A006130 -id:A006130 - cp's OEIS frontend

A116390 Expansion of 1/(2sqrt(1-4x^2)-x-1).

1, 1, 5, 9, 33, 73, 233, 569, 1693, 4353, 12477, 32985, 92637, 248673, 690549, 1869513, 5158881, 14033161, 38587193, 105246041, 288818305, 788939769, 2162574513, 5912375033, 16196093881, 44300854441, 121311490937

Offset: 0

Paul Barry, Feb 12 2006

Hankel transform is 4^n. - Paul Barry, Jan 19 2011

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

Row sums of number triangle A116389.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2*Sqrt(1-4*x^2)-x-1) )); // G. C. Greubel, May 23 2019

CoefficientList[Series[1/(2*Sqrt[1-4*x^2]-x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)

my(x='x+O('x^30)); Vec(1/(2*sqrt(1-4*x^2)-x-1)) \\ G. C. Greubel, May 23 2019

(1/(2*sqrt(1-4*x^2)-x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 23 2019

a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..floor(n/2)} (-1)^(k-j)*C(k,j) *C(i+(j-1)/2,i)*C(j,n-2*i)*4^i.
a(n) = Sum_{k=0..floor((n+1)/2)} (C(n,k) - C(n,k-1))*A006130(n-2*k). - Paul Barry, Jan 19 2011
Starting with offset 1, let M = an infinite tridiagonal matrix with [1,0,0,0,...] in the main diagonal and [2,1,1,1,...] in the super and subdiagonals. Let V = vector [1,0,0,0,...]. The sequence = iterates of M*V as to the leftmost column. - Gary W. Adamson, Jun 08 2011
D-finite with recurrence: -3*n*a(n) + 2*n*a(n-1) + (29*n-36)*a(n-2) + 8*(3-n)*a(n-3) + 68*(3-n)*a(n-4)=0. - R. J. Mathar, Aug 09 2012
a(n) ~ (1+2/sqrt(13)) * (1+2*sqrt(13))^n / 3^(n+1). - Vaclav Kotesovec, Feb 03 2014

A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 5, 1, 8, 10, 3, 13, 20, 9, 1, 21, 38, 22, 4, 34, 71, 51, 14, 1, 55, 130, 111, 40, 5, 89, 235, 233, 105, 20, 1, 144, 420, 474, 256, 65, 6, 233, 744, 942, 594, 190, 27, 1, 377, 1308, 1836, 1324, 511, 98, 7, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 987, 3970

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045). Column 0 yields the Fibonacci numbers (A000045); the other columns yield convolved Fibonacci numbers (A001629, A001628, A001872, A001873, etc.). Sum_{k=0..floor(n/2)} k*T(n,k) = A073371(n-2).
Triangle T(n,k), with zeros omitted, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 24 2012
Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - Philippe Deléham, Feb 06 2012
Diagonal sums are A000073(n+2) (tribonacci numbers). - Philippe Deléham, Feb 16 2014
Number of induced subgraphs of the Fibonacci cube Gamma(n-1) that are isomorphic to the hypercube Q_k. Example: row n=4 is 5, 5, 1; indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; it has 5 vertices (i.e., Q_0's), 5 edges (i.e., Q_1's) and 1 square (i.e., Q_2). - Emeric Deutsch, Aug 12 2014
Row n gives the coefficients of the polynomial p(n,x) defined as the numerator of the rational function given by f(n,x) = 1 + (x + 1)/f(n-1,x), where f(x,0) = 1. Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). - Clark Kimberling, Oct 22 2014

			Triangle starts:
   1;
   1;
   2,  1;
   3,  2;
   5,  5,  1;
   8, 10,  3;
  13, 20,  9,  1;
  21, 38, 22,  4;
From _Philippe Deléham_, Jan 24 2012: (Start)
Triangle (1, 1, -1, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  2,  0,  0;
   5,  5,  1,  0,  0;
   8, 10,  3,  0,  0,  0;
  13, 20,  9,  1,  0,  0,  0;
  21, 38, 22,  4,  0,  0,  0,  0; (End)
From _Clark Kimberling_, Oct 22 2014: (Start)
Here are the first 4 polynomials p(n,x) as in Comment and generated by Mathematica program:
  1
  2 +  x
  3 + 2x
  5 + 5x + x^2. (End)

C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH: Commun. Math. Comput. Chem, 68 (2012) 3-30. See Eq. (9). - From _N. J. A. Sloane_, Dec 23 2012
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105. - _Emeric Deutsch_, Aug 12 2014

Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371.

Cf. A109466, A119473.

G:=1/(1-z-(1+t)*z^2): Gser:=simplify(series(G,z=0,19)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

p[x_, n_] := 1 + (x + 1)/p[x, n - 1]; p[x_, 1] = 1;
Numerator[Table[Factor[p[x, n]], {n, 1, 20}]]  (* Clark Kimberling, Oct 22 2014 *)

G.f.: 1/(1-z-(1+t)z^2).
Sum_{k=0..n} T(n,k)*x^k = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, and -13, respectively. - Philippe Deléham, Jan 24 2012
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Jan 24 2012
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
T(n,k) = Sum_{i=k..floor(n/2)} binomial(n-i,i)*binomial(i,k). See Corollary 3.3 in the Klavzar et al. link. - Emeric Deutsch, Aug 12 2014

A136689 Triangular sequence of q-Fibonacci polynomials for s=3: F(x,n) = xF(x,n-1) + sF(x,n-2).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 6, 0, 1, 9, 0, 9, 0, 1, 0, 27, 0, 12, 0, 1, 27, 0, 54, 0, 15, 0, 1, 0, 108, 0, 90, 0, 18, 0, 1, 81, 0, 270, 0, 135, 0, 21, 0, 1, 0, 405, 0, 540, 0, 189, 0, 24, 0, 1, 243, 0, 1215, 0, 945, 0, 252, 0, 27, 0, 1, 0, 1458, 0, 2835, 0, 1512, 0

Offset: 1

Roger L. Bagula, Apr 06 2008

Row sums: 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ... = A006130(n-1).

			Triangle begins:
    1;
    0,   1;
    3,   0,    1;
    0,   6,    0,   1;
    9,   0,    9,   0,   1;
    0,  27,    0,  12,   0,   1;
   27,   0,   54,   0,  15,   0,   1;
    0, 108,    0,  90,   0,  18,   0,  1;
   81,   0,  270,   0, 135,   0,  21,  0,  1;
    0, 405,    0, 540,   0, 189,   0, 24,  0, 1;
  243,   0, 1215,   0, 945,   0, 252,  0, 27, 0, 1;
  ...

Nathaniel Johnston, Rows n=1..36 of triangle, flattened
J. Cigler, q-Fibonacci polynomials, Fibonacci Quarterly 41 (2003) 31-40.

Cf. A136688, A136705.

A136689 := proc(n) option remember: if(n<=1)then return n: else return x*procname(n-1)+3*procname(n-2): fi: end:
seq(seq(coeff(A136689(n), x, m), m=0..n-1), n=1..10); # Nathaniel Johnston, Apr 27 2011

s=2; F[x_, n_]:= F[x, n]= If[n<2, n, x*F[x, n-1] + s*F[x, n-2]]; Table[
CoefficientList[F[x, n], x], {n,10}]//Flatten
F[n_, x_, s_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^Binomial[j+1, 2]*x^(n-2*j-1) *s^j, {j, 0, Floor[(n-1)/2]}]; Table[CoefficientList[F[n,x,3,1], x], {n, 1, 10}]//Flatten (* G. C. Greubel, Dec 16 2019 *)

def f(n,x,s,q): return sum( q_binomial(n-j-1, j, q)*q^binomial(j+1,2)*x^(n-2*j-1)*s^j for j in (0..floor((n-1)/2)))
def A136689_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(n,x,3,1) ).list()
[A136689_list(n) for n in (1..10)] # G. C. Greubel, Dec 16 2019

F(x,n) = x*F(x,n-1) + s*F(x,n-2), where F(x,0)=0, F(x,1)=1 and s=3.

A176737 Expansion of 1 / (1-4x^2-3x^3). (4,3)-Padovan sequence.

Original entry on oeis.org

1, 0, 4, 3, 16, 24, 73, 144, 364, 795, 1888, 4272, 9937, 22752, 52564, 120819, 278512, 640968, 1476505, 3399408, 7828924, 18027147, 41513920, 95595360, 220137121, 506923200, 1167334564, 2688104163, 6190107856, 14254420344, 32824743913, 75588004944, 174062236684

Offset: 0

Wolfdieter Lang, Jun 26 2010

See A000931 (Padovan), and the W. Lang link given there.

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

Cf. A053088 ((3,2)-Padovan).

CoefficientList[Series[1/(1-4*x^2-3*x^3),{x,0,40}],x] (* or *) LinearRecurrence[ {0,4,3},{1,0,4},40] (* Harvey P. Dale, Jan 21 2013 *)

Vec(1 / ((1 + x)*(1 - x - 3*x^2)) + O(x^40)) \\ Colin Barker, Dec 25 2017

O.g.f.: 1/((1-x-3*x^2)*(1+x)) = (2-3*x)/(1-x-3*x^2) -1/(1+x).
a(n) = 2*b(n) - 3*b(n-1) - (-1)^n, n>=0, with b(n):=A006130(n) ((1,3)-Fibonacci), b(-1):=0.
From Wolfdieter Lang, Aug 26 2010: (Start)
a(n) = a(n-1) + 3*a(n-2) + (-1)^n, n>=2, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x)= x*(1 + 3*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in an Aug 24 2010 email to the author.)
(End)
a(n) = 4*a(n-2) + 3*a(n-3) for n>2. - Harvey P. Dale, Jan 21 2013
a(n) = (-1)^(n+1)*A140165(n+2)-(-1)^n. - R. J. Mathar, Apr 22 2013
a(n) = ((-1)^(1+n) + (2^(-n)*((-2+sqrt(13))*(1+sqrt(13))^n + (1-sqrt(13))^n*(2+sqrt(13)))) / sqrt(13)). - Colin Barker, Dec 25 2017

A362298 Number of tilings of a 4 X n rectangle using dominos and 2 X 2 right triangles.

Original entry on oeis.org

1, 1, 19, 55, 472, 2023, 13249, 66325, 392299, 2088856, 11877025, 64803157, 362823607, 1998759703, 11123273896, 61509329983, 341492705365, 1891193243713, 10489893539203, 58127214942544, 322296397820593, 1786338231961609, 9903234373856059, 54893955008138983

Offset: 0

Gerhard Kirchner, Apr 19 2023

Triangles only occur as pairs forming 2 X 2 squares. For program code and additional details, see A362297.

			a(2) = 19.
Partitions of a 2 X 2 square (triangles or dominos):
   ___    ___    ___    ___
  |  /|  |\  |  |___|  | | |
  |/__|  |__\|  |___|  |_|_|
       2t            2d
   ___ ___    ___ ___    ___ ___    _ ___ _    _______
  |2t |2t |  |2t |2d |  |2d |2t |  | |2t | |  |only d |
  |___|___|  |___|___|  |___|___|  |_|___|_|  |_______|
    4 ways +   4 ways +  4 ways  +   2 ways +  5 ways  = 19 ways
Only dominos: A005178(3) = 5.

Index entries for linear recurrences with constant coefficients, signature (4,18,-48,-42,99).

Column k=2 of A362297.

Cf. A351322, A352432, A352433, A006130, A362299.

LinearRecurrence[{4,18,-48,-42,99},{1,1,19,55,472},24] (* Stefano Spezia, Apr 20 2023 *)

a(n) = 4*a(n-1) + 18*a(n-2) - 48*a(n-3) - 42*a(n-4) + 99*a(n-5).

G.f.: (9*x^3-3*x^2-3*x+1)/(-99*x^5+42*x^4+48*x^3-18*x^2-4*x+1).

A110111 Sequence associated to the recurrence b(n) = b(n-1) + 3*b(n-2).

Original entry on oeis.org

0, 1, 7, 133, 1330, 18430, 210490, 2673223, 31940881, 394918819, 4788779380, 58709030380, 715296121540, 8745656280829, 106717441265323, 1303667366328817, 15915556720909510, 194371775990116810

Offset: 0

Paul Barry, Jul 12 2005

In general, let T(n,k) be the solution to T(n,k) = T(n-1,k) + k*T(n-2,k) for n >= 2 with T(0,k) = 0 and T(1,k) = 1 for all k. Then, for fixed k, S(n,k) = T(n,k) * T(n+1,k) * T(n+2,k)/(k+1) has g.f. x/((1 + k*x - k^3*x^2) * (1 - (3*k + 1)*x - k^3*x^2)) (cf. A110112). For the current sequence, a(n) = S(n,k=3) = T(n,3) * T(n+1,3) * T(n+2,3)/4. [Edited by Petros Hadjicostas, Dec 26 2019]

Index entries for linear recurrences with constant coefficients, signature (7,84,-189,-729).

Cf. A006130, A110112.

CoefficientList[Series[x/((1+3x-27x^2)(1-10x-27x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{7,84,-189,-729},{0,1,7,133},30] (* Harvey P. Dale, Jun 29 2022 *)

G.f.: x/((1 + 3*x - 27*x^2) * (1 - 10*x - 27*x^2)).
a(n) = 7*a(n-1) + 84*a(n-2) - 189*a(n-3) - 729*a(n-4) for n >= 4.
a(n) = b(n) * b(n+1) * b(n+2)/4, where b(n) = (((1 + sqrt(13))/2 )^n - ((1 - sqrt(13))/2)^n)/sqrt(13). [Corrected by Petros Hadjicostas, Dec 26 2019]
a(n) = A006130(n-1) * A006130(n) * A006130(n+1)/4 for n >= 1.

A112883 A skew Jacobsthal-Pascal matrix.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 2, 5, 0, 0, 1, 7, 11, 0, 0, 0, 3, 16, 21, 0, 0, 0, 1, 12, 41, 43, 0, 0, 0, 0, 4, 34, 94, 85, 0, 0, 0, 0, 1, 18, 99, 219, 171, 0, 0, 0, 0, 0, 5, 60, 261, 492, 341, 0, 0, 0, 0, 0, 1, 25, 195, 678, 1101, 683, 0, 0, 0, 0, 0, 0, 6, 95, 576, 1692, 2426, 1365, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Oct 05 2005

T(n,n) is A001045(n), row sums are A006130, column sums are A002605. Compare with [0,1,-1,0,0,..] DELTA [1,2,-2,0,0,...] where DELTA is the operator defined in A084938. A skewed version of the Riordan array (1/(1-x-2x^2),x/(1-x-2x^2)) (A073370).

Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

			Rows begin
  1;
  0, 1;
  0, 1, 3;
  0, 0, 2, 5;
  0, 0, 1, 7, 11;
  0, 0, 0, 3, 16, 21;
  0, 0, 0, 1, 12, 41, 43;
  0, 0, 0, 0,  4, 34, 94,  85;
  0, 0, 0, 0,  1, 18, 99, 219, 171;
  0, 0, 0, 0,  0,  5, 60, 261, 492,  341;
  0, 0, 0, 0,  0,  1, 25, 195, 678, 1101, 683;

Cf. A111006.

From Philippe Deléham: (Start)
G.f.: 1/(1-yx(1-x)-2x^2*y*2);
Number triangle T(n, k) = Sum_{j=0..2k-n} C(n-k+j, n-k)*C(j, 2k-n-j)*2^(2k-n-j);
T(n, k) = A073370(k, n-k); T(n, k) = T(n-1, k-1) + T(n-2, k-1) + 2*T(n-2, k-2). (End)

A143646 Catalan transform of the 3-Fibonacci sequence A006190.

Original entry on oeis.org

0, 1, 4, 18, 83, 387, 1815, 8541, 40276, 190182, 898844, 4250780, 20111394, 95181166, 450565602, 2133227418, 10101126723, 47834649675, 226540406571, 1072931019393, 5081776592061, 24069823974879, 114009427284309

Offset: 0

Sergio Falcon, Oct 27 2008

nonn

Paul Barry, A Catalan Transform and Related Transformations of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.

Cf. A006190, A143464.

q=50 #change q for more terms
[0]+[sum((k/n)*binomial(2*n-k-1,n-k)*lucas_number1(k,3,-1) for k in [0..n]) for n in [1..q]] # Tom Edgar, Mar 09 2014

a(n) = Sum_{k=0..n} A039599(n,k)*A006130(k-1) with A006130(-1) = 0. - Philippe Deléham, Nov 01 2008

For n>0, a(n) = sum_{k=0..n} (k/n)*C(2n-k-1,n-k)*A006190(k). - Tom Edgar, Mar 09 2014

A172347 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 7, 28, 7, 1, 1, 19, 133, 133, 19, 1, 1, 40, 760, 1330, 760, 40, 1, 1, 97, 3880, 18430, 18430, 3880, 97, 1, 1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1, 1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A006130 and its partial products c(n) = 1, 1, 1, 4, 28, 532, 21280, 2064160, 447922720, 227544741760... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 10, 44, 306, 2932, 44816, 1034844, 36119056, 1882089488,...

			1;
1, 1;
1, 1, 1;
1, 4, 4, 1;
1, 7, 28, 7, 1;
1, 19, 133, 133, 19, 1;
1, 40, 760, 1330, 760, 40, 1;
1, 97, 3880, 18430, 18430, 3880, 97, 1;
1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1;
1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508, 1;
1, 1159, 588772, 31940881, 442609351, 931809160, 442609351, 31940881, 588772, 1159, 1;

Cf. A010048 (m=1), A015109 (m=2), A172349 (m=4).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

A206800 Riordan array (1/(1-3x+x^2), x(1-x)/(1-3*x+x^2)).

Original entry on oeis.org

1, 3, 1, 8, 5, 1, 21, 19, 7, 1, 55, 65, 34, 9, 1, 144, 210, 141, 53, 11, 1, 377, 654, 534, 257, 76, 13, 1, 987, 1985, 1905, 1111, 421, 103, 15, 1, 2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1, 6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1

Offset: 0

Philippe Deléham, Feb 12 2012

			Triangle begins :
1
3, 1
8, 5, 1
21, 19, 7, 1
55, 65, 34, 9, 1
144, 210, 141, 53, 11, 1
377, 654, 534, 257, 76, 13, 1
987, 1985, 1905, 1111, 421, 103, 15, 1
2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1
6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1
Triangle (0,3,-1/3,1/3,0,0,0,0,0,...) DELTA (1,0,-1/3,1/3,0,0,0,0,...) begins :
1
0, 1
0, 3, 1
0, 8, 5, 1
0, 21, 19, 7, 1
0, 55, 65, 34, 9, 1...

Subtriangle of the triangle given by (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Antidiagonal sums are A072264(n).

Cf. A000045, A001519, A001906, A001870,

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1).
G.f.: 1/(1-(y+3)*x+(y+1)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n* A015587(n+1), (-1)^n*A190953(n+1), (-1)^n*A015566(n+1), (-1)*A189800(n+1), (-1)^n*A015541(n+1), (-1)^n*A085939(n+1), (-1)^n*A015523(n+1), (-1)^n*A063727(n), (-1)^n*A006130(n), A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n) for x = -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

cp's OEIS Frontend

A116390 Expansion of 1/(2*sqrt(1-4*x^2)-x-1).

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A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

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A136689 Triangular sequence of q-Fibonacci polynomials for s=3: F(x,n) = x*F(x,n-1) + s*F(x,n-2).

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A176737 Expansion of 1 / (1-4*x^2-3*x^3). (4,3)-Padovan sequence.

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A362298 Number of tilings of a 4 X n rectangle using dominos and 2 X 2 right triangles.

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A110111 Sequence associated to the recurrence b(n) = b(n-1) + 3*b(n-2).

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A112883 A skew Jacobsthal-Pascal matrix.

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A143646 Catalan transform of the 3-Fibonacci sequence A006190.

A116390 Expansion of 1/(2sqrt(1-4x^2)-x-1).

A136689 Triangular sequence of q-Fibonacci polynomials for s=3: F(x,n) = xF(x,n-1) + sF(x,n-2).

A176737 Expansion of 1 / (1-4x^2-3x^3). (4,3)-Padovan sequence.

A206800 Riordan array (1/(1-3x+x^2), x(1-x)/(1-3*x+x^2)).