A257075 -id:A257075 - cp's OEIS frontend

A086953 Binomial transform of (-1)^mod(n,3) (A257075).

1, 0, 0, 2, 6, 12, 22, 42, 84, 170, 342, 684, 1366, 2730, 5460, 10922, 21846, 43692, 87382, 174762, 349524, 699050, 1398102, 2796204, 5592406, 11184810, 22369620, 44739242, 89478486, 178956972, 357913942, 715827882, 1431655764, 2863311530, 5726623062

Offset: 0

Paul Barry, Jul 25 2003

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A024493, A024495, A131370, A131708, A257075.

Join[{1, a = 0, b = 0}, Table[c = 2^n - a + b; a = b; b = c, {n, 1, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
LinearRecurrence[{3,-3,2},{1,0,0},40] (* Harvey P. Dale, Aug 02 2017 *)

a(n+3)/2 = A024495(n+2). - corrected by Vladimir Shevelev, Aug 08 2017
a(n) = 0^n + Sum{k=0..floor((n-1)/3)} C(n-1, 3*k+2).
a(n) = Sum{k=0..n} C(n, k)(-1)^mod(k, 3).
G.f.: (1 - 3*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - Paul Barry, Dec 14 2004
From Vladimir Shevelev, Aug 02 2017: (Start)
a(n) = A024493(n) - A131708(n) + A024495(n);
a(n) = A024495(n) if and only if n == 1 (mod 3);
a(n) = A024495(n) - 1 if and only if n == 2 or 3 (mod 6);
a(n) = A024495(n) + 1 if and only if n == 0 or 5 (mod 6);
a(3*k+1) = 2*A024495(3*k). (End)
a(n) = A131370(n+1)/2. - Rick L. Shepherd, Aug 02 2017
3*a(n) = 2^n + 2*A057079(n+2). - R. J. Mathar, Aug 04 2017

A008611 a(n) = a(n-3) + 1, with a(0)=a(2)=1, a(1)=0.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, 23, 22, 23, 24, 23, 24, 25, 24, 25, 26, 25, 26, 27, 26, 27, 28

Offset: 0

N. J. A. Sloane, Mar 15 1996

Molien series of 2-dimensional representation of cyclic group of order 3 over GF(2).
One step back, two steps forward.
The crossing number of the graph C(n, {1,3}), n >= 8, is [n/3] + n mod 3, which gives this sequence starting at the first 4. [Yang Yuansheng et al.]
A Chebyshev transform of A078008. The g.f. is the image of (1-x)/(1-x-2*x^2) (g.f. of A078008) under the Chebyshev transform A(x)-> (1/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 15 2004
A047878 is an essentially identical sequence. - Anton Chupin, Oct 24 2009
Rhyme scheme of Dante Alighieri's "Divine Comedy." - David Gaita, Feb 11 2011
A194960 results from deleting the first four terms of A008611. Note that deleting the first term or first four terms of A008611 leaves a concatenation of segments (n, n+1, n+2); for related concatenations, see
  A008619, (n,n+1) after deletion of first term;
  A053737, (n,n+1,n+2,n+3) beginning with n=0;
  A053824, (n to n+4) beginning with n=0. - Clark Kimberling, Sep 07 2011
It appears that a(n) is the number of roots of x^(n+1) + x + 1 inside the unit circle. - Michel Lagneau, Nov 02 2012
Also apparently for n >= 2: a(n) is the largest remainder r that results from dividing n+2 by 1..n+2 more than once, i.e., a(n) = max(i, A072528(n+2,i)>1). - Ralf Stephan, Oct 21 2013
Number of n-element subsets of [n+1] whose sum is a multiple of 3. a(4) = 1: {1,2,4,5}. - Alois P. Heinz, Feb 06 2017
It appears that a(n) is the number of roots of the Fibonacci polynomial F(n+2,x) strictly inside the unit circle of the complex plane. - Michel Lagneau, Apr 07 2017
For the proof of the preceding conjecture see my comments under A008615 and A049310. Chebyshev S(n,x) = i^n*F(n+1,-i*x), with i = sqrt(-1). - Wolfdieter Lang, May 06 2017
The sequence is the interleaving of three sequences: the positive integers (A000027), the nonnegative integers (A001477), and the positive integers, in that order. - Guenther Schrack, Nov 07 2020
a(n) is the number of multiples of 3 between n and 2n. - Christian Barrientos, Dec 20 2021
a(n) is the least number of football games a team has to play to be able to get n-1 points, where a win is 3 points, a draw is 1 point, and a loss is 0 points. - Sigurd Kittilsen, Dec 01 2022

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

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 103.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Frederik Glitzner and David Manlove, Perspectives on Unsolvability in Roommates Markets, arXiv:2505.06717 [cs.GT], 2025. See p. 13.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 447.
Gerard P. Michon, Counting Polyhedra.
Yang Yuansheng et al., The crossing number of C(n; {1,3}), Discr. Math. 289 (2004), 107-118.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for Molien series.

Cf. A000027, A001477, A008619, A047878, A049310, A049347, A053737, A053824, A058207, A058788, A072528, A078008, A194960, A257075, A330396.

a008611 n = n' + mod r 2 where (n', r) = divMod (n + 1) 3
a008611_list = f [1,0,1] where f xs = xs ++ f (map (+ 1) xs)
-- Reinhard Zumkeller, Nov 25 2013

[(n-1)-2*Floor((n-1)/3): n in [0..90]]; // Vincenzo Librandi, Aug 21 2011

with(numtheory): for n from 1 to 70 do:it:=0:
y:=[fsolve(x^n+x+1, x, complex)] : for m from 1 to nops(y) do : if abs(y[m])< 1 then it:=it+1:else fi:od: printf(`%d, `,it):od:
A008611:=n->(n-1)-2*floor((n-1)/3); seq(A008611(n), n=0..50); # Wesley Ivan Hurt, May 18 2014

With[{nn=30},Riffle[Riffle[Range[nn],Range[0,nn-1]],Range[nn],3]] (* or *) RecurrenceTable[{a[0]==a[2]==1,a[1]==0,a[n]==a[n-3]+1},a,{n,90}] (* Harvey P. Dale, Nov 06 2011 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 0, 1, 2}, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
a[ n_] := Quotient[n - 1, 3] + Mod[n + 2, 3]; (* Michael Somos, Jan 23 2014 *)

{a(n) = (n-1) \ 3 + (n+2) % 3}; /* Michael Somos, Jan 23 2014 */

a(n) = a(n-3) + 1.
a(n) = (n-1) - 2*floor((n-1)/3).
G.f.: (1 + x^2 + x^4)/(1 - x^3)^2.
After the initial term, has form {n, n+1, n+2} for n=0, 1, 2, ...
From Paul Barry, Mar 18 2004: (Start)
a(n) = Sum_{k=0..n} (-1)^floor(2*(k-2)/3);
a(n) = 4*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/9 + (n+1)/3. (End)
From Paul Barry, Oct 15 2004: (Start)
G.f.: (1 - x + x^2)/((1 + x + x^2)*(x-1)^2);
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*A078008(n-2k)*(-1)^k. (End)
a(n) = -a(-2-n) for all n in Z.
Euler transform of length 6 sequence [0, 1, 2, 0, 0, -1]. - Michael Somos, Jan 23 2014
a(n) = ((n-1) mod 3) + floor((n-1)/3). - Wesley Ivan Hurt, May 18 2014
PSUM transform of A257075. - Michael Somos, Apr 15 2015
a(n) = A194960(n-3), n >= 0, with extended A194960. See the a(n) formula two lines above. - Wolfdieter Lang, May 06 2017
From Guenther Schrack, Nov 07 2020: (Start)
a(n) = (3*n + 3 + 2*(w^(2*n)*(1 - w) + w^n*(2 + w)))/9, where w = (-1 + sqrt(-3))/2, a primitive third root of unity;
a(n) = (n + 1 + 2*A049347(n))/3;
a(n) = (2*n - A330396(n-1))/3. (End)
E.g.f.: (3*exp(x)*(1 + x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, May 06 2022
Sum_{n>=2} (-1)^n/a(n) = 3*log(2) - 1. - Amiram Eldar, Sep 10 2023

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

A131561 Period 3: repeat [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, 1, 1, -1, 1, 1, -1, 1, 1

Offset: 0

Paul Curtz, Aug 27 2007

Other than the first term, this sequence represents numerators in a fraction expansion of log(2) - Pi/8. - Mohammad K. Azarian, Sep 27 2011

Also, the arithmetic function uhat(n,3,3) as defined in A291041. - Robert Price, Aug 25 2017

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

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

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

Cf. A257075, A291041.

&cat [[1, 1, -1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

A131561 := proc(n) op((n mod 3)+1,[1,1,-1]) ; end: seq(A131561(n), n=0..120); # R. J. Mathar, Oct 18 2007

Table[(-1)^Mod[n-1,3], {n, 0, 120}] (* Michael De Vlieger, Mar 07 2015 *)
PadRight[{},120,{1,1,-1}] (* Harvey P. Dale, Mar 15 2021 *)

a(n)=1-2*(n%3==2) /* Jaume Oliver Lafont, Mar 24 2009 */

a(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
G.f.: (1+x-x^2)/(1-x^3). - Jaume Oliver Lafont, Mar 24 2009
a(n) = (-1)^((n-1) mod 3). - Christopher Richmond, Oct 07 2011
a(n) = a(n-1)^2 - a(n-1) - a(n-2), for a(0),a(1) = 1,1; or same repeating pattern with 1,-1 or -1,1 as initial values. - Richard R. Forberg, Jun 13 2013
a(n+1) = A257075(n) for all n in Z. - Michael Somos, May 13 2015
a(n) = a(n-3) for n>2. - Wesley Ivan Hurt, Jul 02 2016
Product_{n >= 1} (1 + a(n-1)*x^n) = 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + ... = Sum_{n >= 0} x^A001318(n), a companion identity to Euler's pentagonal number theorem. - Peter Bala, Aug 30 2017
E.g.f.: (exp(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Oct 19 2024

Edited by N. J. A. Sloane, Sep 15 2007

A257076 Expansion of (1 - x^3) / (1 - x + x^2) in powers of x.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 15 2015

			G.f. = 1 + x - 2*x^3 - 2*x^4 + 2*x^6 + 2*x^7 - 2*x^9 - 2*x^10 + 2*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (1, -1).

Cf. A109265, A130772, A184334, A257075.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x^3)/(1-x+x^2))); // G. C. Greubel, Aug 03 2018

a[ n_] := SeriesCoefficient[ (1 - x^3) / (1 - x + x^2), {x, 0, n}];
Join[{1, 1},LinearRecurrence[{1, -1},{0, -2},76]] (* Ray Chandler, Aug 10 2015 *)

{a(n) = n++; if( n<3, n>0, 2 * (n%3>0) * (-1)^(n\3))};

{a(n) = if( n<0, 0, polcoeff( (1 - x^3) / (1 - x + x^2) + x * O(x^n), n))};

Euler transform of length 6 sequence [ 1, -1, -2, 0, 0, 1].
G.f.: (1 - x^2) * (1 - x^3)^2 / ((1 - x) * (1 - x^6)).
a(n) = -a(n+3) if n>1.
a(n) = A109265(n-1) if n>0.
Convolution inverse of A257075.
a(n) = A130772(n) for n>1. - R. J. Mathar, Apr 19 2015
a(n) = A184334(n+1) if n>1. - Michael Somos, Sep 01 2015

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A086953 Binomial transform of (-1)^mod(n,3) (A257075).

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A008611 a(n) = a(n-3) + 1, with a(0)=a(2)=1, a(1)=0.

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

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A131561 Period 3: repeat [1, 1, -1].

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A257076 Expansion of (1 - x^3) / (1 - x + x^2) in powers of x.

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