A077947 -id:A077947 - cp's OEIS frontend

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 2, 0, 1, 4, 6, 5, 2, 0, 1, 5, 10, 11, 7, 2, 1, 1, 6, 15, 21, 18, 9, 3, 0, 1, 7, 21, 36, 39, 27, 12, 3, 0, 1, 8, 28, 57, 75, 66, 39, 15, 3, 1, 1, 9, 36, 85, 132, 141, 105, 54, 18, 4, 0, 1, 10, 45, 121, 217, 273, 246, 159, 72, 22, 4, 0, 1, 11, 55, 166

Offset: 0

Alford Arnold, May 05 2006

The fourth diagonal is 1, 2, 5, 11, 21, ..., which is 1 + A000292. The fifth diagonal is 0, 2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, ..., which is A051743.
The array A007318 is generated by placing A000012 on both edges with the same Pascal-like recurrence, and the array A059259 uses edges defined by A000012 and A059841. - R. J. Mathar, Jan 21 2008
From Michael A. Allen, Nov 30 2021: (Start)
T(n,n-k) is the (n,k)-th entry of the (1/(1-x^3), x/(1-x)) Riordan array.
Sums of rows give A077947.
Sums of antidiagonals give A079962. (End)

			The table begins
  1
  1  0
  1  1  0
  1  2  1  1
  1  3  3  2  0
  1  4  6  5  2  0
  1  5 10 11  7  2  1
  1  6 15 21 18  9  3  0

Cf. A000292, A079978, A008620, A079998, A051743, A077947.

Generalization of A007318, A059259.

A000012 := proc(n) 1 ; end: A079978 := proc(n) if n mod 3 = 0 then 1; else 0 ; fi ; end: A118923 := proc(n,k) if k = 0 then A000012(n); elif k = n then A079978(n) ; else A118923(n-1,k)+A118923(n-1,k-1) ; fi ; end: for n from 0 to 15 do for k from 0 to n do printf("%d, ",A118923(n,k)) ; od: od: # R. J. Mathar, Jan 21 2008

Flatten@Table[CoefficientList[Series[1/((1 + x*y + x^2*y^2)(1 - x - x*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n + 1, k + 1]], {n, 0, 11}, {k, 0, n}] (* Michael A. Allen, Nov 30 2021 *)

From Michael A. Allen, Nov 30 2021: (Start)
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/3)} binomial(n-3*j,n-k)/(n-3*j).
G.f.: 1/((1+x*y+(x*y)^2)*(1-x-x*y)). (End)

Edited and extended by R. J. Mathar, Jan 21 2008

Offset changed by Michael A. Allen, Nov 30 2021

A122552 a(0)=a(1)=a(2)=1, a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.

Original entry on oeis.org

1, 1, 1, 4, 7, 13, 28, 55, 109, 220, 439, 877, 1756, 3511, 7021, 14044, 28087, 56173, 112348, 224695, 449389, 898780, 1797559, 3595117, 7190236, 14380471, 28760941, 57521884, 115043767, 230087533, 460175068, 920350135, 1840700269, 3681400540

Offset: 0

Philippe Deléham, Sep 20 2006

Equals INVERT transform of (1, 0, 3, 0, 3, 0, 3, ...). - Gary W. Adamson, Apr 27 2009
No term is divisible by 3. - Vladimir Joseph Stephan Orlovsky, Mar 24 2011
For n > 3, a(n) is the number of quaternary sequences of length n-1 starting with q(0) = 0, in which all triples (q(i), q(i+1), q(i+2)) contain digits 0 and 3; cf. A294627. - Wojciech Florek, Jul 30 2018
For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a domino. - Greg Dresden and Bora Bursalı, Aug 31 2023

			It is shown in A294627 that there are 42 quaternary sequences (i.e., build from four digits 0, 1, 2, 3) and having both 0 and 3 in every (consecutive) triple. Only a(5=4+1) = 13 of them start with 0: 003x, 030x, 03y0, 0y30, 0330, where x = 0, 1, 2, 3 and y = 1, 2.

Wojciech Florek, Table of n, a(n) for n = 0..2000
Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A294627.

a:=[1,1,1];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+2*a[n-3]; od; a; # Muniru A Asiru, Jul 30 2018

seq(coeff(series((1-x^2)/(1-x-x^2-2*x^3), x,n+1),x,n),n=0..40); # Muniru A Asiru, Aug 02 2018

LinearRecurrence[{1, 1, 2}, {1, 1, 1}, 40]
CoefficientList[ Series[(x^2 - 1)/(2x^3 + x^2 + x - 1), {x, 0, 35}], x] (* Robert G. Wilson v, Jul 30 2018 *)

Vec((1-x^2)/(1-x-x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Jan 17 2012

from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(1,1,1,1,1,2); [next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008

a(3*n) = 2*a(3*n-1)+2, a(3*n+1) = 2*a(3*n)-1, a(3*n+2) = 2*a(3*n+1)-1, a(0)=1.
G.f.: (1-x^2)/(1-x-x^2-2*x^3).
a(n) = ((-1)^n*A130815(n+2) + 3*2^n)/7. - R. J. Mathar, Nov 30 2008
From Paul Curtz, Oct 02 2009: (Start)
a(n) = A140295(n+2)/4.
a(n+1) - 2a(n) = period 3: repeat -1,-1,2 = -A061347.
a(n) - a(n-1) = 0,0,3,3,6,15,27,54,111,... = 3*A077947.
a(n) - a(n-2) = 0,3,6,9,21,42,81,....
a(n) - a(n-3) = 3,6,12,24,... = A007283 = 3*A000079.
a(3n) + a(3n+1) + a(3n+2) = 3,24,192,... = A103333(n+1) = A140295(3n) + A140295(3n+1) + A140295(3n+2).
See A078010, A139217, A139218. (End)

Corrected by T. D. Noe, Nov 01 2006, Nov 07 2006

Typo in definition corrected by Paul Curtz, Oct 02 2009

A186575 Expansion of (1 + 2x + 6x^2)/(1 - x - x^2 - 2*x^3) in powers of x.

Original entry on oeis.org

1, 3, 10, 15, 31, 66, 127, 255, 514, 1023, 2047, 4098, 8191, 16383, 32770, 65535, 131071, 262146, 524287, 1048575, 2097154, 4194303, 8388607, 16777218, 33554431, 67108863, 134217730, 268435455, 536870911, 1073741826, 2147483647, 4294967295

Offset: 0

Vladimir Kruchinin, Feb 23 2011

From Kai Wang, May 23 2020: (Start)
Let f(t) = t^3 + u*t^2 + v*t + w and {x,y,z} be the simple roots of f(t).
For n >= 0, let p(n) = x^n/((x-y)*(x-z)) + y^n/((y-x)*(y-z)) + z^n/((z-x)*(z-y)) and q(n) = x^n + y^n + z^n.
Then for n >= 0, q(n) = 3*p(n+2) + 2*u*p(n+1) + v*p(n).
In this case, f(t) = t^3 - t^2 - t - 2. q(n) = 3*p(n+2) - 2*p(n+1) - p(n).
p(n) = {0, 0, 1, 1, 2, 5, 9,...}, q(n) = {3, 1, 3, 10, 15, 31,...}.
a(n) = q(n+1), A077939(n) = p(n+2). (End)

			G.f. = 1 + 3*x + 10*x^2 + 15*x^3 + 31*x^4 + 66*x^5 + 127*x^6 + 255*x^7 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Gamaliel Cerda-Morales, A note on Modified Third-order Jacobsthal numbers, arXiv:1905.00725 [math.CO], 2019. See pp. 3-4.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
Kai Wang, Closed Forms and Generating Functions For Power Sums, 2020.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A099837.

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3))); // Marius A. Burtea, Jan 31 2020

CoefficientList[Series[(1+2x+6x^2)/(1-x-x^2-2x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 14 2011 *)

Vec((1 + 2*x + 6*x^2) / ((1 - 2*x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, May 03 2019

polsym(polrecip(1 - x - x^2 - 2*x^3),44)[^1] \\ Joerg Arndt, Jun 23 2020

a(n+1) = n*Sum_{k=1..n} Sum_{j=n-3*k..k} 2^(k-j)*binomial(j,n-3*k+2*j)*binomial(k,j)/k.
G.f.: [log(1/(1 - x - x^2 - 2*x^3))]', (x + x^2 + 2*x^3)^k = Sum_{n>=k} Sum_{j=n-3*k..k} 2^(k-j)*binomial(j,n-3*k+2*j)*binomial(k,j)*x^n (see link).
a(n) = 2^(n+1) + A099837(n+1). - R. J. Mathar, Mar 18 2011
a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n>2. - Colin Barker, May 03 2019
From Kai Wang, May 23 2020: (Start)
a(n) = 3*A077947(n+1) - 2*A077947(n) - A077947(n-1).
A077947(n) = (-8*a(n+3) + 27*a(n+2) - a(n+1))/147. (End)

More terms from Harvey P. Dale, Mar 14 2011

A096669 Rectangular array T(n,k) read by antidiagonals; generating function of row n is 1/F(n,x), where F(n,x) is the polynomial 1 - x - x^2 - 2x^3 -...- F(n+1)x^n and F(n+1) is the (n+1)st Fibonacci number, for n=0,1,2,...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 5, 5, 2, 1, 1, 1, 8, 9, 5, 2, 1, 1, 1, 13, 18, 12, 5, 2, 1, 1, 1, 21, 37, 24, 12, 5, 2, 1, 1, 1, 34, 73, 52, 29, 12, 5, 2, 1, 1, 1, 55, 146, 115, 62, 29, 12, 5, 2, 1, 1, 1, 89, 293, 251, 140, 70, 29, 12, 5, 2, 1, 1, 1, 144, 585, 542, 321, 156, 70

Offset: 1

Clark Kimberling, Jul 03 2004

			Rows begin:
1 1 1 1 1 ... = A000012, with g.f. 1/(1-x)
1 1 2 3 5 ... = A000045, with g.f. 1/(1-x-x^2)
1 1 2 5 9 ... = A077947, with g.f. 1/(1-x-x^2-2*x^3)

Cf. A000045, A096670. Rows converge to A000129.

A152732 a(n) + a(n+1) + a(n+2) = 2^n.

Original entry on oeis.org

0, 0, 2, 2, 4, 10, 18, 36, 74, 146, 292, 586, 1170, 2340, 4682, 9362, 18724, 37450, 74898, 149796, 299594, 599186, 1198372, 2396746, 4793490, 9586980, 19173962, 38347922, 76695844, 153391690, 306783378, 613566756, 1227133514, 2454267026, 4908534052

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

0 + 0 + 2 = 2^1; 0 + 2 + 2 = 2^2; 2 + 2 + 4 = 2^3; 2 + 4 + 10 = 2^4; ...

With a(0)=1, a(n) is the number of length n strings in the language over alphabet {0,1} generated by the regular expression: ((0+1)(0*(11)*)*10)*. - Geoffrey Critzer, Jan 25 2014

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

Cf. A152728, A152729, A152730, A152731, A152725, A152726, A000212.

I:=[0,0,2]; [n le 3 select I[n] else Self(n-1) +Self(n-2) +2*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 01 2018

k0=k1=0;lst={k0,k1};Do[kt=k1;k1=2^n-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{1, 1, 2}, {0, 0, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

concat([0,0],Vec(2/(1-2*x)/(1+x+x^2)+O(x^99))) \\ Charles R Greathouse IV, Feb 24 2012

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = 2*A077947(n-3).
G.f.: 2*x^3/((1-2*x)*(1+x+x^2)). (End)
a(n) = (1/21)*(3*2^n + 18*cos((2*n*Pi)/3) + 2*sqrt(3)*sin((2*n*Pi)/3)). - Zak Seidov, Dec 12 2008

A213948 Triangle, by rows, generated from the INVERT transforms of (1, 1, 2, 4, 8, 16, ...).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 4, 4, 1, 7, 10, 8, 8, 1, 12, 24, 20, 16, 16, 1, 20, 52, 56, 40, 32, 32, 1, 33, 112, 144, 112, 80, 64, 64, 1, 54, 238, 344, 320, 224, 160, 128, 128, 1, 88, 496, 828, 848, 640, 448, 320, 256, 256

Offset: 1

Gary W. Adamson, Jun 25 2012

Row sums = A001519, the odd-indexed Fibonacci terms. The triangle is a companion to A213947, having row sums of the even-indexed Fibonacci terms.

			First few rows of the triangle are:
  1;
  1,   1;
  1,   2,   2;
  1,   4,   4,   4;
  1,   7,  20,   8,   8;
  1,  12,  24,  20,  16,  16;
  1,  20,  52,  56,  40,  32,  32;
  1,  33, 112, 144, 112,  80,  64,  64;
  1,  54, 238, 344, 320, 224, 160, 128, 128;
  1,  88, 496, 828, 848, 640, 448, 320, 256, 256;
  ...

Cf. A001519, A213947, A000071 (2nd column), A020714 (subdiagonal), A005009 (subdiagonal).

read("transforms") ;
A213948i := proc(n,k)
    if n = 1 then
        L := [1,seq(0,i=0..k)] ;
    else
        L := [1,seq(2^i,i=0..n-2),seq(0,i=0..k)] ;
    end if;
    INVERT(L) ;
    op(k,%) ;
end proc:
A213948 := proc(n,k)
    if k = 1 then
        1;
    else
        A213948i(k,n)-A213948i(k-1,n) ;
    end if;
end proc: # R. J. Mathar, Jun 30 2012

Create an array in which the n-th row is the INVERT transform of the first n terms in the sequence (1, 1, 2, 4, 8, 16, 32, ...):
  1,    1,    1,    1,    1,    1,
  1,    2,    3,    5,    8,   13,    (essentially A000045)
  1,    2,    5,    9,   18,   37,    (essentially A077947)
  1,    2,    5,   13,   26,   57,
Terms of the n-th row of the triangle are the finite differences downwards the n-th column of this array.

A349842 Expansion of 1/((1 - 2x)(1 + x + x^2 + x^3 + x^4)).

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 33, 66, 132, 264, 529, 1057, 2114, 4228, 8456, 16913, 33825, 67650, 135300, 270600, 541201, 1082401, 2164802, 4329604, 8659208, 17318417, 34636833, 69273666, 138547332, 277094664, 554189329, 1108378657, 2216757314, 4433514628, 8867029256, 17734058513

Offset: 0

Michael A. Allen, Dec 13 2021

Number of ways to tile an n-board (an n X 1 array of 1 X 1 cells) using squares, dominoes, trominoes, tetrominoes, black pentominoes, and white pentominoes.

Row sums of A349841.

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,2).

Row sums of triangles in the same family as A349841: A000079, A001045, A077947, A115451.

CoefficientList[Series[(1 - x)/((1 - x^5)(1 - 2x)), {x, 0, 35}], x]

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + 2*a(n-5) + delta(n,0), a(n<0)=0.
a(n) = 2*a(n-1) + a(n-5) - 2*a(n-6) + delta(n,0) - delta(n,1), a(n<0)=0.
G.f.: 1/(1-x-x^2-x^3-x^4-2*x^5).

A341905 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2.

Original entry on oeis.org

3, 0, 2, 8, 10, 22, 48, 90, 182, 368, 730, 1462, 2928, 5850, 11702, 23408, 46810, 93622, 187248, 374490, 748982, 1497968, 2995930, 5991862, 11983728, 23967450, 47934902, 95869808, 191739610, 383479222, 766958448, 1533916890, 3067833782, 6135667568, 12271335130

Offset: 0

Michael De Vlieger, Jun 04 2021

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A001045, A077947, A186575, A226308.

a:= n-> (<<0|1|0>, <0|0|1>, <2|1|1>>^n. <<3, 0, 2>>)[1,1]:
seq(a(n), n=0..34);  # Alois P. Heinz, Jun 04 2021

LinearRecurrence[{1, 1, 2}, {3, 0, 2}, 35] (* or *)
CoefficientList[Series[(-3 + 3 x + x^2)/(-1 + x + x^2 + 2 x^3), {x, 0, 34}], x]

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

a(n) = (10*2^(n-1) + 13*A049347(n) - 9*A079978(n+1) + 3)/7. - Greg Dresden, Jun 20 2021

A104579 A Padovan-Jacobsthal convolution triangle.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 1, 4, 3, 0, 1, 4, 3, 6, 4, 0, 1, 5, 12, 6, 8, 5, 0, 1, 6, 16, 24, 10, 10, 6, 0, 1, 13, 24, 34, 40, 15, 12, 7, 0, 1, 16, 53, 60, 60, 60, 21, 14, 8, 0, 1, 25, 72, 135, 120, 95, 84, 28, 16, 9, 0, 1, 42, 126, 200, 275, 210, 140, 112, 36, 18, 10, 0, 1, 57, 220, 381

Offset: 0

Paul Barry, Mar 16 2005

First column is A052947. Row sums are A077947. Diagonal sums are A052907.

			Rows begin {1},{0,1},{1,0,1},{2,2,0,1},{1,4,3,0,1},{4,3,6,4,0,1},..

Riordan array (1/(1-x^2-2x^3), x/(1-x^2-2x^3))

T(n,k) = T(n-1,k-1)+T(n-2,k)+2*T(n-3,k), T(0,0)=1, T(n,k)=0 if k>n or if k<0. - Philippe Deléham, Jan 08 2014

A191239 Triangle T(n,k) = coefficient of x^n in expansion of (x+x^2+2*x^3)^k.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 0, 5, 3, 1, 0, 4, 9, 4, 1, 0, 4, 13, 14, 5, 1, 0, 0, 18, 28, 20, 6, 1, 0, 0, 12, 49, 50, 27, 7, 1, 0, 0, 8, 56, 105, 80, 35, 8, 1, 0, 0, 0, 56, 161, 195, 119, 44, 9, 1, 0, 0, 0, 32, 210, 366, 329, 168, 54, 10, 1, 0, 0, 0, 16, 200, 581, 721, 518, 228, 65, 11, 1, 0, 0, 0, 0, 160, 732, 1337, 1288, 774, 300, 77, 12, 1, 0, 0, 0, 0, 80, 780, 2045, 2716, 2142, 1110, 385, 90, 13, 1

Offset: 1

Vladimir Kruchinin, May 27 2011

Riordan Array (1,x+x^2+2*x^3) without first column.
Riordan Array (1+x+2*x^3,x+x^2+2*x^3) numbering triangle (0,0).
Bell Polynomial of second kind B(n,k){1,2,12,0,0,0,...,0}=n!/k!*T(n,k).
For the g.f. 1/(1-x-x^2-2*x^3) we have a(n)=sum(k=1..n, T(n,k)) (see A077947)
For more formulas see preprints.

			Triangle begins:
  1,
  1,1,
  2,2,1,
  0,5,3,1,
  0,4,9,4,1,
  0,4,13,14,5,1,
  0,0,18,28,20,6,1,

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2013.

T(n,k):=sum(binomial(j,n-3*k+2*j)*2^(k-j)*binomial(k,j),j,0,k);

T(n,k) = Sum_{j=0..k} binomial(j,n-3*k+2*j)*2^(k-j)*binomial(k,j).

cp's OEIS Frontend

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

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A122552 a(0)=a(1)=a(2)=1, a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.

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A186575 Expansion of (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3) in powers of x.

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A096669 Rectangular array T(n,k) read by antidiagonals; generating function of row n is 1/F(n,x), where F(n,x) is the polynomial 1 - x - x^2 - 2*x^3 -...- F(n+1)*x^n and F(n+1) is the (n+1)st Fibonacci number, for n=0,1,2,...

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A152732 a(n) + a(n+1) + a(n+2) = 2^n.

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A213948 Triangle, by rows, generated from the INVERT transforms of (1, 1, 2, 4, 8, 16, ...).

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A349842 Expansion of 1/((1 - 2*x)*(1 + x + x^2 + x^3 + x^4)).

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A186575 Expansion of (1 + 2x + 6x^2)/(1 - x - x^2 - 2*x^3) in powers of x.

A096669 Rectangular array T(n,k) read by antidiagonals; generating function of row n is 1/F(n,x), where F(n,x) is the polynomial 1 - x - x^2 - 2x^3 -...- F(n+1)x^n and F(n+1) is the (n+1)st Fibonacci number, for n=0,1,2,...

A349842 Expansion of 1/((1 - 2x)(1 + x + x^2 + x^3 + x^4)).