A131713 -id:A131713 - cp's OEIS frontend

A143978 a(n) = floor(2n(n+1)/3).

1, 4, 8, 13, 20, 28, 37, 48, 60, 73, 88, 104, 121, 140, 160, 181, 204, 228, 253, 280, 308, 337, 368, 400, 433, 468, 504, 541, 580, 620, 661, 704, 748, 793, 840, 888, 937, 988, 1040, 1093, 1148, 1204, 1261, 1320, 1380, 1441, 1504, 1568, 1633, 1700, 1768, 1837

Offset: 1

Clark Kimberling, Sep 06 2008

Second diagonal of array A143979, which counts certain unit squares in a lattice. First diagonal: A030511.
Convolution of A042965 with A000012, convolution of A131534 with A000027, and convolution of A106510 with A000217. - L. Edson Jeffery, Jan 24 2015
From Miquel A. Fiol, Aug 31 2024: (Start)
a(n+1) is the maximum number N of vertices of a circulant digraph with steps +-s1, s2, and diameter n.
Depending on the value of n, the following table shows the values of N, s1, and s2:
 n |     3*r     |   3*r-1   |   3*r-2   |
 N | 6*r^2+6*r+1 | 6*r^2+2*r | 6*r^2-2*r |
s1 |      1      |     r     |     r     |
s2 |    6*r+3    |   3*r+1   |   3*r-1   |
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Bruno Berselli, Illustration of the initial terms.
P. Morillo and Miquel A. Fiol, El diámetro de ciertos digrafos circulantes de triple paso, Stochastica X (1986), no. 3, 233-249.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000217, A030511, A042965 (first differences), A106510, A131534, A143979.

Cf. A049347, A102283, A103115, A131713.

A143978:= n-> (6*n*(n+1) -1 + `mod`(n+2,3) - `mod`(n+1,3))/9;
seq(A143978(n), n=1..60); # G. C. Greubel, May 27 2020

Table[(6*n^2 +6*n -1  + Mod[n+2, 3] - Mod[n+1, 3])/9, {n, 60}] (* G. C. Greubel, May 27 2020 *)
Table[Floor[2n (n+1)/3],{n,60}] (* or *) LinearRecurrence[{2,-1,1,-2,1},{1,4,8,13,20},60] (* Harvey P. Dale, Aug 12 2025 *)

From R. J. Mathar, Oct 05 2009: (Start)
G.f.: x*(1 + x)^2/((1 + x + x^2)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). (End)
a(n) = Sum_{k=1..(n+1)} A042965(k). - Klaus Purath, May 23 2020
From G. C. Greubel, May 27 2020: (Start)
a(n) = (ChebyshevU(n, -1/2) - ChebyshevU(n-1, -1/2) + (6*n^2 + 6*n -1))/9.
a(n) = (JacobiSymbol(n+1, 3) - JacobiSymbol(n, 3) + (6*n^2 + 6*n -1))/9.
a(n) = (A102283(n+1) - A102283(n) + A103115(n+1))/9
a(n) = (A131713(n) + A103115(n+1))/9. (End)
Sum_{n>=1} 1/a(n) = 3/2 + (tan(Pi/(2*sqrt(3)))-1)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 27 2022
E.g.f.: exp(-x/2)*(exp(3*x/2)*(6*x^2 + 12*x - 1) + cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Apr 05 2023

A056530 Sequence remaining after third round of Flavius Josephus sieve; remove every fourth term of A047241.

Original entry on oeis.org

1, 3, 7, 13, 15, 19, 25, 27, 31, 37, 39, 43, 49, 51, 55, 61, 63, 67, 73, 75, 79, 85, 87, 91, 97, 99, 103, 109, 111, 115, 121, 123, 127, 133, 135, 139, 145, 147, 151, 157, 159, 163, 169, 171, 175, 181, 183, 187, 193, 195, 199, 205, 207, 211, 217, 219, 223, 229, 231

Offset: 1

Henry Bottomley, Jun 19 2000

Numbers {1, 3, 7} mod 12: A017533, A017557, A017605 interleaved.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for sequences related to the Josephus Problem

We have A000027 after 0 rounds of sieving, A005408 after 1 round of sieving, A047241 after 2 rounds, A056530 after 3 rounds, A056531 after 4 rounds, A000960 after all rounds. After n rounds the remaining sequence comprises A002944(n) numbers mod A003418(n+1), i.e. 1/(n+1) of them.

LinearRecurrence[{1,0,1,-1},{1,3,7,13},60] (* Harvey P. Dale, Oct 19 2022 *)

From Chai Wah Wu, Jul 24 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
G.f.: x*(5*x^3 + 4*x^2 + 2*x + 1)/(x^4 - x^3 - x + 1). (End)
a(n) = 4*n - (13 + 2*A131713(n))/3. - R. J. Mathar, Jun 22 2020

A166587 A signed variant of the Motzkin numbers.

Original entry on oeis.org

1, 1, -1, 2, -4, 9, -21, 51, -127, 323, -835, 2188, -5798, 15511, -41835, 113634, -310572, 853467, -2356779, 6536382, -18199284, 50852019, -142547559, 400763223, -1129760415, 3192727797, -9043402501, 25669818476, -73007772802

Offset: 0

Paul Barry, Oct 17 2009

Hankel transform is A131713. Binomial transform is A166588.

[a(n+1)] = [1,-1,2,-4,9,...] is the inverse binomial transform of A126120. - Philippe Deléham, Nov 29 2009

			G.f. = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 21*x^6 + 51*x^7 - 127*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..500

f:= gfun:-rectoproc({3*n*a(n)+(-3-2*n)*a(1+n)+(-3-n)*a(n+2)=0,a(0) = 1, a(1) = 1}, a(n),remember):
map(f, [$0..100]); # Robert Israel, May 17 2016
with(PolynomialTools): CoefficientList(convert(taylor((1 + 3*x - sqrt(1 + 2*x - 3*x^2))/2/x, x = 0, 33), polynom), x); # Taras Goy, Aug 07 2017

CoefficientList[Series[(1 + 3*t - Sqrt[1 + 2*t - 3*t^2])/(2 t), {t, 0, 50}], t] (* G. C. Greubel, May 17 2016 *)

G.f.: (1+3x-sqrt(1+2x-3x^2))/(2x); (1+3x)/(1+2x-x^2/(1+x-x^2/(1+x-x^2/(1+x-x^2/(1+...))))) (continued fraction).
a(n) = 0^n + Sum_{k=0..n} binomial(n-1, k-1)*(-3)^(n-k)*A000108(k).
G.f.: (1+3*x-sqrt(1+2*x-3*x^2))/(2x) = (3-1/G(0))/2 ; G(k) = 1+2*x/(1-x/(1-x/(1+2*x/(1+x/(2+x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
Conjecture: n*(n+1)*a(n) + n*(n+1)*a(n-1) - (5*n-3)*(n-2)*a(n-2) + 3*(n-2)*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012
G.f. G(x) satisfies (3 x^2 - 2 x^2 - x) G'(x) - (x+1) G(x) + 3 x + 1 = 0, from which follows 3*n*a(n) + (-3-2*n)*a(1+n) + (-3-n)*a(n+2) = 0 as well as Mathar's conjecture. - Robert Israel, May 17 2016
E.g.f.: 1 + Integral (exp(-x) * BesselI(1,2*x) / x) dx. - Ilya Gutkovskiy, Jun 01 2020

A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

Original entry on oeis.org

0, 1, 5, 11, 18, 28, 40, 53, 69, 87, 106, 128, 152, 177, 205, 235, 266, 300, 336, 373, 413, 455, 498, 544, 592, 641, 693, 747, 802, 860, 920, 981, 1045, 1111, 1178, 1248, 1320, 1393, 1469, 1547, 1626, 1708, 1792, 1877, 1965, 2055, 2146, 2240, 2336, 2433, 2533, 2635

Offset: 1

Jörg Zurkirchen, Apr 05 2014

Difference table of a(n), with a(0)=0 and offset=0:
0,   0,  1,  5, 11, 18, 28, 40, 53, 69, ...
0,   1,  4,  6,  7, 10, 12, 13, 16, 18, ...   =  A047234(n+1)
1,   3,  2,  1,  3,  2,  1,  3,  2,  1, ...   =  A130784
2,  -1, -1,  2, -1, -1,  2, -1, -1,  2, ...   = -A131713(n+1)
-3,  0,  3, -3,  0,  3, -3,  0,  3, -3; ...   =  A099838(n+4)
3,   3, -6,  3,  3, -6,  3,  3, -6,  3, ...
0,  -9,  9,  0, -9,  9,  0, -9,  9,  0, ...
-9, 18, -9, -9, 18, -9, -9, 18, -9, -9, ...
First column: see A057682. - Paul Curtz, Nov 11 2014
Diameter of the chamber graph Γ(Alt(2n+1)). Definition of this graph:
Each vertex v is a sequence (v[1],v[2],...,v[n]) of length n, where each v[i] is a 2-subset of {1,2,...,2n+1} and v[i] and v[j] are disjoint unless i=j.
Vertices u and v are connected iff either:
u and v are identical except for their first elements u[1] and v[1], or
u and v are identical except for some i for which u[i]=v[i+1] and v[i]=u[i+1] - Tim Crinion, 17 Feb 2019

			For n = 3 an example of a honeycomb with the greatest minimal difference of a(3) = 5 is:
.         __
.      __/ 7\__
.   __/15\__/13\__
.  / 4\__/ 2\__/ 1\
.  \__/10\__/ 8\__/
.  /18\__/16\__/14\
.  \__/ 5\__/ 3\__/
.  /12\__/11\__/ 9\
.  \__/19\__/17\__/
.     \__/ 6\__/
.        \__/
.

22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 100 terms from Jörg Zurkirchen)
Tim Crinion, Chamber Graphs of some geometries related to the Petersen Graph, 2013.
Fédération Suisse des Jeux Mathématiques, 22nd Championship of Mathematical and Logical Games - Quarter Final 2008, 18 problems in French; problem number 18 was decisive to creating this sequence. See following pdf for an English version of problem 18.
Jörg Zurkirchen, Honeycomb.pdf
Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Cf. A042965, A047234, A057682, A099838, A130784, A131713.

[n*(n-1)-Floor((n+1)/3): n in [1..60]]; // Vincenzo Librandi, Nov 12 2014

A240438:=n->n*(n-1)-floor((n+1)/3); seq(A240438(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014

Table[n (n - 1) - Floor[(n + 1)/3], {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[x (x + 1) (2 x + 1) / ((1 - x)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2014 *)
LinearRecurrence[{2, -1, 1, -2, 1},{0, 1, 5, 11, 18},52] (* Ray Chandler, Sep 24 2015 *)

a(n) = n*(n-1)-floor((n+1)/3).
G.f.: -x^2*(x+1)*(2*x+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Apr 08 2014
a(n+3) = a(n) + 6*n+5. - Paul Curtz, Nov 11 2014
a(n) = n^2 - (A042965(n+1)=0, 1, 3, 4, ...). - Paul Curtz, Nov 11 2014
a(n+1) = a(n) + A047234(n+1). - Paul Curtz, Nov 11 2014

A152731 a(n) + a(n+1) + a(n+2) = n^6, a(1)=a(2)=0.

Original entry on oeis.org

0, 0, 1, 63, 665, 3368, 11592, 31696, 74361, 156087, 300993, 542920, 927648, 1515416, 2383745, 3630375, 5376505, 7770336, 10990728, 15251160, 20803993, 27944847, 37017281, 48417776, 62600832, 80084368, 101455425, 127375983

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

nonn

0 + 0 + 1 = 1^6; 0 + 1 + 63 = 2^6; ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (6,-15,21,-21,21,-21,15,-6,1).

Cf. A152728, A152729, A152730, A152725, A152726, A000212.

m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(1+x)*(x^4+56*x^3+246*x^2+56*x+1)/((1-x)^7*(1 +x+ x^2)))); // G. C. Greubel, Sep 01 2018

k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^6-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{6, -15, 21, -21, 21, -21, 15, -6, 1}, {0, 0, 1, 63, 665, 3368, 11592, 31696, 74361}, 5000]
CoefficientList[Series[x^2*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)),{x, 0, 5000}], x] (* Stefano Spezia, Sep 02 2018 *)

x='x+O('x^30); concat([0,0], Vec(x^3*(1+x)*(x^4+56*x^3 +246*x^2 +56*x+1)/((1-x)^7*(1+x+x^2)))) \\ G. C. Greubel, Sep 01 2018

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = -26*n/3 + 20*n^3/3 - 5*n^2 + 7/3 - 2*n^5 + n^6/3 + 5*n^4/3 - 7*A131713(n)/3.
G.f.: x^3*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)). (End)

A166588 Partial sums of A097331; binomial transform of A166587.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 5, 10, 10, 24, 24, 66, 66, 198, 198, 627, 627, 2057, 2057, 6919, 6919, 23715, 23715, 82501, 82501, 290513, 290513, 1033413, 1033413, 3707853, 3707853, 13402698, 13402698, 48760368, 48760368, 178405158, 178405158, 656043858

Offset: 0

Paul Barry, Oct 17 2009

Hankel transform is A131713. The Hankel transform of the sequence 1,1,2,2,... is A128017(n+3). A155587 doubled.

G. C. Greubel, Table of n, a(n) for n = 0..1000

CoefficientList[Series[(1+2*x-Sqrt[1-4*x^2])/(2*x*(1-x)), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

G.f.: (1+2x-sqrt(1-4x^2))/(2x(1-x))=((1+x^2*c(x^2))/(1-x)-1)/x, c(x) the g.f. of A000108.
a(n) = Sum_{k=0..n} C(n,k)*A166587(k).
Conjecture: (-n-1)*a(n) + (n+1)*a(n-1) + 4*(n-2)*a(n-2) + 4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012
a(n) ~ 2^(n+1/2) * (3-(-1)^n) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 08 2014

A301696 Partial sums of A219529.

Original entry on oeis.org

1, 6, 17, 33, 54, 81, 113, 150, 193, 241, 294, 353, 417, 486, 561, 641, 726, 817, 913, 1014, 1121, 1233, 1350, 1473, 1601, 1734, 1873, 2017, 2166, 2321, 2481, 2646, 2817, 2993, 3174, 3361, 3553, 3750, 3953, 4161, 4374, 4593, 4817, 5046, 5281, 5521, 5766

Offset: 0

N. J. A. Sloane, Mar 25 2018

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

Cf. A219529.

A301696:= n-> (8*(3*n*(n+1) +1) + `mod`(n+2, 3) - `mod`(n+1, 3))/9;
seq(A301696(n), n=0..60); # G. C. Greubel, May 27 2020

Table[(Mod[n+2, 3] - Mod[n+1, 3] + 8*(3*n*(n+1) +1))/9, {n,0,60}] (* G. C. Greubel, May 27 2020 *)

Vec((1 + x)^4 / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Mar 26 2018

[(24*n*(n+1)+8 + (n+2)%3 - (n+1)%3 )/9 for n in (0..60)] # G. C. Greubel, May 27 2020

From Colin Barker, Mar 26 2018: (Start)
G.f.: (1 + x)^4 / ((1 - x)^3*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4. (End)
From G. C. Greubel, May 27 2020: (Start)
a(n) = (ChebyshevU(n, -1/2) - ChebyshevU(n-1, -1/2) + 8*(3*n*(n+1) +1))/9.
a(n) = ( A131713(n) + 8*A028896(n) + 8 )/9. (End)

A301775 Number of odd chordless cycles in the (2n+1)-web graph.

Original entry on oeis.org

0, 12, 30, 74, 200, 522, 1362, 3572, 9350, 24474, 64080, 167762, 439202, 1149852, 3010350, 7881194, 20633240, 54018522, 141422322, 370248452, 969323030, 2537720634, 6643838880, 17393796002, 45537549122, 119218851372, 312119004990, 817138163594, 2139295485800

Offset: 1

Eric W. Weisstein, Mar 26 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1).

Cf. A301774.

I:=[0,12,30,74,200]; [n le 5 select I[n] else  2*Self(n-1)+Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 27 2018

Rest @ CoefficientList[Series[2 x^2*(6 + 3 x + x^2 - x^3)/((1 - 3 x + x^2) (1 + x + x^2)), {x, 0, 29}], x] (* Michael De Vlieger, Mar 26 2018 *)
Join[{0}, LinearRecurrence[{2, 1, 2, -1}, {12, 30, 74, 200}, 30]] (* Vincenzo Librandi, Mar 27 2018 *)
Join[{0}, Table[LucasL[2 n + 1] + Cos[2 n Pi/3] - Sqrt[3] Sin[2 n Pi/3], {n, 2, 20}]] (* Eric W. Weisstein, Mar 27 2018 *)

Vec(2*(6 + 3*x + x^2 - x^3)/((1 - 3*x + x^2)*(1 + x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 26 2018

From Andrew Howroyd, Mar 26 2018: (Start)
a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4) for n > 5.
G.f.: 2*x^2*(6 + 3*x + x^2 - x^3)/((1 - 3*x + x^2)*(1 + x + x^2)).
(End)
a(n) = A002878(n) + cos(2*n*Pi/3) - sqrt(3)*sin(2*n*Pi/3) for n > 1. - Eric W. Weisstein, Mar 27 2018
a(n) = A002878(n) + A131713(n), n>1. - R. J. Mathar, Apr 17 2018

Terms a(10) and beyond from Andrew Howroyd, Mar 26 2018

A008806 Expansion of (1+x^3)/((1-x^2)^2*(1-x^3)).

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 6, 6, 9, 10, 12, 14, 17, 18, 22, 24, 27, 30, 34, 36, 41, 44, 48, 52, 57, 60, 66, 70, 75, 80, 86, 90, 97, 102, 108, 114, 121, 126, 134, 140, 147, 154, 162, 168, 177, 184, 192, 200, 209, 216, 226, 234, 243, 252, 262, 270, 281, 290, 300, 310, 321

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Beckwith, Vadim Ponomarenko and Rob Pratt, Building Two Piles of Equal Height: 11183, The American Mathematical Monthly, 114 (2007), 551-552.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

a:=[1,0,2,2,3,4];; for n in [7..70] do a[n]:=a[n-1]+a[n-2]-a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^3)/((1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series((1+x^3)/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 12 2019

CoefficientList[Series[(1+x^3)/((1-x^2)^2*(1-x^3)), {x,0,70}], x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1}, {1,0,2,2,3,4}, 70] (* G. C. Greubel, Sep 12 2019 *)

Vec((1+x^3)/((1-x^2)^2*(1-x^3)) +O(x^70)) \\ Charles R Greathouse IV, Sep 26 2012; modified by G. C. Greubel, Sep 12 2019

def A008806_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^3)/((1-x^2)^2*(1-x^3))).list()
A008806_list(70) # G. C. Greubel, Sep 12 2019

From R. J. Mathar, Nov 08 2010: (Start)
a(n) = (16*A131713(n) +29 +24*n +6*n^2 +27*(-1)^n)/72.
G.f.: (1 -x +x^2)/( (1+x)*(1+x+x^2)*(1-x)^3 ). (End)
a(n) = floor((6*n^2+24*n+61+27*(-1)^n)/72). - Tani Akinari, Jul 24 2013

Terms a(52) onward added by G. C. Greubel, Sep 12 2019

A147534 a(n) is congruent to (1,1,2) mod 3.

Original entry on oeis.org

1, 1, 2, 4, 4, 5, 7, 7, 8, 10, 10, 11, 13, 13, 14, 16, 16, 17, 19, 19, 20, 22, 22, 23, 25, 25, 26, 28, 28, 29, 31, 31, 32, 34, 34, 35, 37, 37, 38, 40, 40, 41, 43, 43, 44, 46, 46, 47, 49, 49, 50, 52, 52, 53, 55, 55, 56, 58, 58, 59, 61, 61, 62, 64, 64, 65, 67, 67, 68, 70, 70, 71

Offset: 1

Giovanni Teofilatto, Nov 06 2008

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

Cf. A004396 for a(n) congruent to (0, 1, 1) mod 2.

Cf. A131713.

I:=[1,1,2]; [n le 3 select I[n] else Self(n-3)+3: n in [1..70]]; // Vincenzo Librandi, Jul 25 2016

a:=n->add(chrem( [n,j], [1,3] ), j=1..n): seq(a(n)+1, n=-1..70); # Zerinvary Lajos, Apr 08 2009

LinearRecurrence[{1,0,1,-1},{1,1,2,4},80] (* Harvey P. Dale, Dec 09 2012 *)

a(n) = a(n-3)+3 = n-2/3-A131713(n)/3. G.f.: x*(1+x^2+x^3)/((1-x)^2*(1+x+x^2)). [R. J. Mathar, Nov 07 2008]
a(1)=1, a(2)=1, a(3)=2, a(4)=4, a(n)=a(n-1)+a(n-3)-a(n-4) for n>4. - Harvey P. Dale, Dec 09 2012
a(n) = (3*n - 2 - cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/3. - Wesley Ivan Hurt, Jul 24 2016
a(n) = 1 + floor((n-1)/3) + floor(2*(n-1)/3). - Wesley Ivan Hurt, Jul 25 2016
a(n) = n - sign((n-1) mod 3). - Wesley Ivan Hurt, Sep 25 2017

cp's OEIS Frontend

A143978 a(n) = floor(2*n*(n+1)/3).

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A056530 Sequence remaining after third round of Flavius Josephus sieve; remove every fourth term of A047241.

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A166587 A signed variant of the Motzkin numbers.

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A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

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A152731 a(n) + a(n+1) + a(n+2) = n^6, a(1)=a(2)=0.

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A166588 Partial sums of A097331; binomial transform of A166587.

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A301696 Partial sums of A219529.

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A143978 a(n) = floor(2n(n+1)/3).