A100063 -id:A100063 - cp's OEIS frontend

A380561 Rectangular array R read by descending antidiagonals: (row 1) = (R(1,k)) = (A006337(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = (1, 1, 2, 1, 1, 2, 1, 1, 2, ...) = (A100063(n)) for n >= 1. See Comments.

1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Clark Kimberling, Jan 27 2025

For present purposes, all sequences to be considered consist entirely of 1s and 2s. If u and v are such sequences (infinite or finite), we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array (in which each row after the first is an inverse runlength sequence of the preceding row) is determined by its first column. In this array, the first column is the periodic sequence with period 1,1,2. There are three limiting sequences: A378283, A378284, A378285. No two rows are identical.

See A380560 for a guide to related sequences.

			Corner:
    1  2  1  2  1  1  2  1  2  1  1  2  1  2  1  2  1  1  2
    1  2  2  1  2  2  1  2  1  1  2  1  1  2  1  2  2  1  2
    2  1  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2
    1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1
    1  2  1  1  2  1  1  2  2  1  2  1  1  2  1  2  2  1  1
    2  1  1  2  1  2  2  1  2  1  1  2  2  1  2  2  1  2  1
    1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1
    1  2  1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1
    2  1  1  2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2

Cf. A000002, A100063 (column 1), A006337, A380560.

invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = Flatten[Table[Nest[Flatten[# /. {1 -> {1, 2}, 2 -> {1, 1, 2}}] &, {1},  n], {n, 3}]];(* A006337 *)
rows = {row1};
col = PadRight[{}, 30, {1, 1, 2}];
Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
v[n_, k_] := rows[[n]][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*sequence*)
(* Peter J. C. Moses, Nov 20 2024 *)

A061347 Period 3: repeat [1, 1, -2].

Original entry on oeis.org

1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2

Offset: 1

Jason Earls, Jun 07 2001

WARNING: It is unclear whether this sequence should start at offset 1 (as written) or offset 0 (in analogy to many similar sequences, which seems to be assumed in many of the given formulas).
Inverse binomial transform of A057079. - Paul Barry, May 15 2003
The unsigned version, with g.f. (1 + x + 2*x^2)/(1 - x^3), has a(n) = 4/3 -cos(2*Pi*n/3)/3 - sqrt(3)*sin(2*Pi*n/3)/3 = gcd(Fib(n+4), Fib(n+1)). - Paul Barry, Apr 02 2004
a(n) = L(n-2,-1), where L is defined as in A108299; see also A010892 for L(n,+1). - Reinhard Zumkeller, Jun 01 2005
From the Taylor expansion of log(1 + x + x^2) at x = 1, Sum_{k > 0} a(k)/k = log(3) = A002391. This is case n = 3 of the general expression Sum_{k > 0} (1-n*!(k mod n))/k = log(n). - Jaume Oliver Lafont, Oct 16 2009
If used with offset zero, a non-simple continued fraction representation of 2+sqrt(2). - R. J. Mathar, Mar 08 2012
Periodic sequences of this type can be also calculated by a(n) = c + floor(q/(p^m-1)*p^n) mod p, where c is a constant, q is the number representing the periodic digit pattern and m is the period length. c, p and q can be calculated as follows: Let D be the array representing the number pattern to be repeated, m = size of D, max = maximum value of elements in D, min = minimum value of elements in D. Than c := min, p := max - min + 1 and q := p^m*Sum_{i=1..m} (D(i)-min)/p^i. Example: D = (1, 1, -2), c = -2, p = 4 and q = 60 for this sequence. - Hieronymus Fischer, Jan 04 2013
This is the Dirichlet inverse of A117997. - Petros Hadjicostas, Jul 25 2020

			G.f.: x + x^2 - 2*x^3 + x^4 + x^5 - 2*x^6 + x^7 + x^8 - 2*x^9 + ... - _Michael Somos_, Nov 27 2019

W. Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
Ralph E. Griswold, Shaft Sequences, 2001 (see also here).
Tanya Khovanova, Recursive Sequences.
W. O. J. Moser, Cyclic binary strings without long runs of like (alternating) bits, Fibonacci Quart. 31(1) (1993), 2-6.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Apart from signs, same as A057079 (also bin. Transf), A100063. Cf. A000045, A010892 for the rules a(n) = a(n - 1) + a(n - 2), a(n) = a(n - 1) - a(n - 2). a(n) = - a(n - 1) + a(n - 2) gives a signed version of Fibonacci numbers.
Alternating row sums of A130777: repeat(1,-2,1).
Cf. A000032, A002391, A007040, A100886, A108299, A117997, A320769.

Flat(List([1..50],n->[1,1,-2])); # Muniru A Asiru, Aug 02 2018

&cat [[1, 1, -2]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([1, 1, -2]), n=1..50); # Wesley Ivan Hurt, Jul 01 2016

a[n_] := {1, 1, -2}[[Mod[n - 1, 3] + 1]]; Table[a[n], {n, 108}] (* Jean-François Alcover, Jul 19 2013 *)
PadRight[{}, 90, {1, 1, -2}] (* After Harvey P. Dale, or *)
CoefficientList[ Series[(2x + 1)/(x^2 + x + 1), {x, 0, 89}], x]  (* or *)
LinearRecurrence[{-1, -1}, {1, 1}, 90] (* Robert G. Wilson v, Jul 30 2018 *)

a(n)=1-3*!(n%3) \\ Jaume Oliver Lafont, Oct 16 2009

def A061347():
    x, y = -1, -1
    while True:
        yield -x
        x, y = y, -x -y
a = A061347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

With offset zero, a(n) = A057079(2n). a(n) = -a(n-1) - a(n-2) with a(0) = a(1) = 1.
From Mario Catalani (mario.catalani(AT)unito.it), Jan 07 2003: (Start)
G.f.: x*(1 + 2*x)/(1 + x + x^2).
a(n) = (-1)^floor(2n/3) + ((-1)^floor((2n-1)/3) + (-1)^floor((2n+1)/3))/2. (End)
a(n) = -2*cos(2*Pi*n/3). - Jaume Oliver Lafont, May 06 2008
Dirichlet g.f.: zeta(s)*(1-1/3^(s-1)). - R. J. Mathar, Feb 09 2011
a(n) = n * Sum_{k=1..n} binomial(k,n-k)/k*(-1)^(k+1). - Dmitry Kruchinin, Jun 03 2011
a(n) = -2 + floor(110/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013
a(n) = -2 + floor(20/21*4^(n+1)) mod 4. - Hieronymus Fischer, Jan 04 2013
a(n) = a(n-3) for n > 3. - Wesley Ivan Hurt, Jul 01 2016
E.g.f.: 2 - 2*cos(sqrt(3)*x/2)*exp(-x/2). - Ilya Gutkovskiy, Jul 01 2016
a(n) = (-1)^n*hypergeom([-n/2-1, -n/2-3/2], [-n-2], 4). - Peter Luschny, Dec 17 2016
a(n) = A000032(n) - A007040(n), for n > 1. - Wojciech Florek, Feb 20 2018

Better definition from M. F. Hasler, Jan 13 2013

A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

Original entry on oeis.org

1, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1

Offset: 0

Paul Barry, Oct 27 2004

A transform of (-1)^n.
Row sums of Riordan array ((1-x)/(1+x), x/(1+x)^2), A110162.
Let b(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)(-1)^(n-2k). Then a(n) = b(n) - b(n-2) = A049347(n) - A049347(n-2) (n > 0). The g.f. 1/(1+x) of (-1)^n is transformed to (1-x^2)/(1+x+x^2) under the mapping G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). Partial sums of A099838.
A(n) = a(n+3) (or a(n) if a(0) is replaced by 2) appears, together with B(n) = A049347(n) in the formula 2*exp(2*Pi*n*i/3) = A(n) + B(n)*sqrt(3)*i, n >= 0, with i = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014

			G.f. = 1 - x - x^2 + 2*x^3 - x^4 - x^5 + 2*x^6 - x^7 - x^8 + 2*x^9 - x^10 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A061347, A100051, A100063, A098554, A110162.

A099837 := proc(n)
    option remember;
    if n <=2 then
        op(n+1,[1,-1,-1]) ;
    else
        -procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A099837(n),n=0..80) ; # R. J. Mathar, Apr 26 2022

a[0] = 1; a[n_] := Mod[n+2, 3] - Mod[n, 3]; A099837 = Table[a[n], {n, 0, 71}](* Jean-François Alcover, Feb 15 2012, after Michael Somos *)
LinearRecurrence[{-1, -1}, {1, -1, -1}, 50] (* G. C. Greubel, Aug 08 2017 *)

A099837(n) := block(
        if n = 0 then 1 else [2,-1,-1][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = [2, -1, -1][n%3 + 1] - (n == 0)}; /* Michael Somos, Jan 19 2012 */

Vec((1-x^2)/(1+x+x^2) + O(x^20)) \\ Felix Fröhlich, Aug 08 2017

G.f.: (1-x^2)/(1+x+x^2).
Euler transform of length 3 sequence [-1, -1, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 3 sequence [-1, 0, 3]. - Michael Somos, Mar 22 2011
a(n) = -b(n) where b(n) = A061347(n) is multiplicative with b(3^e) = -2 if e > 0, b(p^e) = 1 otherwise. - Michael Somos, Jan 19 2012
a(n) = a(-n). a(n) = c_3(n) if n > 1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
G.f.: (1 - x) * (1 - x^2) / (1 - x^3). a(n) = -a(n-1) - a(n-2) unless n = 0, 1, 2. - Michael Somos, Jan 19 2012
Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)*(3^(1-s)-1). - R. J. Mathar, Apr 11 2011
a(n+3) = R(n,-1) for n >= 0, with the monic Chebyshev T-polynomials R with coefficient table A127672. - Wolfdieter Lang, Feb 27 2014
For n > 0, a(n) = 2*cos(n*Pi/3)*cos(n*Pi). - Wesley Ivan Hurt, Sep 25 2017
From Peter Bala, Apr 20 2024: (Start)
a(n) is equal to the n-th order Taylor polynomial (centered at 0) of 1/c(x)^(2*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. Cf. A333093.
Row sums of the Riordan array A110162. (End)

A132367 Period 6: repeat [1, 1, 2, -1, -1, -2].

Original entry on oeis.org

1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2

Offset: 0

Paul Curtz, Nov 09 2007

Nonsimple continued fraction expansion of 1+1/sqrt(3) = 1 + A020760. - R. J. Mathar, Mar 08 2012

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

Cf. A020760, A061347, A100051, A100063, A122876.

&cat[[1, 1, 2, -1, -1, -2]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A132367:=n->[1, 1, 2, -1, -1, -2][(n mod 6)+1]: seq(A132367(n), n=0..100); # Wesley Ivan Hurt, Jun 19 2016

PadRight[{}, 120, {1,1,2,-1,-1,-2}] (* Harvey P. Dale, Jul 28 2012 *)

a(n)=[1,1,2,-1,-1,-2][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n) = cos(Pi*n/3)/3+sqrt(3)*sin(Pi*n/3)+2*(-1)^n/3. - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 19 2016: (Start)
G.f.: (1+x+2*x^2)/(1+x^3).
a(n) + a(n-3) = 0 for n>2. (End)

A101825 G.f.: x*(1+x)^2/(1-x^3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 17 2007

Period 3. Used in A042965.

First differences are essentially A049347. The binomial transform yields 0, 1, 4, 10,... i.e. A111927 with the first two zeros removed. - R. J. Mathar, May 14 2008

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

LinearRecurrence[{0,0,1},{0,1,2,1},120] (* Harvey P. Dale, Aug 01 2016 *)

x='x+O('x^50); Vec(x*(1+x)^2/(1-x^3)) \\ G. C. Greubel, May 03 2017

a(n) = A100063(n+1), n>0. - R. J. Mathar, Sep 01 2008

A107751 a(n) = A107750(n+1) - A107750(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Reinhard Zumkeller, May 23 2005

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A100063, A099837, A100051, A107688.

A107751[ nmax_ ] := Length /@ Split[ (Total[ 1 - IntegerDigits[ #1, 2 ] ] &) /@ Range[ 0, nmax ] ]; A107751[ 200 ] (* Peter Pein (petsie(AT)dordos.net), Oct 12 2007 *)

up_to = 65537;
A107751list(up_to) = { my(v=vector(up_to)); v[1]=v[2]=v[3]=v[4]=1; for(n=5,up_to,v[n] = (0^(v[n-1]-1) + 0^(v[n-2]-1))); (v); };
v107751 = A107751list(up_to);
A107751(n) = if(!n,1,v107751[n]); \\ Antti Karttunen, Dec 23 2018

a(n) = if n<=4 then 1 else 0^(a(n-1)-1) + 0^(a(n-2)-1).

cp's OEIS Frontend

A380561 Rectangular array R read by descending antidiagonals: (row 1) = (R(1,k)) = (A006337(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = (1, 1, 2, 1, 1, 2, 1, 1, 2, ...) = (A100063(n)) for n >= 1. See Comments.

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A061347 Period 3: repeat [1, 1, -2].

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A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

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Maxima

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A132367 Period 6: repeat [1, 1, 2, -1, -1, -2].

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A101825 G.f.: x*(1+x)^2/(1-x^3).

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A107751 a(n) = A107750(n+1) - A107750(n).

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A378708 Circumference of the n-Lucas cube graph.

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