A057078 -id:A057078 - cp's OEIS frontend

A047878 a(n) is the least number of knight's moves from corner (0,0) to n-th diagonal of unbounded chessboard.

0, 3, 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

Offset: 0

Clark Kimberling

Apart from initial terms, same as A008611. - Anton Chupin, Oct 24 2009

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A008611, A049604, A057078, A194960, A330396.

I:=[2, 1, 2, 3]; [0,3] cat [n le 4 select I[n] else Self(n-1) +Self(n-3) -Self(n-4): n in [1..81]]; // G. C. Greubel, Oct 22 2022

LinearRecurrence[{1,0,1,-1},{0,3,2,1,2,3},80] (* Harvey P. Dale, Sep 01 2018 *)
Join[{0,3}, Table[(n+2 -2*ChebyshevU[2*n, 1/2])/3, {n,2,75}]] (* G. C. Greubel, Oct 22 2022 *)

concat(0, Vec(x*(2*x^4-2*x^3-x^2-x+3)/((x-1)^2*(x^2+x+1)) + O(x^100))) \\ Colin Barker, May 04 2014

(Sage) [0,3]+[(n+2 - 2*chebyshev_U(2*n, 1/2))/3 for n in (2..75)] # G. C. Greubel, Oct 22 2022

a(n) = Min_{i=0..n} A049604(i,n-i).
a(3n) = n, a(3n+1) = n+1, a(3n+2) = n+2 for n >= 1.
From Colin Barker, May 04 2014: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.
G.f.: x*(3-x-x^2-2*x^3+2*x^4) / ((1-x)^2*(1+x+x^2)). (End)
From Guenther Schrack, Nov 19 2020: (Start)
a(n) = a(n-3) + 1, for n > 4 with a(0) = 0, a(1) = 3, a(2) = 2, a(3) = 1, a(4) = 2;
a(n) = (3*n + 6 - 2*(w^(2*n)*(2 + w) + w^n*(1 - w)))/9, for n > 1 with a(0) = 0, a(1) = 3, where w = (-1 + sqrt(-3))/2, a primitive third root of unity;
a(n) = (n + 2 - 2*A057078(n))/3 for n > 1;
a(n) = A194960(n-2) for n > 2;
a(n) = (2*n + 2 - A330396(n))/3 for n > 1. (End)

A104538 Expansion of g.f. (1 + 2x) / (1 + 2x + 4*x^2).

Original entry on oeis.org

1, 0, -4, 8, 0, -32, 64, 0, -256, 512, 0, -2048, 4096, 0, -16384, 32768, 0, -131072, 262144, 0, -1048576, 2097152, 0, -8388608, 16777216, 0, -67108864, 134217728, 0, -536870912, 1073741824, 0, -4294967296, 8589934592, 0, -34359738368, 68719476736, 0, -274877906944, 549755813888, 0

Offset: 0

Paul Barry, Mar 13 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-4).

Cf. A057078, A287479.

CoefficientList[Series[(1+2x)/(1+2x+4x^2),{x,0,50}],x] (* or *) LinearRecurrence[ {-2,-4},{1,0},50] (* Harvey P. Dale, Sep 18 2022 *)

a(n) = 2^n*polchebyshev(2*n, 2, 1/2); \\ Michel Marcus, Aug 05 2018

Vec((1 + 2*x) / (1 + 2*x + 4*x^2) + O(x^40)) \\ Colin Barker, Aug 08 2018

a(n) = 2^n*A057078(n), where A057078(n) = U(2n, 1/2), U(n, x) Chebyshev polynomial of second kind.
a(n) = b(n) + b(n+1) where b(n) = 0, 1, -1, -3, 11, -11, -21, 85, -85, ... is the inverse binomial transform of A287479(n). - Paul Curtz, Aug 05 2018
a(n) = (((-1+i*sqrt(3))^n*(-i+sqrt(3)) + (-1-i*sqrt(3))^n*(i+sqrt(3)))) / (2*sqrt(3)) where i=sqrt(-1). - Colin Barker, Aug 08 2018
E.g.f.: exp(-x)*(sqrt(3)*cos(sqrt(3)*x) + sin(sqrt(3)*x))/sqrt(3). - Stefano Spezia, Jul 15 2024

A131476 a(n) = floor(n^3/3).

Original entry on oeis.org

0, 0, 2, 9, 21, 41, 72, 114, 170, 243, 333, 443, 576, 732, 914, 1125, 1365, 1637, 1944, 2286, 2666, 3087, 3549, 4055, 4608, 5208, 5858, 6561, 7317, 8129, 9000, 9930, 10922, 11979, 13101, 14291, 15552, 16884, 18290, 19773, 21333, 22973

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A000982, A008619, A004526, A007590, A036487.

[Floor(n^3/3): n in [0..50]]; // Vincenzo Librandi, May 08 2011

Table[Quotient[n^3,3],{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, May 07 2011 *)

a(n)=n^3\3 \\ Charles R Greathouse IV, May 08 2011

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: x^2*(2 + 3*x + x^3)/((1 - x)^4*(1 + x + x^2)).
a(n) = (A057078(n) - A024001(n))/3. (End)
a(n) = (3*n^3 + 3*cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) - 3)/9. - Vladimir Reshetnikov, Oct 09 2016
a(n) = (n - 1)*n*(n + 1)/3 + floor(n/3). - Bruno Berselli, Jun 08 2017

A131737 Essentially even numbers followed by duplicated odd numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45

Offset: 0

Paul Curtz, Sep 19 2007

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

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

Cf. A004396.

A131737 := proc(n): (1/9)*add(5*(k mod 3)+2*((k+1) mod 3)-((k+2) mod 3),k=0..n)-1+(binomial(2*n,n) mod 2)+(binomial((n+1)^2,n+3) mod 2) end: seq( A131737(n),n=0..74); # Johannes W. Meijer, Jun 27 2011

Join[{0, 1}, LinearRecurrence[{1, 0, 1, -1}, {1, 1, 2, 3}, 68]] (* Georg Fischer, Feb 27 2019 *)
Insert[Flatten[Table[If[EvenQ[n],n,{n,n}],{n,0,70}]],1,2] (* Harvey P. Dale, Sep 04 2020 *)

{a(n) = (n==0) + (n==1) + (n\3)*2 + (n%3) - 1}; /* Michael Somos, Jan 11 2011 */

a(0)=0. a(1)=a(2)=1. a(3*n)=A005408(n-1). a(3*n+1)=a(3*n)+1. a(3*n+2)=a(3*n)+2, n>0.
O.g.f.: x*(1+x^4)/((1-x)^2*(x^2+x+1)). a(n)=(2*n-2-A057078(n))/3, n>1. - R. J. Mathar, Jul 16 2008
Euler transform of length 8 sequence [ 1, 0, 1, 1, 0, 0, 0, -1]. - Michael Somos, Jan 11 2011
0 = a(n) - a(n+1) - a(n+3) + a(n+4) if n>1. - Michael Somos, Nov 11 2015
a(n) = floor((2*n-1)/3) for n > 1. - Werner Schulte, Feb 27 2019

Edited by R. J. Mathar, Jul 16 2008

A182895 Number of (1,0)-steps at level 0 in all weighted lattice paths in L_n.

Original entry on oeis.org

0, 1, 3, 7, 19, 50, 130, 341, 893, 2337, 6119, 16020, 41940, 109801, 287463, 752587, 1970299, 5158310, 13504630, 35355581, 92562113, 242330757, 634430159, 1660959720, 4348449000, 11384387281, 29804712843, 78029751247, 204284540899

Offset: 0

Emeric Deutsch, Dec 12 2010

nonn

The members of L_n are paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

			a(3) = 7. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 0+0+2+2+3=7 (1,0)-steps at level 0.

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1)

Cf. A182893.

G:=z*(1+z)/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..28);

LinearRecurrence[{2,1,2,-1},{0,1,3,7},30] (* Harvey P. Dale, Jan 05 2022 *)

a(n) = Sum_{k>=0} k*A182893(n,k).
G.f.: z(1+z)/[(1+z+z^2)(1-3z+z^2)].
a(n) = (A000032(2n+1) - A010892(2n))/4. - John M. Campbell, Dec 30 2016
4*a(n) = -A057078(n) +A002878(n). - R. J. Mathar, Jul 26 2022

A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).

Original entry on oeis.org

1, -2, 2, 1, 2, 6, 9, 14, 26, 41, 66, 110, 177, 286, 466, 753, 1218, 1974, 3193, 5166, 8362, 13529, 21890, 35422, 57313, 92734, 150050, 242785, 392834, 635622, 1028457, 1664078, 2692538, 4356617, 7049154, 11405774, 18454929, 29860702, 48315634, 78176337

Offset: 0

Paul Curtz, Jun 04 2013

a(n+1)/a(n) -> the golden ratio, A001622.

a(3*n)+a(3*n+1)+a(3*n+2) = 1,9,49,217,929,... = b(n), and b(n+1)-b(n) = 8*A015448(n+1).

			a(2)=-2+1+3=2, a(3)=2-2+1=1, a(4)=1+2-1=2, a(5)=2+1+3=6.
a(0)=F(-3)+F(n)-1=2+0-1=1,  a(1)=-1+1-2=-2, a(2)=1+1-0=2.
a(3)=1+4*0=1, a(4)=-2+4*1=2, a(5)=2+4*1=6, a(6)=1+4*2=9.

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

Cf. A156279, A022120, A022353.

I:=[1,-2,2,1,2]; [n le 5 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jun 05 2013

CoefficientList[Series[(2 x^4 + 3 x^2 - 3 x + 1) / (x^5 + x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{1, 1, 1, -1, -1}, {1, -2, 2, 1, 2}, 40] (* Hugo Pfoertner, Feb 12 2024 *)

Vec((2*x^4+3*x^2-3*x+1)/(x^5+x^4-x^3-x^2-x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 04 2013

a(n) = F(n-3) + F(n) - A010872(n+1).
a(n+3) = a(n) + 4*F(n).
G.f.: (2*x^4+3*x^2-3*x+1)/( (x-1)*(x^2+x-1)*(1+x+x^2) ). [Charles R Greathouse IV, Jun 04 2013]
a(n) = A057078(n+1) +2*A212804(n) -1. - R. J. Mathar, Jun 26 2013

a(23) corrected by Charles R Greathouse IV, Jun 04 2013

More terms from Bruno Berselli, Jun 04 2013

A350668 Numbers congruent to 2, 4, and 6 modulo 9: positions of 2 in A159955.

Original entry on oeis.org

2, 4, 6, 11, 13, 15, 20, 22, 24, 29, 31, 33, 38, 40, 42, 47, 49, 51, 56, 58, 60, 65, 67, 69, 74, 76, 78, 83, 85, 87, 92, 94, 96, 101, 103, 105, 110, 112, 114, 119, 121, 123, 128, 130, 132, 137, 139, 141, 146, 148

Offset: 0

Wolfdieter Lang, Jan 29 2022

This sequence, together with A350666 and A350667, gives a 3-set partition of the nonnegative integers.
This sequence {a(n)}_{n>=0} gives the indices of the row sequences of array A = A347834, that are modulo 6 periodic with period length 3, namely
{A347834(a(n), m) mod 6}_{m >= 0} = {repeat(3, 1, 5)}.

			Rows of array {A347834(a(n), m)}_{m >= 0}, with modulo 6 congruence:
n = 0: row 2: {3, 13, 53, 213, 853, 3413, 13653, ...} mod 6 = {repeat(3, 1, 5)},
n = 1: row 4: {9, 37, 149, 597, 2389, 9557, ...} (mod 6) = {repeat(3, 1, 5)},
...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A049310, A159955, A347834, A350666, A350667.

Select[Range[0, 150], MemberQ[{2, 4, 6}, Mod[#, 9]] &] (* Amiram Eldar, Jan 29 2022 *)
LinearRecurrence[{1,0,1,-1},{2,4,6,11},80] (* Harvey P. Dale, Jul 12 2024 *)

A159955(a(n)) = 2.
Trisection: a(3*k) = 2 + 9*k, a(3*k + 1) = 4 + 9*k, and a(3*k + 3) = 6 + 9*k, for k >= 0.
G.f.: (2 + 2*x + 2*x^2 + 3*x^3)/((1 - x)*(1 - x^3)).
a(n) = 1 + 3*n + cos(2*n*Pi/3) + sin(2*n*Pi/3)/sqrt(3). - Stefano Spezia, Jan 30 2022
a(n) = 1 + 3*n + S(2*n, 1) = 1+3*n+A057078(n), with the Chebyshev S polynomials from A049310, using the partial fraction decomposition of the g.f., or the previous formula.

A375373 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^2) ).

Original entry on oeis.org

1, -2, 4, -4, 6, -4, 9, -4, 16, 0, 28, 16, 57, 58, 132, 172, 322, 476, 817, 1272, 2112, 3360, 5496, 8832, 14353, 23158, 37540, 60668, 98238, 158876, 257145, 415988, 673168, 1089120, 1762324, 2851408, 4613769, 7465138, 12078948, 19544044, 31623034, 51167036

Offset: 0

Seiichi Manyama, Aug 13 2024

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,0,4,6,4,1).

Cf. A093040, A375372.

CoefficientList[Series[1/((1+x)^2(1-x^2(1+x)^2)),{x,0,50}],x] (* or *) LinearRecurrence[{-2,0,4,6,4,1},{1,-2,4,-4,6,-4},50] (* Harvey P. Dale, Dec 11 2024 *)

my(N=50, x='x+O('x^N)); Vec(1/((1+x)^2*(1-x^2*(1+x)^2)))

a(n) = sum(k=0, n\2, binomial(2*k-2, n-2*k));

a(n) = -2*a(n-1) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k-2,n-2*k).
2*a(n) = 2*(-1)^n*(n+1) +A212804(n)-A057078(n). - R. J. Mathar, Aug 14 2024

A008612 Molien series of 2-dimensional representation of SL(2,3).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 3, 3, 4, 4, 3, 5, 4, 4, 5, 5, 4, 6, 5, 5, 6, 6, 5, 7, 6, 6, 7, 7, 6, 8, 7, 7, 8, 8, 7, 9, 8, 8, 9, 9, 8, 10, 9, 9, 10, 10, 9, 11, 10, 10, 11, 11, 10, 12, 11, 11, 12, 12, 11, 13, 12, 12, 13, 13, 12, 14

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 2 ).

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

[2*Floor(n/3)-n/2+(3+(-1)^n)/4: n in [0..100]]; // Vincenzo Librandi, Oct 23 2014

(1+x^12)/(1-x^6)/(1-x^8);seq(coeff(series(%,x,2*n+1),x,2*n), n=0..100);

CoefficientList[Series[(1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Oct 23 2014 *)

Vec((1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)) + O(x^100)) \\ Colin Barker, Jan 07 2014

def A008612_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^6)/((1-x^3)*(1-x^4)) ).list()
A008612_list(100) # G. C. Greubel, Feb 06 2020

From Colin Barker, Jan 07 2014: (Start)
a(n) = a(n-2) + a(n-3) - a(n-5).
G.f.: (1-x^2+x^4) / ((1-x)^2*(1+x)*(1+x+x^2)). (End)
a(n) ~ n/6 (first difference is 6-periodic). - Ralf Stephan, Apr 29 2014
a(n) = 2*floor(n/3) -n/2 +(3+(-1)^n)/4. - Tani Akinari, Oct 23 2014
12*a(n) = 1 +2*n +3*(-1)^n +8*A057078(n). - R. J. Mathar, Jan 14 2021

A008647 Expansion of g.f.: (1+x^9)/((1-x^4)*(1-x^6)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Molien series of binary octahedral group of order 48. Also Molien series for W_1 - W_3 of shadow of singly-even binary self-dual code.

T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.

T. D. Noe, Table of n, a(n) for n = 0..1000
Eiichi Bannai, Etsuko Bannai, Michio Ozeki and Yasuo Teranishi, On the ring of simultaneous invariants for the Gleason-MacWilliams group, European J. Combin. 20 (1999), 619-627.
J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1334.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,-1).
Index entries for Molien series

a:=[1,0,0,0,1,0,1];; for n in [8..80] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # G. C. Greubel, Sep 06 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x^9)/((1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 06 2019

g:= proc(n) local m, r; m:= iquo(n, 12, 'r'); irem(r+1,2) *(m+1) -`if`(r=2, 1, 0) end: a:= n-> g(n) +`if`(n>8, g(n-9), 0); seq(a(n), n=0..100); # Alois P. Heinz, Oct 06 2008

CoefficientList[Series[(1+x^9)/((1-x^4)*(1-x^6)),{x,0,80}],x] (* or *) LinearRecurrence[{0,0,1,1,0,0,-1}, {1,0,0,0,1,0,1}, 80] (* Harvey P. Dale, Oct 10 2011 *)

a(n)=(9*(-1)^n+2*(n+7)+6*(-1)^(n\2))\24 \\ Charles R Greathouse IV, Feb 10 2017

def A008647_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^9)/((1-x^4)*(1-x^6))).list()
A008647_list(80) # G. C. Greubel, Sep 06 2019

G.f.: (1 - x^3 + x^6) / ( (1+x)*(1+x+x^2)*(1+x^2)*(1-x)^2 ).
G.f.: (1+x^6+x^9+x^15)/((1-x^4)*(1-x^12)).
It appears that the first differences have period 12. Hence in blocks of 12, the sequence is {1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0}+k for k=0,1,2,... - T. D. Noe, May 23 2008
a(n) = (6*A057077(n) +8*A057078(n) +1 +2*n +9*(-1)^n)/24. - R. J. Mathar, Jun 28 2009
a(n) = a(n-3) + a(n-4) - a(n-7), a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=0, a(6)=1. - Harvey P. Dale, Oct 10 2011
a(n) = floor((9*(-1)^n+2*(n+7)+6*(-1)^floor(n/2))/24). - Tani Akinari, Jun 17 2013
a(n) = floor(n/2) + floor(n/3) + floor(n/4) - n + 1. - Ridouane Oudra, Mar 21 2021

cp's OEIS Frontend

A047878 a(n) is the least number of knight's moves from corner (0,0) to n-th diagonal of unbounded chessboard.

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A104538 Expansion of g.f. (1 + 2*x) / (1 + 2*x + 4*x^2).

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A131476 a(n) = floor(n^3/3).

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A131737 Essentially even numbers followed by duplicated odd numbers.

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A182895 Number of (1,0)-steps at level 0 in all weighted lattice paths in L_n.

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A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).

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A350668 Numbers congruent to 2, 4, and 6 modulo 9: positions of 2 in A159955.

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A104538 Expansion of g.f. (1 + 2x) / (1 + 2x + 4*x^2).