A099254 -id:A099254 - cp's OEIS frontend

A011779 Expansion of 1/((1-x)^3*(1-x^3)^2).

1, 3, 6, 12, 21, 33, 51, 75, 105, 145, 195, 255, 330, 420, 525, 651, 798, 966, 1162, 1386, 1638, 1926, 2250, 2610, 3015, 3465, 3960, 4510, 5115, 5775, 6501, 7293, 8151, 9087, 10101, 11193, 12376, 13650

Offset: 0

Emeric Deutsch

The Ca2 and Ze4 triangle sums of A139600 are related to the sequence given above, e.g., Ze4(n) = A011779(n-1) - A011779(n-2) - A011779(n-4) + 3*A011779(n-5), with A011779(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Project Euler, Problem 577. Counting hexagons.
Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).

Cf. A011779, A049347, A099254, A139600, A236770 (first trisection, except 0).

R:=PowerSeriesRing(Integers(), 60);
Coefficients(R!( 1/((1-x)^3*(1-x^3)^2) )); // G. C. Greubel, Oct 22 2024

CoefficientList[Series[1 / ((1 - x)^3 (1 - x^3)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)

Vec(1/((1-x)^3*(1-x^3)^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

a(n)=1/216 * n^4 + 1/12 * n^3 + 37/72 * n^2 + [5/4, 139/108, 131/108][1+n%3] * n + [1, 10/9, 7/9][1+n%3] \\ Yurii Ivanov, Jul 06 2021

def A011779_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^3*(1-x^3)^2) ).list()
A011779_list(60) # G. C. Greubel, Oct 22 2024

a(n) = (1/216)*((208 + 270*n + 111*n^2 + 18*n^3 + n^4) - 8*(-1)^n*(A099254(n) + A099254(n-1)) + 16*(A049347(n) + 2*A049347(n-1)) ). - G. C. Greubel, Oct 22 2024

A128502 Convolution array for Chebyshev's S(n,x)=U(n,x/2) polynomials.

Original entry on oeis.org

1, 2, 3, -2, 4, -6, 5, -12, 3, 6, -20, 12, 7, -30, 30, -4, 8, -42, 60, -20, 9, -56, 105, -60, 5, 10, -72, 168, -140, 30, 11, -90, 252, -280, 105, -6, 12, -110, 360, -504, 280, -42, 13, -132, 495, -840, 630, -168, 7, 14, -156, 660, -1320, 1260, -504, 56, 15, -182, 858, -1980, 2310, -1260, 252, -8, 16, -210, 1092

Offset: 0

Wolfdieter Lang Apr 04 2007

S1(n,x):=sum(S(n-k,x)*S(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)).
The unsigned column sequences, m>=0, divided by (m+1) give Pascal triangle column sequences for m+1.
G.f. for column m sequence: ((-1)^m)*(m+1)*(x^(2*m))/(1-x)^(m+2), m>=0.
Row polynomials P1(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).
Written as a triangle with increasing powers of x this is A294519. - Wolfdieter Lang, Nov 12 2017

			[1];[2];[3,-2],[4,-6];[5,-12,3];[6,-20,12];[7,-30,30,-4];[8,-42,60,-20];...
n=4: [5,-12,3] stands for the polynomial S1(4,x) = 5*x^4-12*x^2+3 = 2*(S(4,x)*1+S(3,x)*S(1,x))+S(2,x)*S(2,x).
n=4: [5,-12,3] stands also for the row polynomial P1(4,x) = 5-12*x+3*x^2.

W. Lang, First 15 rows and more.

Row sums (signed array) give A099254. Unsigned row sums are A001629(n+2).
Cf. A115139 (with offset n>=0 is S(n, x) array, decreasing powers of x).
Cf. A294519 (as triangle).

a(n,m)=binomial(n-m,m)*(n+1-m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)=binomial(n+1-m,m+1)*(m+1)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S1(n,x): 1/(1-x*z+z^2)^2.
G.f. for P1(n,x): 1/(1-z+x*z^2)^2.

A186731 a(3n) = 2n, a(3n+1) = n, a(3n+2) = n+1.

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 4, 2, 3, 6, 3, 4, 8, 4, 5, 10, 5, 6, 12, 6, 7, 14, 7, 8, 16, 8, 9, 18, 9, 10, 20, 10, 11, 22, 11, 12, 24, 12, 13, 26, 13, 14, 28, 14, 15, 30, 15, 16, 32, 16, 17, 34, 17, 18, 36, 18, 19, 38, 19, 20, 40, 20, 21, 42, 21, 22, 44, 22, 23, 46, 23, 24, 48

Offset: 0

Philippe Deléham, Jan 21 2012

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Column k = 2 of triangle in A198295.
Cf. A000027, A001477, A005843, A011655, A185395, A185292.
Cf. A002264, A008130.

I:=[0,0,1,2,1,2]; [n le 6 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..80]]; // Vincenzo Librandi, Apr 28 2015

f:= gfun:-rectoproc({a(n)=2*a(n-3)-a(n-6), seq(a(i) = [0,0,1,2,1,2][i+1],i=0..5)},a(n),remember):
map(f, [$0..100]); # Robert Israel, Apr 01 2016

CoefficientList[Series[(x*(1 + x)/(1 - x^3))^2, {x, 0, 100}], x] (* Wesley Ivan Hurt, Apr 28 2015 *)
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {0, 0, 1, 2, 1, 2}, 100] (* Vincenzo Librandi, Apr 28 2015 *)

vector(50,n,n--;(n+1+n*0^(n%3)-(n+1)%3)/3) \\ Derek Orr, Apr 28 2015

G.f.: (x*(1+x)/(1-x^3))^2.
a(n) = |A099254(n-2)| = |A099470(n-1)|. - R. J. Mathar, May 02 2013
From Wesley Ivan Hurt, Apr 28 2015: (Start)
a(n) = 2*a(n-3)-a(n-6).
a(n) = (n+1+n*0^mod(n,3)-mod(n+1,3))/3. (End)
E.g.f.: (4/9)*x*exp(x) - (x/9)*exp(-x/2)*cos(sqrt(3)*x/2) - (sqrt(3)/9)*(2+x)*exp(-x/2)*sin(sqrt(3)*x/2). - Robert Israel, Apr 01 2016
From Ridouane Oudra, Nov 24 2024: (Start)
a(n) = n^3/6 - n/6 - (n^2 + 3*n/2 - 5/2)*floor(n/3) + (3*n/2 + 9/2)*floor(n/3)^2.
a(n) = t(n+1)*t(n+3) - t(n-1)*t(n+1), where t(n) = A002264(n).
a(n) = A008130(n+1) - A008130(n-1). (End)
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/2. - Amiram Eldar, May 10 2025

More terms from Vincenzo Librandi, Apr 28 2015

A099470 A sequence generated from the Quadrifoil.

Original entry on oeis.org

-1, -2, -1, 2, 4, 2, -3, -6, -3, 4, 8, 4, -5, -10, -5, 6, 12, 6, -7, -14, -7, 8, 16, 8, -9, -18, -9, 10, 20, 10, -11, -22, -11, 12, 24, 12, -13, -26, -13, 14, 28, 14, -15, -30, -15, 16, 32, 16, -17, -34, -17, 18, 36, 18, -19, -38, -19, 20, 40, 20, -21, -42

Offset: 1

Gary W. Adamson, Oct 17 2004

a(3n - 1) = 2n (unsigned; n = 1, 2, 3...). In A099471, a(3n) = (2n + 1), unsigned. Odifreddi, p. 135: "Since the trefoil has polynomial x^2 - x + 1 and the quadrifoil (or flat knot) is the sum of two trefoils, its polynomial is (x^2 - x + 1)^2 = x^4 - 2x^3 + 3x^2 - 2x + 1."

Coefficient of x of the characteristic polynomial of the n X n matrix with 1's along the superdiagonal, main diagonal and subdiagonal. - John M. Campbell, Sep 14 2011

			a(7) = -3 since M^7 * [1 0 0 0] = [2 4 2 -3].

Piergiorgio Odifreddi, "The Mathematical Century; The 30 Greatest Problems of the Last 100 Years", Princeton University Press, 2000, page 135.

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

Cf. A099254, A099471.

I:=[-1,-2,-1,2]; [n le 4 select I[n] else 2*Self(n-1)-3*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Sep 09 2016

Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1,#2] + KroneckerDelta[#1,#2+1] + KroneckerDelta[#1,#2-1] &, {n,n}], x], x], {n,75}] (* John M. Campbell, Sep 14 2011 *)
Table[(3 n Cos[Pi n/3] - Sqrt[3] (3 n + 4) Sin[Pi n/3])/9, {n, 20}] (* Vladimir Reshetnikov, Sep 08 2016 *)
LinearRecurrence[{2, -3, 2, -1}, {-1, -2, -1, 2}, 90] (* Vincenzo Librandi, Sep 09 2016 *)

M = the 4 X 4 companion matrix to the Quadrafoil polynomial x^4 - 2x^3 + 3x^2 - 2x + 1: [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 2 -3 2]. a(n) = rightmost term in M^n * [1 0 0 0].

O.g.f.: -x/(x^2-x+1)^2. a(n) = 2*a(n-1)-3*a(n-2)+2*a(n-3)-a(n-4) = -A099254(n-1). - R. J. Mathar, Apr 06 2008, Apr 23 2009

A128214 Expansion of (1+2x+3x^2)/(1+x+x^2)^2.

Original entry on oeis.org

1, 0, 0, -2, 3, 0, -5, 6, 0, -8, 9, 0, -11, 12, 0, -14, 15, 0, -17, 18, 0, -20, 21, 0, -23, 24, 0, -26, 27, 0, -29, 30, 0, -32, 33, 0, -35, 36, 0, -38, 39, 0, -41, 42, 0, -44, 45, 0, -47, 48, 0, -50, 51, 0, -53, 54, 0, -56, 57, 0, -59, 60, 0, -62, 63, 0, -65

Offset: 0

Paul Barry, Feb 19 2007

Binomial transform is A128213.

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

Cf. A128213.

CoefficientList[Series[(1 + 2 x + 3 x^2)/(1 + x + x^2)^2, {x, 0, 50}], x] (* Wesley Ivan Hurt, Mar 15 2015 *)
LinearRecurrence[{-2,-3,-2,-1},{1,0,0,-2},70] (* Harvey P. Dale, Jul 16 2021 *)

Vec((1+2*x+3*x^2)/(1+x+x^2)^2 + O(x^80)) \\ Michel Marcus, Mar 16 2015

G.f.: (1+2x+3x^2)/(1+x+x^2)^2.
a(n) = (1-n)*cos(2*Pi*n/3)+(n-1)*sin(2*Pi*n/3)/sqrt(3).
a(n) = (-1)^n*( A099254(n)-2*A099254(n-1)+3*A099254(n-2) ). - R. J. Mathar, Mar 21 2011
From Wesley Ivan Hurt, Mar 15 2015: (Start)
a(n) + 2*a(n-1) + 3*a(n-2) + 2*a(n-3) + a(n-4) = 0.
a(n) = (n-1) * ((n-2)^2 mod 3) * (-1)^floor((2n-2)/3). (End)

More terms from Wesley Ivan Hurt, Mar 15 2015

A147621 The 3rd Witt transform of A000292.

Original entry on oeis.org

0, 0, 0, 0, 4, 26, 120, 455, 1456, 4122, 10608, 25194, 55980, 117572, 235144, 450681, 832048, 1485800, 2575368, 4345965, 7158060, 11532402, 18209100, 28224105, 43008120, 64512240, 95365920, 139075245, 200268432, 284997384, 401107356

Offset: 0

R. J. Mathar, Nov 08 2008

The 2nd Witt transform is essentially in A032094.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Index entries for linear recurrences with constant coefficients, signature (8,-28,60,-102,168,-258,336,-393,452,-484,452,-393, 336,-258,168,-102,60,-28,8,-1).

Cf. A000292, A032094, A049347, A099254, A128504.

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) )); // G. C. Greubel, Oct 24 2022

CoefficientList[Series[x^4(2*x^2 - x + 2)(2*x^4 - 2*x^3 + 9*x^2 - 2*x+2)/((1-x)^12 * (1 + x + x^2)^4), {x, 0, 40}],  x] (* Vincenzo Librandi  Dec 13 2012 *)

def A147621_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) ).list()
A147621_list(40) # G. C. Greubel, Oct 24 2022

G.f.: x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4).

a(n) = (1/729)*(b(n) + c(n)), where b(n) = n*(n+3)*(n+6)*(3*n^8 +72*n^7 +618*n^6 + 2052*n^5 +207*n^4 -11772*n^3 -14268*n^2 +9648*n -232960)/492800 and c(n) = 9*A049347(n) +5*A049347(n-1) +9*(-1)^n*(A099254(n) -A099254(n-1)) -18(-1)^n*A128504(n) +27*(-1)^n*Sum_{k=0..n} A099254(n-k)*A099254(k-1). - G. C. Greubel, Oct 24 2022

A200067 Maximum sum of all products of absolute differences and distances between element pairs among the integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 12, 20, 30, 45, 63, 84, 112, 144, 180, 225, 275, 330, 396, 468, 546, 637, 735, 840, 960, 1088, 1224, 1377, 1539, 1710, 1900, 2100, 2310, 2541, 2783, 3036, 3312, 3600, 3900, 4225, 4563, 4914, 5292, 5684, 6090, 6525, 6975, 7440, 7936, 8448

Offset: 0

Alois P. Heinz, Nov 13 2011

Also the maximum sum of weighted inversions among the compositions of n where weights are products of absolute differences and distances between the element pairs which are not in sorted order.

a(n) is divisible by at least one triangular number >1 for n>=4. Thus 3 is the only prime in this sequence.

			a(2) =  0: [1,1]-> 0, [2]-> 0; the maximum is 0.
a(3) =  1: [1,1,1]-> 0, [2,1]-> 1, [3]-> 0; the maximum is 1.
a(4) =  3: [1,1,1,1]-> 0, [2,1,1]-> 1+2 = 3, [2,2]->0, [3,1]->2, [4]->0.
a(5) =  6: [2,1,1,1]-> 1+2+3 = 6, [3,1,1]-> 2 + 2*2 = 2*(1+2) = 6.
a(6) = 12: [3,1,1,1]-> 2 + 2*2 + 2*3 = 2*(1+2+3) = 12.
a(7) = 20: [3,1,1,1,1]-> 2 + 2*2 + 2*3 + 2*4 = 2*(1+2+3+4) = 20.
a(8) = 30: [3,1,1,1,1,1]-> 2*(1+2+3+4+5) = 30, [4,1,1,1,1]-> 3*(1+2+3+4) = 30.

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

Cf. A000217, A004396, A200068.

a:= n-> (k-> (n-k-1)*k*(k+1)/2)(max(0, floor((2*n-1)/3))):
seq(a(n), n=0..50);

a[n_] := Max[Table[(n-k-1)*k*(k+1)/2, {k, 0, n}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 22 2013, after Alois P. Heinz *)

G.f.: x^3*(1+x)*(1+x^2)/((1+x+x^2)^2*(x-1)^4).
a(n) = max_{k=0..n} (n-k-1)*k*(k+1)/2.
a(n) = (n-k-1)*k*(k+1)/2 with k = max(0, floor((2*n-1)/3)), or k = A004396(n-1) for n>0.
27*a(n) = (2*n-1)*(n^2-n-1) - A132677(n) - 3*(-1)^n*A099254(n-1). - R. J. Mathar, Mar 14 2025

A147618 The 3rd Witt transform of A000217.

Original entry on oeis.org

0, 0, 0, 0, 3, 15, 54, 165, 429, 999, 2145, 4290, 8100, 14586, 25194, 41985, 67830, 106590, 163431, 245157, 360525, 520749, 740025, 1036035, 1430703, 1950975, 2629575, 3506085, 4628052, 6052068, 7845255, 10086780, 12869340, 16301142

Offset: 0

R. J. Mathar, Nov 08 2008

The 2nd Witt transform of A000217 is essentially in A032092.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Index entries for linear recurrences with constant coefficients, signature (6,-15,23,-33,51,-64,63,-63,64,-51,33,-23,15,-6,1).

Cf. A000217, A032092, A049347, A099254, A128504.

R:=PowerSeriesRing(Integers(), 30); [0,0,0,0] cat Coefficients(R!( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) )); // G. C. Greubel, Oct 24 2022

CoefficientList[Series[3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3), {x,0,40}], x] (* Vincenzo Librandi, Dec 13 2012 *)

def A147618_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) ).list()
A147618_list(30) # G. C. Greubel, Oct 24 2022

G.f.: 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3).

a(n) = (1/81)*(n*(n+3)*(3*n^6 +27*n^5 +45*n^4 -135*n^3 -288*n^2 +108*n -2000)/4480 +2*A049347(n) +A049347(n-1) +(-1)^n*(A099254(n) -2*A099254(n- 1)) -3*(-1)^n*(A128504(n) -2*A128504(n-1))). - G. C. Greubel, Oct 24 2022

A291014 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^3)^2.

Original entry on oeis.org

0, 0, 2, 6, 12, 23, 48, 105, 228, 486, 1026, 2161, 4548, 9555, 20026, 41874, 87384, 182043, 378648, 786429, 1631120, 3378750, 6990510, 14447045, 29826156, 61516455, 126761190, 260978922, 536870916, 1103567983, 2266788288, 4652881233, 9544371772, 19565962134

Offset: 0

Clark Kimberling, Aug 23 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A291000 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,22,-21,12,-4).

Cf. A000012, A010892, A033453, A099254, A289780, A291000.

R:=PowerSeriesRing(Integers(), 50); [0,0] cat Coefficients(R!( x^2*(2-6*x+6*x^2-3*x^3)/((1-2*x)*(1-x+x^2))^2 )); // G. C. Greubel, Jun 05 2023

z = 60; s = x/(1-x); p = (1 - s^3)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291014 *)
LinearRecurrence[{6,-15,22,-21,12,-4}, {0,0,2,6,12,23}, 50] (* G. C. Greubel, Jun 05 2023 *)

def A291014_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(2-6*x+6*x^2-3*x^3)/((1-2*x)*(1-x+x^2))^2 ).list()
A291014_list(50) # G. C. Greubel, Jun 05 2023

G.f.: x^2*(2 - 6*x + 6*x^2 - 3*x^3)/( (1-2*x)*(1-x+x^2) )^2.
a(n) = 6*a(n-1) - 15*a(n-2) + 22*a(n-3) - 21*a(n-4) + 12*a(n-5) - 4*a(n-6) for n >= 7.
a(n) = (1/9)*( 2^(n-1)*(n + 8) - 3*(A099254(n) - A099254(n-1)) - A010892(n) - 5*A010892(n-1) ). - G. C. Greubel, Jun 05 2023

A136161 a(n) = 2*a(n-3) - a(n-6), starting a(0..5) = 0, 5, 2, 1, 3, 1.

Original entry on oeis.org

0, 5, 2, 1, 3, 1, 2, 1, 0, 3, -1, -1, 4, -3, -2, 5, -5, -3, 6, -7, -4, 7, -9, -5, 8, -11, -6, 9, -13, -7, 10, -15, -8, 11, -17, -9, 12, -19, -10, 13, -21, -11, 14, -23, -12, 15, -25, -13, 16, -27, -14

Offset: 0

Paul Curtz, Mar 16 2008

Consider the general recurrence a(n) = k*a(n-1) + (5-2*k)*a(n-2) + (2-k)*a(n-3). The coefficients, in k, can be used to form the triple (k, 5-2*k, 2-k). Each triple is associated with a sequence, for example (0, 5, 2) leads to A111108, A112685, ..., (1, 3, 1) leads to A051927, A097075, ..., and so on. This sequence is formed from the triples {(0, 5, 2), (1, 3, 1), (2, 1, 0), (3, -1, -1), (4, -3, -2), ...}, for k >= 0. (Comment modified by G. C. Greubel, Dec 31 2023).

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

Cf. A097076, A099254, A112685, A135138, A135997, A137241.

Cf. A051927, A097075, A111108, A112685.

I:=[0,5,2,1,3,1]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..60]]; // G. C. Greubel, Dec 26 2023

LinearRecurrence[{0,0,2,0,0,-1},{0,5,2,1,3,1},60] (* Harvey P. Dale, Aug 16 2012 *)
Table[PadRight[{n, 5-2*n, 2-n}], {n,0,20}]//Flatten (* _G. C. Greubel, Dec 26 2023 *)

Vec(x*(5+2*x+x^2-7*x^3-3*x^4)/((1-x)^2*(1+x+x^2)^2+O(x^99))) \\ Charles R Greathouse IV, Jul 06 2011

def a(n): # a = A136161
    if n<6: return (0,5,2,1,3,1)[n]
    else: return 2*a(n-3) - a(n-6)
[a(n) for n in range(61)] # G. C. Greubel, Dec 26 2023

G.f.: x*(5+2*x+x^2-7*x^3-3*x^4) / ( (1-x)^2*(1+x+x^2)^2 ). - R. J. Mathar, Jul 06 2011
a(3n) = n.
a(3n+1) = 5 - 2*n.
a(3n+3) = 2 - n.
a(n) = (1/9)*( 27 - 2*(n+1) - 34*ChebyshevU(n, -1/2) + (-1)^n*(9*A099254(n) - 6*A099254(n-1)) ). - G. C. Greubel, Dec 26 2023

cp's OEIS Frontend

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A099470 A sequence generated from the Quadrifoil.

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A128214 Expansion of (1+2x+3x^2)/(1+x+x^2)^2.

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A147621 The 3rd Witt transform of A000292.

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A200067 Maximum sum of all products of absolute differences and distances between element pairs among the integer partitions of n.

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