A131531 -id:A131531 - cp's OEIS frontend

A141325 a(n) = A000045(n) + A131531(n+3).

1, 1, 1, 1, 3, 5, 9, 13, 21, 33, 55, 89, 145, 233, 377, 609, 987, 1597, 2585, 4181, 6765, 10945, 17711, 28657, 46369, 75025, 121393, 196417, 317811, 514229, 832041, 1346269, 2178309, 3524577, 5702887, 9227465, 14930353, 24157817, 39088169

Offset: 0

Paul Curtz, Aug 03 2008

Michael De Vlieger, Table of n, a(n) for n = 0..4786
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,1).

Cf. A014437, A140096, A140413.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2+x^4)/((1+x^3)*(1-x-x^2)) )); // G. C. Greubel, Jun 11 2019

LinearRecurrence[{1,1,-1,1,1}, {1,1,1,1,3}, 40] (* Jean-François Alcover, Aug 16 2017 *)
Table[Fibonacci@ n + Boole[Mod[n, 3] == 0] - 2 Boole[Mod[n, 6] == 3], {n, 0, 40}] (* Michael De Vlieger, Aug 16 2017 *)

my(x='x+O('x^40)); Vec((1-x^2+x^4)/((1+x^3)*(1-x-x^2))) \\ G. C. Greubel, Jun 11 2019

((1-x^2+x^4)/((1+x^3)*(1-x-x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 11 2019

G.f.: (1-x^2+x^4)/((1+x)*(1-x+x^2)*(1-x-x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

Definition corrected by R. J. Mathar, Sep 16 2009

More terms from R. J. Mathar, Sep 27 2009

A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, -1, -2, -3, 0, 0, 1, 3, 6, -1, -1, -1, -2, -5, -11, 0, 1, 2, 3, 5, 10, 21, 0, 0, -1, -3, -6, -11, -21, -42, 1, 1, 1, 2, 5, 11, 22, 43, 85, 0, -1, -2, -3, -5, -10, -21, -43, -86, -171, 0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, -1, -1, -1, -2, -5, -11, -22, -43, -85, -170, -341, -683, 0, 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365

Offset: 0

Paul Curtz, Nov 07 2009

The array contains A131708(0) in diagonal 0, then -A024495(0..1) in diagonal 1, then A024493(0..2) in diagonal 2, then -A131708(0..3), then A024495(0..4), then -A024493(0..5).

			The table starts in row n=0 with columns k >= 0 as:
0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0 A131531
0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1 A092220
1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2 A131556
-3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3 A164359
6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5
-11, 10, -11, 11, -10, 11, -11, 10, -11, 11, -10, 11, -11, 10, -11
21, -21, 22, -21, 21, -22, 21, -21, 22, -21, 21, -22, 21, -21, 22

Cf. A167617 (antidiagonal sums).

A131531 := proc(n) op((n mod 6)+1,[0,0,1,0,0,-1]) ; end proc:
A167613 := proc(n,k) option remember; if n= 0 then A131531(k); else procname(n-1,k+1)-procname(n-1,k) ; end if;end proc: # R. J. Mathar, Dec 17 2010

nmax = 13;
A131531 = Table[{0, 0, 1, 0, 0, -1}, {nmax}] // Flatten;
T[n_] := T[n] = Differences[A131531, n];
T[n_, k_] := T[n][[k]];
Table[T[n-k, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 20 2023 *)

T(0,k) = A131531(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.
T(n,n) = A001045(n). T(n,n+1) = -A001045(n). T(n,n+2) = A078008(n).
T(n,0) = -T(n,3) = (-1)^(n+1)*A024495(n).
T(n,1) = (-1)^(n+1)*A131708(n).
T(n,2) = (-1)^n*A024493(n).
T(n,k+6) = T(n,k).
a(n) = A131708(0), -A024495(0,1), A024493(0,1,2), -A131708(0,1,2,3), A024495(0,1,2,3,4), -A024493(0,1,2,3,4,5).

A113405 Expansion of x^3/(1 - 2x + x^3 - 2x^4) = x^3/( (1-2x)(1+x)*(1-x+x^2) ).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 14, 28, 57, 114, 228, 455, 910, 1820, 3641, 7282, 14564, 29127, 58254, 116508, 233017, 466034, 932068, 1864135, 3728270, 7456540, 14913081, 29826162, 59652324, 119304647, 238609294, 477218588, 954437177, 1908874354, 3817748708

Offset: 0

Paul Barry, Oct 28 2005

A transform of the Jacobsthal numbers. A059633 is the equivalent transform of the Fibonacci numbers.
Paul Curtz, Aug 05 2007, observes that the inverse binomial transform of 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,... gives the same sequence up to signs. That is, the extended sequence is an eigensequence for the inverse binomial transform (an autosequence).
The round() function enables the closed (non-recurrence) formula to take a very simple form: see Formula section. This can be generalized without loss of simplicity to a(n) = round(b^n/c), where b and c are very small, incommensurate integers (c may also be an integer fraction). Particular choices of small integers for b and c produce a number of well-known sequences which are usually defined by a recurrence - see Cross Reference. - Ross Drewe, Sep 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

From Ross Drewe, Sep 03 2009: (Start)
Other sequences a(n) = round(b^n / c), where b and c are very small integers:
A001045 b = 2; c = 3
A007910 b = 2; c = 5
A016029 b = 2; c = 5/3
A077947 b = 2; c = 7
abs(A078043) b = 2; c = 7/3
A007051 b = 3; c = 2
A015518 b = 3; c = 4
A034478 b = 5; c = 2
A003463 b = 5; c = 4
A015531 b = 5; c = 6
(End)

[Round(2^n/9): n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

A010892 := proc(n) op((n mod 6)+1,[1,1,0,-1,-1,0]) ; end proc:
A113405 := proc(n) (2^n-(-1)^n)/9 -A010892(n-1)/3; end proc: # R. J. Mathar, Dec 17 2010

CoefficientList[Series[x^3/(1-2x+x^3-2x^4),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,-1,2},{0,0,0,1},40] (* Harvey P. Dale, Apr 30 2011 *)

a(n)=2^n\/9 \\ Charles R Greathouse IV, Jun 05 2011

def A113405(n): return ((1<Chai Wah Wu, Apr 17 2025

a(n) = 2a(n-1) - a(n-3) + 2a(n-4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A001045(k).
a(n) = Sum_{k=0..n} binomial((n+k)/2,k)*A001045((n-k)/2)*(1+(-1)^(n-k))/2.
a(3n) = A015565(n), a(3n+1) = 2*A015565(n), a(3n+2) = 4*A015565(n). - Paul Curtz, Nov 30 2007
From Paul Curtz, Dec 16 2007: (Start)
a(n+1) - 2a(n) = A131531(n).
a(n) + a(n+3) = 2^n. (End)
a(n) = round(2^n/9). - Ross Drewe, Sep 03 2009
9*a(n) = 2^n + (-1)^n - 3*A010892(n). - R. J. Mathar, Mar 24 2018

Edited by N. J. A. Sloane, Dec 13 2007

A088911 Period 6: repeat [1, 1, 1, 0, 0, 0].

Original entry on oeis.org

1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Oct 22 2003

For periodic sequences having a period of 2*k and composed of k ones followed by k zeros we have a(n) = floor(((n+k) mod 2*k)/k). Sequences of this form are A000035(n+1) (k=1), A133872(n) (k=2), this sequence (k=3), A131078(n) (k=4), and A112713(n-1) (k=5). - Gary Detlefs, May 17 2011

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A000035, A010701, A079979, A133872, A131078, A112713.

&cat [[1, 1, 1, 0, 0, 0]^^30]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 1, 1, 0, 0, 0]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

CoefficientList[Series[(1 + x + x^2)/(1 - x^6), {x, 0, 50}], x]
Flatten[Table[{1,1,1,0,0,0},{20}]] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=n%6<3 \\ Jaume Oliver Lafont, Mar 17 2009

def A088911(n): return int(n % 6 < 3) # Chai Wah Wu, May 25 2022

G.f.: (1+x+x^2)/(1-x^6) = 1/((1-x)*(1+x)*(1-x+x^2)).
a(n) = a(n-6) for n>=6, a(0)=a(1)=a(2)=1, a(3)=a(4)=a(5)=0.
a(n) = ((-1)^floor((5*n + 2)/3) + 1)/2 = ( (-1)^floor(n/3) + 1 )/2. [Simplified by Bruno Berselli, Jul 09 2013]
a(n) = Sum_{k=0..floor(n/2)} U(n-2k, 1/2). - Paul Barry, Nov 15 2003
From Paul Barry, Mar 14 2004: (Start)
Partial sums of expansion of 1/(1+x^3), see A131531.
a(n) = 2*sin(Pi*n/3 + Pi/6)/3 + cos(Pi*n)/6 + 1/2. (End)
a(n) = floor(((n+3) mod 6)/3).
a(n) = floor((5*n-1)/3) mod 2. - Gary Detlefs, May 17 2011
a(n) = 1/2 + cos(Pi*n/3)/3 + sin(Pi*n/3)/sqrt(3) + (-1)^n/6. - R. J. Mathar, Oct 08 2011
a(n) = floor(((n+2)^2)/3) mod 2. - Wesley Ivan Hurt, Jun 29 2013
a(n) = A079979(n) + A079979(n-1) + A079979(n-2). - R. J. Mathar, Jul 10 2015
a(n) = a(n-1) - a(n-3) + a(n-4) for n > 3. - Wesley Ivan Hurt, Jul 05 2016
a(n) = 2*floor(n/6) - floor(n/3) + 1. - Ridouane Oudra, Dec 14 2021
E.g.f.: (2*cosh(x) + exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + sinh(x))/3. - Stefano Spezia, Aug 04 2025

A130151 Period 6: repeat [1, 1, 1, -1, -1, -1].

Original entry on oeis.org

1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1

Offset: 0

Paul Curtz, Aug 03 2007

			G.f. = 1 + x + x^2 - x^3 - x^4 - x^5 + x^6 + x^7 + x^8 - x^9 - x^10 - x^11 + ...
G.f. = q + q^3 + q^5 - q^7 - q^9 - q^11 + q^13 + q^15 + q^17 - q^19 - q^21 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A131561, A131531, A257075.

Cf. A057077, A143621, A143622, A143628, A143629, A143630. - Peter Bala, Aug 28 2008

&cat [[1, 1, 1, -1, -1, -1]^^20]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 1, 1, -1, -1, -1]), n=0..30); # Wesley Ivan Hurt, Jul 05 2016

a[ n_] := (-1)^Quotient[n, 3]; (* Michael Somos, Apr 24 2014 *)
PadRight[{}, 100, {1, 1, 1, -1, -1, -1}] (* Wesley Ivan Hurt, Jul 05 2016 *)

{a(n) = (-1) ^ (n\3)}; /* Michael Somos, Feb 26 2011 */

a(n+6) = a(n), a(0)=a(1)=a(2)=-a(3)=-a(4)=-a(5)=1.
a(n) = ((-1)^n * (4 * (cos((2*n + 1)*Pi/3) + cos(n*Pi)) + 1) - 4) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 01 2007
a(n) = (-1)^n * (4 * cos((2*n + 1) * Pi/3) + 1) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 02 2007
G.f.: (1+x+x^2)/((1+x)*(x^2-x+1)). - R. J. Mathar, Nov 14 2007
a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4) for n>3. - Paul Curtz, Nov 22 2007
a(n) = (-1)^floor(n/3). Compare with A057077, A143621 and A143622. Define E(k) = Sum_{n >= 0} a(n)*n^k/n! for k = 0,1,2,... . Then E(k) is an integral linear combination of E(0), E(1) and E(2) (a Dobinski-type relation). Precisely, E(k) = A143628(k)*E(0) + A143629(k)*E(1) + A143630(k)*E(2). - Peter Bala, Aug 28 2008
Euler transform of length 6 sequence [1, 0, -2, 0, 0, 1]. - Michael Somos, Feb 26 2011
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = -(-1)^e if e>0, b(p^e) = 1 if p == 1 (mod 4), b(p^e) = (-1)^e if p == 3 (mod 4) and p>3. - Michael Somos, Feb 26 2011
a(n + 3) = a(-1 - n) = -a(n) for all n in Z. - Michael Somos, Feb 26 2011
a(n) = (-1)^n * A257075(n) for all n in Z. - Michael Somos, Apr 15 2015
G.f.: 1 / (1 - x / (1 + 2*x^2 / (1 + x / (1 + x / (1 - x))))). - Michael Somos, Apr 15 2015
From Wesley Ivan Hurt, Jul 05 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = (cos(n*Pi) + 2*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3)) / 3. (End)
a(n)*a(n-4) = a(n-1)*a(n-3) for all n in Z. - Michael Somos, Feb 25 2020

A092220 Expansion of x(1-x)/ ((1+x)(1-x+x^2)) in powers of x.

Original entry on oeis.org

0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1

Offset: 0

Paul Barry, Feb 25 2004

Period 6: repeat [0, 1, -1, 0, -1, 1]. - Joerg Arndt, Aug 28 2024
Transform of the Jacobsthal numbers A001045 under the Riordan array A102587. - Paul Barry, Jul 14 2005
The BINOMIAL transform generates (-1)^(n+1)*A024495(n+1). - R. J. Mathar, Apr 07 2008

			G.f. = x - x^2 - x^4 + x^5 + x^7 - x^8 - x^10 + x^11 + x^13 - x^14 - x^16 + x^17 + ...

Michael Somos, Rational Function Multiplicative Coefficients, 2014.
Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A001045, A024495, A049310, A102587, A131531, A128834.

seq(2*sin(Pi*n^2/3)/sqrt(3), n=0..100); # Ridouane Oudra, Oct 30 2024

a[ n_] := {1, -1, 0, -1, 1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Aug 25 2014 *)
LinearRecurrence[{0,0,-1},{0,1,-1},120] (* or *) PadRight[{},120,{0,1,-1,0,-1,1}] (* Harvey P. Dale, Mar 30 2016 *)

{a(n) = [0, 1, -1, 0, -1, 1][n%6 + 1]}; /* Michael Somos, Apr 10 2011 */

a(n) = 2*cos(Pi*n/3)/3 - 2(-1)^n/3.
Multiplicative with a(2^e) = -1, a(3^e) = 0, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
From Michael Somos, Apr 10 2011: (Start)
Euler transform of length 6 sequence [-1, 0, -1, 0, 0, 1].
Moebius transform is length 6 sequence [1, -2, -1, 0, 0, 2].
G.f.: x * (1 - x) * (1 - x^3) / (1 - x^6).
a(n) = a(-n), a(n + 3) = -a(n), a(3*n) = 0, for all n in Z. (End)
a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4). - Paul Curtz, Dec 10 2007
a(n) = ( (-1)^floor((n+1)/3) - (-1)^n )/2. - Bruno Berselli, Jul 09 2013
a(n) = S(n-1,-1), n >= 0, with Chebyshev's S-polynomials evaluated at -1 (see A049310). - Wolfdieter Lang, Sep 06 2013
a(n) = A131531(n+2) - A131531(n+1) . - R. J. Mathar, Nov 28 2019
a(n) = A128834(n^2). - Ridouane Oudra, Oct 30 2024
E.g.f.: 2*(exp(x/2)*cos(sqrt(3)*x/2) - cosh(x) + sinh(x))/3. - Stefano Spezia, Oct 31 2024
Dirichlet g.f.: zeta(s) * (1 - 1/2^(s-1)) * (1 - 1/3^s). - Amiram Eldar, Jun 09 2025

A052702 Expansion of (1/2)(1/x^2 - 1/x)(1-x-sqrt(1-2x+x^2-4x^3)) - x.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 13, 26, 52, 108, 226, 472, 993, 2106, 4485, 9586, 20576, 44332, 95814, 207688, 451438, 983736, 2148618, 4702976, 10314672, 22664452, 49887084, 109985772, 242854669, 537004218, 1189032613, 2636096922, 5851266616, 13002628132, 28925389870, 64412505472, 143576017410

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

From Paul Barry, May 24 2009: (Start)
Hankel transform of A052702 is A160705. Hankel transform of A052702(n+1) is A160706.
Hankel transform of A052702(n+2) is -A131531(n+1). Hankel transform of A052702(n+3) is A160706(n+5).
Hankel transform of A052702(n+4) is A160705(n+5). (End)
For n > 1, number of Dyck (n-1)-paths with each descent length one greater or one less than the preceding ascent length. - David Scambler, May 11 2012

Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 10, 19-21.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 654

Cf. A023431.

spec := [S,{B=Prod(C,Z),S=Prod(B,B),C=Union(S,B,Z)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

a[n_] := Sum[Binomial[n-k-2, 2k-1] CatalanNumber[k], {k, 0, n-2}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 11 2022, after Paul Barry *)

x='x+O('x^66);
s='a0+(1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2);
v=Vec(s);  v[1]-='a0;  v
/* Joerg Arndt, May 11 2012 */

Recurrence: {a(1)=0, a(2)=0, a(4)=1, a(3)=0, a(6)=3, a(7)=6, a(5)=2, (-2+4*n)*a(n)+(-7-5*n)*a(n+1)+(8+3*n)*a(n+2)+(-13-3*n)*a(n+3)+(n+6)*a(n+4)}.
From Paul Barry, May 24 2009: (Start)
G.f.: (1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2).
a(n+1) = Sum_{k=0..n-1} C(n-k-1,2k-1)*A000108(k). (End)
a(n) = A023431(n-1)-A023431(n-2). - R. J. Mathar, Jan 13 2025

More terms from Joerg Arndt, May 11 2012

A081374 Size of "uniform" Hamming covers of distance 1, that is, Hamming covers in which all vectors of equal weight are treated the same, included or excluded from the cover together.

Original entry on oeis.org

1, 2, 2, 5, 10, 22, 43, 86, 170, 341, 682, 1366, 2731, 5462, 10922, 21845, 43690, 87382, 174763, 349526, 699050, 1398101, 2796202, 5592406, 11184811, 22369622, 44739242, 89478485, 178956970, 357913942, 715827883, 1431655766, 2863311530, 5726623061

Offset: 1

David Applegate, Aug 22 2003

nonn

Motivation: consideration of the "hats" problem (which boils down to normal hamming covering codes) in the case when the people are indistinguishable or unlabeled.
From Paul Curtz, May 26 2011: (Start)
If we add a(0)=1 in front and build the table of a(n) and iterated differences in further rows we get:
1,    1,  2,  2,  5, 10,
0,    1,  0,  3,  5, 12,
1,   -1,  3,  2,  7,  9,
-2,   4, -1,  5,  2, 13,
6,   -5,  6, -3, 11,  6
-11, 11, -9, 14, -5, 21.
The first column is the inverse binomial transform, which is 1,0 followed by (-1)^n*A083322(n-1), n>=2.
The main diagonal in the table above is A001045, the adjacent upper diagonals are A078008, A048573 and A062092. (End)

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

Cf. A083322.

I:=[1,2,2,5]; [n le 4 select I[n] else 2*Self(n-1)-Self(n-3)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 08 2016

hatwork := proc(n,i,covered) local val, val2; options remember;
# computes the minimum cover of the i-bit through n-bit words.
# if covered is true the i-bit words are already covered (by the (i-1)-bit words)
if (i>n or (i = n and covered)) then 0; elif (i = n and not covered) then 1; else
# one choice is to include the i-bit words in the cover
val := hatwork(n, i+1, true) + binomial(n,i);
# the other choice is not to include the i-bit words in the cover
if (covered) then val2 := hatwork (n, i+1, false); if (val2 < val) then val := val2; fi; else
# if the i-bit words were not covered by (i-1), they must be covered by the (i+1)-bit words
if (i <= n) then val2 := hatwork (n, i+2, true) + binomial(n,i+1); if (val2 < val) then val := val2; fi; fi; fi; val; fi; end proc;
A081374 := proc (n) hatwork(n, 0, false); end proc;

LinearRecurrence[{2,0,-1,2},{1,2,2,5},40] (* Harvey P. Dale, Feb 11 2015 *)

If (n mod 6 = 5) then sum(binomial(n, 3*i+1), i=0..n/3); elif (n mod 6 = 2) then sum(binomial(n, 3*i), i=0..n/3)+1; else sum(binomial(n, 3*i), i=0..n/3); fi;
G.f.: x*(2*x^3-2*x^2+1)/( (1-2*x)*(1+x)*(1-x+x^2) ).
a(n)=2*a(n-1)-a(n-3)+2*a(n-4).
From Paul Curtz, May 26 2011: (Start)
a(n+1) - 2*a(n) has period length 6: repeat 0, -2, 1, 0, 2, -1 (see A080425).
a(n) - A083322(n-1) = A010892(n-1) has period length 6.
a(n) + a(n+3) = 3*2^n = A007283(n).
a(n+6)-a(n) = 21*2^n = A175805(n).
a(n) - A131708(n) = -A131531(n).  (End)

A173432 NW-SE diagonal sums of Riordan array A112468.

Original entry on oeis.org

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

Offset: 1

Mark Dols, Feb 18 2010

Matches Fibonacci-sequence, such that F(n) + a(n) and F(n) - a(n) = always even.

Periodic sequence with period: [1,1,2,1,1,0]. - Philippe Deléham, Oct 11 2011

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

Cf. A000045, A024495, A112468, A131531.

[2*Ceiling(n/6)-2*Floor(n/6)+Floor(n/3)-Ceiling(n/3) : n in [1..100]]; // Wesley Ivan Hurt, Sep 27 2014

A173432:=n->2*ceil(n/6)-2*floor(n/6)+floor(n/3)-ceil(n/3): seq(A173432(n), n=1..100); # Wesley Ivan Hurt, Sep 27 2014

Table[2 Ceiling[n/6] - 2 Floor[n/6] + Floor[n/3] - Ceiling[n/3], {n, 50}] (* Wesley Ivan Hurt, Sep 27 2014 *)

Vec(-x*(x^2+1) / ((x-1)*(x+1)*(x^2-x+1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

a(n) = 1 + A131531(n) with inverse binomial transform: 1, 0, 1, -3, 6, -11, 21, .., a signed variant of A024495. - R. J. Mathar, Mar 04 2010
a(2n+1) = a(2n)-a(2n-1)+2, a(2n) = a(2n-1)-a(2n-2) with a(1) = a(2)=1. - Philippe Deléham, Oct 11 2011
a(n) = a(n-1)-a(n-3)+a(n-4). - Colin Barker, Sep 26 2014
G.f.: -x*(x^2+1) / ((x-1)*(x+1)*(x^2-x+1)). - Colin Barker, Sep 26 2014
a(n) = 2*ceiling(n/6)-2*floor(n/6)+floor(n/3)-ceiling(n/3). - Wesley Ivan Hurt, Sep 27 2014
a(n) = A001045(n) - A111927(n). - Paul Curtz, Dec 16 2020

Corrected and extended by Philippe Deléham, Oct 11 2011

A242563 a(n) = 2a(n-1) - a(n-3) + 2a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.

Original entry on oeis.org

0, 0, 2, 3, 6, 10, 21, 42, 86, 171, 342, 682, 1365, 2730, 5462, 10923, 21846, 43690, 87381, 174762, 349526, 699051, 1398102, 2796202, 5592405, 11184810, 22369622, 44739243, 89478486, 178956970, 357913941, 715827882, 1431655766, 2863311531, 5726623062, 11453246122

Offset: 0

Paul Curtz, May 17 2014

Generally, a(n) is an autosequence if its inverse binomial transform is (-1)^n*a(n). It is of the first kind if the main diagonal is 0's and the first two upper diagonals (just above the main one) are the same. It is of the second kind if the main diagonal is equal to the first upper diagonal multiplied by 2. If the first upper diagonal is an autosequence, the sequence is a super autosequence. Example: A113405. The first upper diagonal is A001045(n). Another super autosequence: 0, 0, 0 followed by A059633(n). The first upper diagonal is A000045(n).
Difference table of a(n):
   0,  0,  2, 3, 6, 10, 21, 42, ...
   0,  2,  1, 3, 4, 11, 21, 44, ...
   2, -1,  2, 1, 7, 10, 23, 41, ...
  -3,  3, -1, 6, 3, 13, 18, 45, ... .
This is an autosequence of the second kind. The main diagonal is 2*A001045(n) = A078008(n). More precisely it is a super autosequence, companion of A113405(n).
a(n+1) mod 10 = period 12: repeat 0, 2, 3, 6, 0, 1, 2, 6, 1, 2, 2, 5.
It is shifted A081374(n+1) mod 10 =
    period 12: repeat 1, 2, 2, 5, 0, 2, 3, 6, 0, 1, 2, 6.
a(n) mod 9 = period 18:
repeat 0, 0, 2, 3, 6, 1, 3, 6, 5, 0, 0, 7, 6, 3, 8, 6, 3, 4 = c(n).
c(n) + c(n+9) = 0, 0, 9, 9, 9, 9, 9, 9, 9.

			G.f. = 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 21*x^6 + 42*x^7 + 86*x^8 + ...

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

Cf. A000032, 1/(n+1), A164555/A027642 (all autosequences of 2nd kind). A007283, A175805.

a[n_] := (m = Mod[n, 6]; 1/3*(2^n + (-1)^n + 1/120*(m-6)*(m+1)*(m^3-29*m+40))); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 19 2014, a non-recursive formula, after Mathematica's RSolve *)
LinearRecurrence[{2, 0, -1, 2}, {0, 0, 2, 3},50] (* G. C. Greubel, Feb 21 2017 *)

concat([0,0], Vec(x^2*(x-2)/((x+1)*(2*x-1)*(x^2-x+1)) + O(x^100))) \\ Colin Barker, May 18 2014

a(n+3) = 3*2^n - a(n), a(0)=a(1)=0, a(2)=2.
a(n) = 2*A113405(n+1) - A113405(n).
a(n+1) = 2*a(n) + period 6: repeat 0, 2, -1, 0, -2, 1. a(0)=0.
a(n) = 2^n - A081374(n+1).
a(n+3) = a(n+1) + A130755(n).
G.f.: x^2*(x-2) / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, May 18 2014
a(n) = A024495(n) + A131531(n).
a(n+6) = a(n) + 21*2^n, a(0)=a(1)=0, a(2)=2, a(3)=3, a(4)=6, a(5)=10.
a(n) = A001045(n) - A092220(n).
a(n+12) = a(n) + 1365*2^n. First 12 values in the Data. (A024495(n+12) = A024495(n) + 1365*2^n).
a(3n)   = A132805(n) = 3*A015565(n).
a(3n+1) = A132804(n) = 6*A015565(n).
a(3n+2) = A132397(n) = 2*A082311(n).
a(n) = 1/3*((-1)^n - 2*cos((n*Pi)/3) + 2^n). - Alexander R. Povolotsky,  Jun 02 2014

More terms from Colin Barker, May 18 2014

cp's OEIS Frontend

A141325 a(n) = A000045(n) + A131531(n+3).

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A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

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A113405 Expansion of x^3/(1 - 2*x + x^3 - 2*x^4) = x^3/( (1-2*x)*(1+x)*(1-x+x^2) ).

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A088911 Period 6: repeat [1, 1, 1, 0, 0, 0].

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A130151 Period 6: repeat [1, 1, 1, -1, -1, -1].

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A092220 Expansion of x*(1-x)/ ((1+x)*(1-x+x^2)) in powers of x.

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A113405 Expansion of x^3/(1 - 2x + x^3 - 2x^4) = x^3/( (1-2x)(1+x)*(1-x+x^2) ).

A092220 Expansion of x(1-x)/ ((1+x)(1-x+x^2)) in powers of x.

A052702 Expansion of (1/2)(1/x^2 - 1/x)(1-x-sqrt(1-2x+x^2-4x^3)) - x.

A242563 a(n) = 2a(n-1) - a(n-3) + 2a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.