A057681 -id:A057681 - cp's OEIS frontend

A057083 Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1 - 3x + 3x^2).

1, 3, 6, 9, 9, 0, -27, -81, -162, -243, -243, 0, 729, 2187, 4374, 6561, 6561, 0, -19683, -59049, -118098, -177147, -177147, 0, 531441, 1594323, 3188646, 4782969, 4782969, 0, -14348907, -43046721, -86093442, -129140163, -129140163, 0

Offset: 0

Wolfdieter Lang, Aug 11 2000

With different sign pattern, see A000748.

Conjecture: Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = A057681(n) - A057682(n)*M + z(n)*M^2, where z(0) = z(1) = 0 and, apparently, z(n+2) = a(n). - Stanislav Sykora, Jun 10 2012

Robert Israel, Table of n, a(n) for n = 0..4170
T. Alden Gassert, Discriminants of simplest 3^n-tic extensions, arXiv preprint arXiv:1409.7829 [math.NT], 2014.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=3, q=-3.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45),lhs, m=3.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (3,-3).

Cf. A000748, A010892, A049310, A057078, A057681, A057682, A129339.

I:=[1,3]; [n le 2 select I[n] else 3*Self(n-1) - 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 23 2018

seq(3^(n/2)*orthopoly[U](n,sqrt(3)/2),n=0..100); # Robert Israel, Nov 21 2016

Join[{a=1,b=3},Table[c=3*b-3*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
CoefficientList[Series[1/(1 - 3 x + 3 x^2), {x, 0, 35}], x] (* Michael De Vlieger, Jul 30 2017 *)

a(n)=([0,1; -3,3]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,3,3) for n in range(1, 37)] # Zerinvary Lajos, Apr 23 2009

a(n) = S(n, sqrt(3))*(sqrt(3))^n with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
a(2*n) = A057078(n)*3^n; a(2*n+1)= A010892(n)*3^(n+1).
G.f.: 1/(1-3*x+3*x^2).
Binomial transform of A057079. a(n) = Sum_{k=0..n} 2*binomial(n, k)*cos((k-1)Pi/3). - Paul Barry, Aug 19 2003
For n > 5, a(n) = -27*a(n-6) - Gerald McGarvey, Apr 21 2005
a(n) = Sum_{k=0..n} A109466(n,k)*3^k. - Philippe Deléham, Nov 12 2008
a(n) = Sum_{k=1..n} binomial(k,n-k) * 3^k *(-1)^(n-k) for n>0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
By the conjecture: Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) - z(n), y(n+1) = y(n) - x(n), z(n+1) = z(n) - y(n). Then a(n) = z(n+2). This recurrence indeed ends up in a repetitive cycle of length 6 and multiplicative factor -27, confirming G. McGarvey's observation. - Stanislav Sykora, Jun 10 2012
G.f.: Q(0) where Q(k) =  1 + k*(3*x+1) + 9*x - 3*x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013
G.f.: G(0)/(2-3*x), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+4) + 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
a(n) = Sum_{k = 0..floor(n/3)} (-1)^k*binomial(n+2,3*k+2). Sykora's conjecture in the Comments section follows easily from this. - Peter Bala, Nov 21 2016
From Vladimir Shevelev, Jul 30 2017: (Start)
a(n) = 2*3^(n/2)*cos(Pi*(n-2)/6);
a(n) = K_2(n+2) - K_1(n+2);
For m,n>=1, a(n+m) = a(n-1)*K_1(m+1) + K_2(n+1)*K_2(m+1) + K_1(n+1)*a(m-1) where K_1 = A057681, K_2 = A057682. (End)

A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j+1).

Original entry on oeis.org

0, 1, 2, 3, 3, 0, -9, -27, -54, -81, -81, 0, 243, 729, 1458, 2187, 2187, 0, -6561, -19683, -39366, -59049, -59049, 0, 177147, 531441, 1062882, 1594323, 1594323, 0, -4782969, -14348907, -28697814, -43046721, -43046721, 0, 129140163, 387420489, 774840978

Offset: 0

N. J. A. Sloane, Oct 20 2000

Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = A057681(n)-a(n)*M+z(n)*M^2, where z(0)=z(1)=0 and, apparently, z(n+2)=A057083(n). - Stanislav Sykora, Jun 10 2012
From Tom Copeland, Nov 09 2014: (Start)
This array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interp. (here t=-2) o.g.f. G(x,t) = x(1-x)/[1+(t-1)x(1-x)] and inverse o.g.f. Ginv(x,t) =  [1-sqrt(1-4x/(1+(1-t)x))]/2 (Cf. A005773 and A091867 and A030528 for more info on this family). (End)
{A057681, A057682, A*}, where A* is A057083 prefixed by two 0's, is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x)} of order 3. For the definitions of {k_i(x)} and the difference analog {K_i (n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Jul 31 2017

			G.f. = x + 2*x^2 + 3*x^3 + 3*x^4 - 9*x^6 - 27*x^7 - 54*x^8 - 81*x^9 + ...
If M^3=1 then (1-M)^6 = A057681(6) - a(6)*M + A057083(4)*M^2 = -18 + 9*M + 9*M^2. - _Stanislav Sykora_, Jun 10 2012

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mark W. Coffey, Reductions of particular hypergeometric functions 3F2 (a, a+1/3, a+2/3; p/3, q/3; +-1), arXiv preprint arXiv:1506.09160 [math.CA], 2015.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3).

Alternating row sums of triangle A030523.

Cf. A000108, A000484, A000748, A005043, A005773, A057083, A057681, A091867.

I:=[0,1,2]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

A057682:=n->add((-1)^j*binomial(n,3*j+1), j=0..floor(n/3)):
seq(A057682(n), n=0..50); # Wesley Ivan Hurt, Nov 11 2014

A[n_] := Array[KroneckerDelta[#1, #2 + 1] - KroneckerDelta[#1, #2] + Sum[KroneckerDelta[#1, #2 -q], {q, n}] &, {n, n}];
Join[{0,1}, Table[(-1)^(n-1)*Total[CoefficientList[ CharacteristicPolynomial[A[(n-1)], x], x]], {n,2,30}]] (* John M. Campbell, Mar 16 2012 *)
Join[{0}, LinearRecurrence[{3,-3}, {1,2}, 40]] (* Jean-François Alcover, Jan 08 2019 *)

{a(n) = sum( j=0, n\3, (-1)^j * binomial(n, 3*j + 1))} /* Michael Somos, May 26 2004 */

{a(n) = if( n<2, n>0, n-=2; polsym(x^2 - 3*x + 3, n)[n + 1])} /* Michael Somos, May 26 2004 */

b=BinaryRecurrenceSequence(3,-3,1,2)
def A057682(n): return 0 if n==0 else b(n-1)
[A057682(n) for n in range(41)] # G. C. Greubel, Jul 14 2023

G.f.: (x - x^2) / (1 - 3*x + 3*x^2).
a(n) = 3*a(n-1) - 3*a(n-2), if n>1.
Starting at 1, the binomial transform of A000484. - Paul Barry, Jul 21 2003
It appears that abs(a(n)) = floor(abs(A000748(n))/3). - John W. Layman, Sep 05 2003
a(n) = ((3+i*sqrt(3))/2)^(n-2) + ((3-i*sqrt(3))/2)^(n-2). - Benoit Cloitre, Oct 27 2003
a(n) = n*3F2(1/3-n/3,2/3-n/3,1-n/3 ; 2/3,4/3 ; 1) for n>=1. - John M. Campbell, Jun 01 2011
Let A(n) be the n X n matrix with -1's along the main diagonal, 1's everywhere above the main diagonal, and 1's along the subdiagonal. Then a(n) equals (-1)^(n-1) times the sum of the coefficients of the characteristic polynomial of A(n-1), for all n>1 (see Mathematica code below). - John M. Campbell, Mar 16 2012
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) - z(n), y(n+1) = y(n) - x(n), z(n+1) = z(n) - y(n). Then a(n) = -y(n). But this recurrence falls into a repetitive cycle of length 6 and multiplicative factor -27, so that a(n) = -27*a(n-6) for any n>6. - Stanislav Sykora, Jun 10 2012
a(n) = A057083(n-1) - A057083(n-2). - R. J. Mathar, Oct 25 2012
G.f.: 3*x - 1/3 + 3*x/(G(0) - 1) where G(k)= 1 + 3*(2*k+3)*x/(2*k+1 - 3*x*(k+2)*(2*k+1)/(3*x*(k+2) + (k+1)/G(k+1)));(continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
G.f.: Q(0,u) -1, where u=x/(1-x), Q(k,u) = 1 - u^2 + (k+2)*u - u*(k+1 - u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
From Vladimir Shevelev, Jul 31 2017: (Start)
For n>=1, a(n) = 2*3^((n-2)/2)*cos(Pi*(n-2)/6);
For n>=2, a(n) = K_1(n) + K_3(n-2);
For m,n>=2, a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_3(n-2)*K_3(m-2), where
K_1 = A057681, K_3 = A057083. (End)

A307039 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -2, 0, 1, 1, 1, 0, -4, 0, 1, 1, 1, 1, -3, -4, 0, 1, 1, 1, 1, 0, -9, 0, 0, 1, 1, 1, 1, 1, -4, -18, 8, 0, 1, 1, 1, 1, 1, 0, -14, -27, 16, 0, 1, 1, 1, 1, 1, 1, -5, -34, -27, 16, 0, 1, 1, 1, 1, 1, 1, 0, -20, -68, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, -6, -55, -116, 81, -32, 0

Offset: 0

Seiichi Manyama, Mar 21 2019

			Square array begins:
   1,  1,   1,    1,    1,   1,   1,  1, ...
   0,  1,   1,    1,    1,   1,   1,  1, ...
   0,  0,   1,    1,    1,   1,   1,  1, ...
   0, -2,   0,    1,    1,   1,   1,  1, ...
   0, -4,  -3,    0,    1,   1,   1,  1, ...
   0,  0, -18,  -14,   -5,   0,   1,  1, ...
   0,  8, -27,  -34,  -20,  -6,   0,  1, ...
   0, 16, -27,  -68,  -55, -27,  -7,  0, ...
   0, 16,   0, -116, -125, -83, -35, -8, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-9 give A000007, A146559, A057681, A099586, A289306, A307040, A307041, A307044, A307045.

Cf. A306846, A306914.

T[n_, k_] := Sum[(-1)^j * Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 13}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n,k*j).

A192080 Expansion of 1/((1-x)^6 - x^6).

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 463, 804, 1365, 2366, 4368, 8736, 18565, 40410, 87381, 184604, 379050, 758100, 1486675, 2884776, 5592405, 10919090, 21572460, 43144920, 87087001, 176565486, 357913941, 723002336, 1453179126, 2906358252

Offset: 0

Bruno Berselli, Jun 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Cf. A006090, A049016, A049017, A057079, A057083, A057681.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), this sequence (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

m:=30; R:=PowerSeriesRing(Integers(),m); Coefficients(R!(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))));

CoefficientList[Series[1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), {x,0,50}], x] (* Vincenzo Librandi, Oct 15 2012 *)
LinearRecurrence[{6,-15,20,-15,6},{1,6,21,56,126},30] (* Harvey P. Dale, Feb 22 2017 *)

makelist(coeff(taylor(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), x, 0, n), x, n), n, 0, 29);

Vec(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))+O(x^99)) \\ Charles R Greathouse IV, Jun 23 2011

def A192080_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^6-x^6) ).list()
A192080_list(51) # G. C. Greubel, Apr 11 2023

a(n) = abs(A006090(n)) = (-1)^n * A006090(n).
G.f.: 1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)).
From G. C. Greubel, Apr 11 2023: (Start)
a(n) = (2^(n+5) + A010892(n) - 2*A010892(n-1) - 27*(A057083(n) - 2*A057083(n-1)))/6.
a(n) = (2^(n+5) + A057079(n+2) - 27*A057681(n+1))/6. (End)

A099587 a(n) = coefficient of x in (1+x)^n mod (1+x^4).

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 0, -14, -48, -116, -232, -396, -560, -560, 0, 1912, 6528, 15760, 31520, 53808, 76096, 76096, 0, -259808, -887040, -2141504, -4283008, -7311552, -10340096, -10340096, 0, 35303296, 120532992, 290992384, 581984768

Offset: 0

Ralf Stephan, Oct 24 2004

{A099586, A099587, A099588, A099589} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x)} of order 4.

For the definition, see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jul 03 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Vladimir Shevelev, Coefficient of x^k in ((x+1)^n modulo x^N+1), seqfan, Thu Jul 20 2017.
G. Tollisen and T. Lengyel, A Congruential Identity and the 2-adic Order of Lacunary Sums of Binomial Coefficients, Integers 4 (2004), #A4.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-2).

Cf. A099586, A099588, A099589.

Cf. A057681, A099586, A139398, A133476, A139714, A139748, A139761.

RecurrenceTable[{a[1]=1, a[2]=2, a[3]=3, a[4]=4, a[n] = 4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4]}, a, {n, 1, 100}] (* G. C. Greubel, Nov 09 2015 *)
a[n_] := n*HypergeometricPFQ[{(1-n)/4, (2-n)/4, (3-n)/4, (4-n)/4}, {1/2, 3/4, 5/4}, -1]; Array[a, 40, 0] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)
LinearRecurrence[{4,-6,4,-2},{0,1,2,3},50] (* Harvey P. Dale, Mar 27 2022 *)

PARI
```
a(n) = polcoeff(((1+x)^n)%(x^4+1),1)
```

G.f.: x*(x-1)^2 / (2*x^4-4*x^3+6*x^2-4*x+1). - Colin Barker, Jul 15 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-4). - G. C. Greubel, Nov 09 2015
From Vladimir Shevelev, Jun 29 2017: (Start)
a(n) = Sum_{k >= 0}(-1)^k*binomial(n,4*k+1).
a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-2)/8)/2), where round(x) is the integer nearest to x.
a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_4(n)*K_3(m) - K_3(n)*K_4(m), where K_1 is A099586, K_3=A099588, and K_4=A099589.
(End)
a(n) = A099589(n+2)-2*A099589(n+1)+A099589(n). - R. J. Mathar, Jun 28 2025

a(0)=0 added by N. J. A. Sloane, Jun 30 2017

A131023 First subdiagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 14, 37, 101, 256, 593, 1267, 2534, 4825, 8921, 16384, 30581, 58975, 117950, 242461, 504605, 1048576, 2156201, 4371451, 8742902, 17308657, 34085873, 67108864, 132623405, 263652487, 527304974, 1059392917, 2133134741

Offset: 1

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

Also first differences of main diagonal A129339.

			For first seven rows of T see A131022 or A129339.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6).

Cf. A131022 (T read by rows), A129339 (main diagonal of T), A131024 (row sums of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

m:=34; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ M[n+1, n]: n in [1..m-1] ];

{m=33; v=concat([1, 2, 3, 4],vector(m-4)); for(n=5, m, v[n]=5*v[n-1]-9*v[n-2]+6*v[n-3]); v}

a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4; for n > 4, a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3).
G.f.: x*(1-3*x+2*x^2+x^3)/((1-2*x)*(1-3*x+3*x^2)).
a(n) = A057681(n-1) + 2^(n-2), a(1) = 1. - Bruno Berselli, Feb 17 2011

A348308 a(n) = Sum_{k=0..floor(n/6)} (-1)^k * binomial(n-3k,3k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, -3, -9, -19, -34, -55, -82, -112, -136, -135, -75, 99, 469, 1147, 2269, 3970, 6325, 9235, 12231, 14166, 12771, 4076, -18244, -63424, -143695, -273223, -464779, -722439, -1027959, -1317915, -1448612, -1146827, 52219, 2870965, 8337370, 17769349, 32615514, 54022692, 81938664

Offset: 0

Seiichi Manyama, Oct 11 2021

sign

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

Cf. A348309, A348310.

Cf. A057681, A100134.

LinearRecurrence[{3, -3, 1, 0, 0, -1}, {1, 1, 1, 1, 1, 1}, 45] (* Amiram Eldar, Oct 11 2021 *)

a(n) = sum(k=0, n\6, (-1)^k*binomial(n-3*k, 3*k));

my(N=66, x='x+O('x^N)); Vec((1-x)^2/((1-x)^3+x^6))

G.f.: (1-x)^2/((1-x)^3 + x^6).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-6).

A103135 Expansion of (-3x^3-18x^2+14x-1)/(3x^4-5x^2+4x-1).

Original entry on oeis.org

1, -10, -27, -55, -82, -83, -3, 238, 721, 1445, 2166, 2153, -55, -6650, -19827, -39599, -59426, -59659, -987, 175550, 528857, 1058701, 1587558, 1583377, -17711, -4811626, -14395275, -28772839, -43168114, -43243139, -317811, 128625934, 386588449, 773494709, 1160083158, 1158736889

Offset: 0

Creighton Dement, Jan 24 2005

A floretion-generated sequence which emerges as a transformation of A000004. a(6n+6)= A103134(n).
It appears that Fib(6n+1) = a(6n+4) - a(6n+5). - Creighton Dement, Jan 31 2005
Floretion Algebra Multiplication Program. FAMP code: 4lesforcycseq[ - .25'i + .5'j - .25i' - .5j' + .5k' - .25'ii' + .75'jj' - .25'kk' + .5'ji' + .25'jk' + .25'kj' + .75e ] Note: vesforcycseq = A000004, 4lesforseq gives A000045, vesseq gives A057681.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,3).

Cf. A103134.

Vec((-3*x^3-18*x^2+14*x-1)/(3*x^4-5*x^2+4*x-1)+O(x^99)) \\ Charles R Greathouse IV, Feb 05 2013

a(n) = -9*A057083(n-1) - Fib(n-2). - Ralf Stephan, May 18 2007

a(n) = 4*a(n-1) - 5*a(n-2) + 3*a(n-4) for n>3. - Colin Barker, May 06 2019

Definition not clear to me. A000004 is the zero sequence! N. J. A. Sloane.

A103312 A transform of the Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 1, 0, -3, -9, -18, -27, -27, 0, 81, 243, 486, 729, 729, 0, -2187, -6561, -13122, -19683, -19683, 0, 59049, 177147, 354294, 531441, 531441, 0, -1594323, -4782969, -9565938, -14348907, -14348907, 0, 43046721, 129140163, 258280326, 387420489, 387420489, 0, -1162261467, -3486784401

Offset: 0

Paul Barry, Jan 30 2005

Apply the Chebyshev transform (1/(1+x^2), x/(1+x^2)) followed by the binomial involution (1/(1-x),-x/(1-x)) (expressed as Riordan arrays) to -A001045(n). All elements are multiples of a power of 3. - Ralf Stephan, Jan 28 2005

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

Cf. A057681.

Join[{0,1},LinearRecurrence[{3,-3},{1,1},50]] (* Harvey P. Dale, Apr 12 2014 *)
CoefficientList[Series[x (1 - x)^2/(1 - 3 x + 3 x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Apr 13 2014 *)

{a(n) = if( n<0, 0, polcoeff( x * (1 - x)^2 / (1 - 3*x + 3*x^2) + x * O(x^n), n))} /* Michael Somos, Sep 29 2007 */

{a(n) = if(n<2, n>0, 3^(n\2-1) * (-1)^((n+1)\6) * (1 + (-1)^((n-1)\3) * (n%3==1)))} /* Michael Somos, Sep 29 2007 */

G.f.: x(1-x)^2/(1-3x+3x^2); a(n)=-sum{j=0..n, (-1)^j*C(n, j)*sum{k=0..floor(j/2), (-1)^k*C(n-k, k)A001045(j-2k)}}.

Recurrence: a(n+2) = 3a(n-1) - 3a(n), starting with 0, 1, 1, 1. - Ralf Stephan, Jan 28 2005

A124394 Inverse of Fine number renewal array.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -2, -2, 0, 1, -3, -4, -3, 0, 1, -4, -5, -6, -4, 0, 1, -5, -4, -6, -8, -5, 0, 1, -6, 0, 0, -6, -10, -6, 0, 1, -7, 8, 14, 8, -5, -12, -7, 0, 1, -8, 21, 36, 36, 20, -3, -14, -8, 0, 1, -9, 40, 63, 72, 65, 36, 0, -16, -9, 0, 1

Offset: 0

Paul Barry, Oct 30 2006

Inverse of A065600. Row sums are A057681(n+1). Diagonal sums are A124395.

			Triangle begins
1,
0, 1,
-1, 0, 1,
-2, -2, 0, 1,
-3, -4, -3, 0, 1,
-4, -5, -6, -4, 0, 1,
-5, -4, -6, -8, -5, 0, 1,
-6, 0, 0, -6, -10, -6, 0, 1

Riordan array ((1-2x)/(1-x)^2,x(1-2x)/(1-x)^2); Number triangle T(n,k)=sum{j=0..k+1, C(k+1,j)C(n-j+k+1,2k+1)(-2)^j};

cp's OEIS Frontend

A057083 Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1 - 3*x + 3*x^2).

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A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1).

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A307039 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k+x^k).

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A192080 Expansion of 1/((1-x)^6 - x^6).

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A099587 a(n) = coefficient of x in (1+x)^n mod (1+x^4).

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A131023 First subdiagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

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A057083 Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1 - 3x + 3x^2).

A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j+1).

A348308 a(n) = Sum_{k=0..floor(n/6)} (-1)^k * binomial(n-3k,3k).

A103135 Expansion of (-3x^3-18x^2+14x-1)/(3x^4-5x^2+4x-1).