 3, 5, 7, 13, 23, 45, 87, 173, 343, 685, 1367, 2733, 5463, 10925, 21847, 43693, 87383, 174765, 349527, 699053, 1398103, 2796205, 5592407, 11184813, 22369623, 44739245, 89478487, 178956973, 357913943, 715827885, 1431655767, 2863311533, 5726623063

Offset: 0

Paul Curtz, Dec 30 2008

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. A001045, A007395, A010684, A078008, A128209.

a:=[2,3,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Apr 02 2019

I:=[2,3,3]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) -2*Self(n-3): n in [1..40]]; // G. C. Greubel, Apr 02 2019

LinearRecurrence[{1,2},{0,1}, 40] + 2 (* Harvey P. Dale, May 26 2014 *)
LinearRecurrence[{2,1,-2},{2,3,3}, 40] (* Georg Fischer, Apr 02 2019 *)

my(x='x+O('x^40)); Vec( (2-x-5*x^2)/((1-x^2)*(1-2*x)) ) \\ G. C. Greubel, Apr 02 2019

def A153643(n): return ((1<Chai Wah Wu, Apr 18 2025

((2-x-5*x^2)/((1-x^2)*(1-2*x))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 02 2019

a(n) = 2 + A001045(n) = A001045(n) + A007395(n) = 1 + A128209(n).
a(n) - A010684(n) = A078008(n), first differences of A001045. - Paul Curtz, Jan 17 2009
G.f.: (2 - x - 5*x^2)/((1+x)*(1-x)*(1-2*x)). - R. J. Mathar, Jan 23 2009
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n >= 3. - Andrew Howroyd, Feb 26 2018

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

A166956 a(n) = 2^n +(-1)^n - 2.

Original entry on oeis.org

0, -1, 3, 5, 15, 29, 63, 125, 255, 509, 1023, 2045, 4095, 8189, 16383, 32765, 65535, 131069, 262143, 524285, 1048575, 2097149, 4194303, 8388605, 16777215, 33554429, 67108863, 134217725, 268435455, 536870909, 1073741823, 2147483645, 4294967295, 8589934589

Offset: 0

Paul Curtz, Oct 25 2009

The inverse binomial transform yields 0,-1,5,-7,17,-31,..., a sign alternating variant of A014551.
In a table of a(n) and higher-order differences in successive rows, the main diagonal contains 0, 4, 8, 16, ... (zero followed by A020707).
Similar to the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero, which begins with 1,3,5,15,29,63,125. - Robert Price, Aug 08 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..240
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,1,-2)

Cf. A166920, A166956, A290660, A290661, A290662.

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

LinearRecurrence[{2,1,-2},{0,-1,3},20] (* G. C. Greubel, May 29 2016 *)

a(n) = A000079(n) - A010684(n).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
G.f.: x*(5*x -1)/((1-x)*(1-2*x)*(1+x)).
E.g.f.: exp(2*x) - 2*exp(x) + exp(-x). - G. C. Greubel, May 29 2016

Edited and extended by R. J. Mathar, Mar 02 2010

A176014 Decimal expansion of (3+sqrt(21))/6.

Original entry on oeis.org

1, 2, 6, 3, 7, 6, 2, 6, 1, 5, 8, 2, 5, 9, 7, 3, 3, 3, 4, 4, 3, 1, 3, 4, 1, 1, 9, 8, 9, 5, 4, 6, 6, 8, 0, 8, 1, 4, 9, 7, 4, 0, 9, 4, 2, 9, 4, 6, 1, 3, 2, 8, 6, 5, 0, 4, 3, 4, 5, 4, 0, 3, 5, 3, 9, 8, 4, 4, 7, 8, 0, 7, 0, 9, 2, 4, 6, 2, 8, 4, 8, 1, 1, 0, 0, 7, 2, 6, 9, 2, 6, 5, 8, 2, 2, 4, 0, 8, 3, 8, 7, 7, 9, 6, 0

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(21))/6 is A010684.

Also greatest eigenvalue of the 6 X 6 matrix [[3 0 0 3 0 0][0 0 0 0 1 0][0 3 0 0 3 0][0 0 0 0 1 0][0 0 3 0 0 3][0 0 0 0 1 0]]/3. It is conjectured that this is lim_{k->infinity} A005186(k+1)/A005186(k), i.e., the asymptotic growth rate of the number of numbers with the same total stopping time in the Collatz iteration. - Hugo Pfoertner, Sep 28 2020

			(3+sqrt(21))/6 = 1.26376261582597333443...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Ratio of successive terms in A005186, illustration of deviation from (3+sqrt(21))/6.
Index entries for algebraic numbers, degree 2.

Cf. A010477 (decimal expansion of sqrt(21)).
Cf. A010684 (repeat 1, 3), A136210, A136211.
Cf. A005186, A006577, A127824, A176866.

RealDigits[(3+Sqrt[21])/6,10,120][[1]] (* Harvey P. Dale, Jul 21 2023 *)

vecmax(mateigen([1,0,0,1,0,0; 0,0,0,0,1/3,0; 0,1,0,0,1,0; 0,0,0,0,1/3,0; 0,0,1,0,0,1; 0,0,0,0,1/3,0],1)[1]) \\ Hugo Pfoertner, Sep 28 2020

A176040 Periodic sequence: Repeat 3, 1.

Original entry on oeis.org

3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3

Offset: 0

Klaus Brockhaus, Apr 07 2010

Interleaving of A010701 and A000012.
Also continued fraction expansion of (3+sqrt(21))/2.
Also decimal expansion of 31/99.
Essentially first differences of A014601.
Inverse binomial transform of 3 followed by A020707.
Second inverse binomial transform of A052919 without initial term 2.
Third inverse binomial transform of A007582 without initial term 1.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + ... is the o.g.f. for A008619. - Peter Bala, Mar 13 2015

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A153284, A010701 (all 3's sequence), A000012 (all 1's sequence), A090458 (decimal expansion of (3+sqrt(21))/2), A010684 (repeat 1, 3), A014601 (congruent to 0 or 3 mod 4), A020707 (2^(n+2)), A052919, A007582 (2^(n-1)*(1+2^n)), A008619.

&cat[ [3, 1]: n in [0..52] ];
[ 2+(-1)^n: n in [0..104] ];

PadRight[{},120,{3,1}] (* or *) LinearRecurrence[{0,1},{3,1},120] (* Harvey P. Dale, Mar 11 2015 *)

a(n) = 2+(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 1.
a(n) = -a(n-1)+4 for n > 0; a(0) = 3.
a(n) = 3*((n+1) mod 2)+(n mod 2).
a(n) = A010684(n+1).
G.f.: (3+x)/((1-x)*(1+x)).
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 3, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+2^(1-s)). (End)

A332439 Primitive period of the partial sums of the periodic unsigned Schick sequence for N = 7 (A130794), taken modulo 14, and the related Euler tour using all regular 14-gon vertices.

Original entry on oeis.org

0, 1, 6, 9, 10, 1, 4, 5, 10, 13, 0, 5, 8, 9, 0, 3, 4, 9, 12, 13, 4, 7, 8, 13, 2, 3, 8, 11, 12, 3, 6, 7, 12, 1, 2, 7, 10, 11, 2, 5, 6, 11

Offset: 0

Wolfdieter Lang, Apr 04 2020

The unsigned Schick sequences SBBseq(N, q0) (BB for Brändli and Beyne) are defined by the recurrence q_j(N) = |N - 2*q_{j-1}(N)|, for odd N >= 3, and j >= 1, with certain initial odd values q0 with gcd(q0, N) = 1. They are periodic with primitive period length ppl(n) = A003558((N-1)/2) (pes = pes(N) in Schick's book). One starts with initial value q0 = 1. If not all odd elements of the smallest positive reduced residue system modulo N (RRSodd(N)) are present then one takes the smallest missing odd number as next initial value q0, etc., until all elements of RRSodd(N) have been reached. The number of necessary distinct initial values q0 is A135303((N-1)/2) (called B = B(N) by Schick).
The present sequence is the instance N = 7. SBBseq(7, 1) = repeat(1,5,3,) = A130794, ppl(7) = 3, RRSodd(7)= {1, 3, 5}, B(7) = 1. The start has been taken as a(0) = 0 with offset 0 (not a(42) = 0, with offset 1). The primitive period length of SBBseq(7, 1) is (2*7)*3 = 42.
The primitive periods of the B(N) sequences, each of length ppl(N), define the set SBB(N). The entries of these periods can be interpreted as odd vertex labels in a regular 2*N-gon, with start vertex V^{(2*N)})_{0} (in Cartesian coordinates (r, 0), with r the radius of the circumscribing circle). The other vertices V^{(2*N)}_k, for k = 1..2*N-1, are taken in the positive (counterclockwise) sense.
In the present case N = 7 there are three directed diagonals (arrows) d_1 = s = arrow(V14_0, V14_1) of length r*0.4450418..., d_3 = arrow(V14_0, V14_3) of length r*1.2469796..., and d_5 = arrow(V14_0, V14_5), of length r*1.8019377... (V14 is shorthand for a 14-gon vertex, and s is the side).
A trail with these three arrows (considered as free vectors) in the order d1, d5, and d3, with tails following the present sequence, leads to an Euler tour, involving all vertices of the 14-gon, each visited thrice. Therefore this is a simple regular digraph with 14 vertices each of degree 6 (out-degree = in-degree = 3), and 42 arrows each used once in the tour from V14_0 to V14_0 (or starting from any other vertex). See the figure in the link.
The vertex-vertex 14 X 14 incidence matrix of this directed Euler graph is cyclic with first row [0,1,0,1,0,1,0_8] (0_k for k zeros) shifted by one step to the right, with last row [1, 0, 1, 0, 1, 0_9].
The 14 triples of positions of the vertex labels k, for k from 0 to 13, in the present sequence are given in A332440.
For N = 3, SBBseq(3, 1) = A000012 (sequence of 1's). The primitive period modulo 6 is [0, 1, 2, 3, 4, 5], and the Euler tour, using as diagonal only the side, is the digraph C_6 (circle graph, trail in the positive sense).
For N=5, SBBseq(5, 1) = repeat(1,3,) = A010684. The primitive period modulo 10 is [0, 1, 4, 5, 8, 9, 2, 3, 6, 7]. The simple digraph, using alternatively the two diagonals d1 (side) and d3 in the regular 10-gon, is regular with degree 2.

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2015-2016.
Wolfdieter Lang, Figure: A directed Euler tour on the regular 14-gon with length 42
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Cf. A000012, A003558, A010684, A135303, A332440, A332441.

get(v, j) = my(x=lift(Mod(j, #v))); if (x==0, x = #v); v[x];
vector(42, k, k--; sum(j=1, k, get([1,5,3], j)) % 14) \\ Michel Marcus, Jun 11 2020

a(n) = (Sum_{j=0..n} A130794(j)) mod 14, for n >= 1 with the periodic sequence SBBseq(7, 1) = repeat(1,5,3,) = A130794, with offset 0, and a(0) = 0 (= a(42)).

A274912 Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 1, 2, 1, 2, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

Omar E. Pol, Jul 11 2016

In the square array we have that:
Antidiagonal sums give A168237.
Odd-indexed rows give A010673.
Even-indexed rows give A010684.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed antidiagonals give the initial terms of A010674.
Even-indexed antidiagonals give the initial terms of A000034.
Main diagonal gives A010674.
This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.
In the triangle we have that:
Row sums give A168237.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed diagonals give A010673.
Even-indexed diagonals give A010684.
Odd-indexed rows give the initial terms of A010674.
Even-indexed rows give the initial terms of A000034.
Odd-indexed antidiagonals give the initial terms of A010673.
Even-indexed antidiagonals give the initial terms of A010684.

			The corner of the square array begins:
0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, ...
0, 2, 0, 2, ...
1, 3, 1, ...
0, 2, ...
1, ...
...
The sequence written as a triangle begins:
0;
1, 2;
0, 3, 0;
1, 2, 1, 2;
0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2, 1, 2;
...

Cf. A000034, A000035, A001477, A010673, A010674, A010684, A010693, A168237, A274913, A274920.

ListTools:-Flatten([seq([[0,3]$i,0,[1,2]$(i+1)],i=0..10)]); # Robert Israel, Nov 14 2016

Table[Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

a(n) = A274913(n) - 1.
From Robert Israel, Nov 14 2016: (Start)
G.f.: 3*x/(1-x^2) - Sum_{k>=0} (2*x^(2*k^2+3*k+1)-x^(2*k^2+5*k+3))/(1+x).
G.f. as triangle: x*(1+2*y+3*x*y)/((1-x^2*y^2)*(1-x^2)). (End)

A274913 Square array read by antidiagonals upwards in which each new term is the least positive integer distinct from its neighbors.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 2, 3, 2, 3, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Omar E. Pol, Jul 11 2016

This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the square array we have that:
Antidiagonal sums give the positive terms of A008851.
Odd-indexed rows give A010684.
Even-indexed rows give A010694.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed antidiagonals give the initial terms of A010685.
Even-indexed antidiagonals give the initial terms of A010693.
Main diagonal gives A010685.
This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the triangle we have that:
Row sums give the positive terms of A008851.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed diagonals give A010684.
Even-indexed diagonals give A010694.
Odd-indexed rows give the initial terms of A010685.
Even-indexed rows give the initial terms of A010693.
Odd-indexed antidiagonals give the initial terms of A010684.
Even-indexed antidiagonals give the initial terms of A010694.

			The corner of the square array begins:
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, ...
1, 3, 1, 3, ...
2, 4, 2, ...
1, 3, ...
2, ...
...
The sequence written as a triangle begins:
1;
2, 3;
1, 4, 1;
2, 3, 2, 3;
1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3, 2, 3;
...

Cf. A000034, A008851, A010684, A010685, A010693, A010694, A010702, A274912, A274921.

Table[1 + Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

a(n) = A274912(n) + 1.

A060590 Numerator of the expected time to finish a random Tower of Hanoi problem with n disks using optimal moves.

Original entry on oeis.org

0, 2, 2, 14, 10, 62, 42, 254, 170, 1022, 682, 4094, 2730, 16382, 10922, 65534, 43690, 262142, 174762, 1048574, 699050, 4194302, 2796202, 16777214, 11184810, 67108862, 44739242, 268435454, 178956970, 1073741822, 715827882, 4294967294

Offset: 0

Henry Bottomley, Apr 05 2001

			a(2)=2 since there are nine equally likely possibilities, with times required of 0,1,1,2,2,3,3,3,3 giving an average of 18/9 = 2/1.

Harry J. Smith, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).
Index entries for sequences related to Towers of Hanoi

Denominator is A010684(n). Cf. A007798, A060586, A060589, A020988 (even bisection).

a(n)={2*(2^n - 1)*(2 - (-1)^n)/3} \\ Harry J. Smith, Jul 07 2009

a(n) = 2*(2^n - 1)*(2 - (-1)^n)/3.
a(2n) = A020988(n-1).
From Ralf Stephan, Mar 07 2003: (Start)
G.f.: (4*x^3+2*x^2+2*x)/(4*x^4-5*x^2+1).
a(n+4) = 5*a(n+2) - 4*a(n). (End)

cp's OEIS Frontend

A074323 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

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A126116 a(n) = a(n-1) + a(n-3) + a(n-4), with a(0)=a(1)=a(2)=a(3)=1.

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A176014 Decimal expansion of (3+sqrt(21))/6.

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A176040 Periodic sequence: Repeat 3, 1.

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A074323 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(1,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.