A190958 -id:A190958 - cp's OEIS frontend

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

1, 2, 4, 1, 8, 4, 16, 12, 1, 32, 32, 6, 64, 80, 24, 1, 128, 192, 80, 8, 256, 448, 240, 40, 1, 512, 1024, 672, 160, 10, 1024, 2304, 1792, 560, 60, 1, 2048, 5120, 4608, 1792, 280, 12, 4096, 11264, 11520, 5376, 1120, 84, 1, 8192, 24576, 28160, 15360

Offset: 1

Clark Kimberling, Feb 18 2012

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is the triangle given by (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012
The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and  along (first layer) skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. - Zagros Lalo, Jul 31 2018
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.414213562373095... (A014176: Decimal expansion of the silver mean, 1+sqrt(2)), when n approaches infinity. - Zagros Lalo, Jul 31 2018

			First seven rows:
1
2
4...1
8...4
16..12..1
32..32..6
64..80..24..1
(2, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, ...) begins:
    1
    2,   0
    4,   1,  0
    8,   4,  0, 0
   16,  12,  1, 0, 0
   32,  32,  6, 0, 0, 0
   64,  80, 24, 1, 0, 0, 0
  128, 192, 80, 8, 0, 0, 0, 0

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358.

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
S. Halici, On some Pell polynomials , Acta Universitatis Apulensis, No. 29/2012, pp. 105-112.
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n

Cf. A028297, A207537, A133156, A038207, A099089.
Cf. A053117, A097610, A091894.
Cf. A013609, A038207.
Cf. A128099.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)
t[n_, k_] := t[n, k] = 2^(n - 2 k) * (n -  k)!/((n - 2 k)! k!) ; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]} ]  // Flatten (* Zagros Lalo, Jul 31 2018 *)

u(n,x) = u(n-1,x)+(x+1)*v(n-1,x), v(n,x) = u(n-1,x)+v(n-1,x), where u(1,x) = 1, v(1,x) = 1. Also, A207538 = |A133156|.
From Philippe Deléham, Mar 04 2012: (Start)
With 0<=k<=n:
Mirror image of triangle in A099089.
Skew version of A038207.
Riordan array (1/(1-2*x), x^2/(1-2*x)).
G.f.: 1/(1-2*x-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190958(n+1), A127357(n), A090591(n), A089181(n+1), A088139(n+1), A045873(n+1), A088138(n+1), A088137(n+1), A099087(n), A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively.
T(n,k) = 2*T(n-1,k) + T(-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n, k) = 0 if k<0 or if k>n. (End)
T(n,k) = A013609(n-k, n-2*k+1). - Johannes W. Meijer, Sep 05 2013
From Tom Copeland, Feb 11 2016: (Start)
A053117 is a reflected, aerated and signed version of this entry. This entry belongs to a family discussed in A097610 with parameters h1 = -2 and h2 = -y.
Shifted o.g.f.: G(x,t) = x / (1 - 2 x - t x^2).
The compositional inverse of G(x,t) is Ginv(x,t) = -[(1 + 2x) - sqrt[(1+2x)^2 + 4t x^2]] / (2tx) = x - 2 x^2 + (4-t) x^3 - (8-6t) x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0).
(End)

A138395 a(n) = 6a(n-1) - 3a(n-2), a(1) = 1, a(2) = 6.

Original entry on oeis.org

1, 6, 33, 180, 981, 5346, 29133, 158760, 865161, 4714686, 25692633, 140011740, 762992541, 4157920026, 22658542533, 123477495120, 672889343121, 3666903573366, 19982753410833, 108895809744900, 593426598236901, 3233872160186706, 17622953166409533

Offset: 1

Gary W. Adamson, Mar 19 2008

a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} avoiding 01, 02 and 03. - Milan Janjic, Dec 17 2015

			a(5) = 981 = 6*a(4) - 3*a(3) = 6*180 - 3*33.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-3).

Cf. A084120, A190958.

I:=[1,6]; [n le 2 select I[n] else 6*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

a[n_]:=(MatrixPower[{{1,2},{1,5}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{6,-3},{1,6},30] (* Harvey P. Dale, Jan 18 2012 *)

Vec(1/(1-6*x+3*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015

A138395=BinaryRecurrenceSequence(6,-3,0,1)
[A138395(n) for n in range(1,30)] # G. C. Greubel, Jan 10 2024

Limit_{n->oo} a(n)/a(n-1) = 3 + sqrt(6) = 5.44948974...
a(n) = ((3+sqrt(6))^n - (3-sqrt(6))^n)/(2*sqrt(6)). - Alexander R. Povolotsky, Apr 01 2008
a(n) = lower left term of n-th power of 2 X 2 matrix [1,2; 1,5].
G.f.: 1/(1 - 6*x + 3*x^2). - Philippe Deléham, Sep 09 2009
a(n) = Chebyshev_U(n, sqrt(3))*(sqrt(3))^n. - Paul Barry, Sep 28 2009

More terms from Philippe Deléham, Sep 09 2009
a(21) and first formula corrected by Klaus Brockhaus, Oct 05 2009
Extended by T. D. Noe, May 23 2011

A190984 a(n) = 9a(n-1) - 7a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 74, 603, 4909, 39960, 325277, 2647773, 21553018, 175442751, 1428113633, 11624923440, 94627515529, 770273175681, 6270065972426, 51038681522067, 415457671891621, 3381848276370120, 27528430784089733, 224082939122216757, 1824047436611322682

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-7).

Cf. A190958 (index to generalized Fibonacci sequences).

[Round(7^((n-1)/2)*Evaluate(ChebyshevU(n), 9/(2*Sqrt(7)))): n in [0..30]]; // G. C. Greubel, Aug 26 2022

Mathematica
```
LinearRecurrence[{9,-7}, {0,1}, 50]
```

A190984 = BinaryRecurrenceSequence(9,-7,0,1)
[A190984(n) for n in (0..30)] # G. C. Greubel, Aug 26 2022

G.f.: x/(1-9*x+7*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (2/sqrt(53))*exp(9*x/2)*sinh(sqrt(53)*x/2). - G. C. Greubel, Aug 26 2022

A190970 a(n) = 5a(n-1) - 9a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 16, 35, 31, -160, -1079, -3955, -10064, -14725, 16951, 217280, 933841, 2713685, 5163856, 1396115, -39494129, -210035680, -694731239, -1583335075, -1664094224, 5929544555, 44624570791, 169756952960, 447163627681, 708005561765, -484444840304

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9).

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 5*Self(n-1) - 9*Self(n-2): n in [1..51]]; // G. C. Greubel, Jun 09 2022

A190970 := proc(n)
    option remember ;
    if n <= 1 then
        n;
    else
        5*procname(n-1)-9*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Mar 23 2023

Mathematica
```
LinearRecurrence[{5,-9}, {0,1}, 50]
```

[3^(n-1)*chebyshev_U(n-1, 5/6) for n in (0..50)] # G. C. Greubel, Jun 09 2022

G.f.: x/(1 - 5*x + 9*x^2). - Philippe Deléham, Oct 12 2011

a(n) = 3^(n-1) * ChebyshevU(n-1, 5/6). - G. C. Greubel, Jun 09 2022

A190972 a(n) = 7a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 7, 46, 301, 1969, 12880, 84253, 551131, 3605158, 23582713, 154263517, 1009096480, 6600884809, 43178904223, 282449675134, 1847611013269, 12085928067481, 79058663432560, 517152859825477, 3382894028480659, 22128799619888182, 144752915253775297

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

a(n+1) equals the number of words of length n over {0,1,2,3,4,5,6} avoiding 01, 02 and 03. - Milan Janjic, Dec 17 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-3).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 7*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

Mathematica
```
LinearRecurrence[{7,-3}, {0,1}, 50]
```

concat(0, Vec(x/(1-7*x+3*x^2) + O(x^100))) \\ Altug Alkan, Dec 18 2015

a(n) = ((7/2 + 1/2*sqrt(37))^n - (7/2 - 1/2*sqrt(37))^n)/sqrt(37). - Giorgio Balzarotti, May 28 2011
G.f.: x/(1 - 7x + 3*x^2). - Philippe Deléham, Oct 12 2011
E.g.f.: (2/sqrt(37))*exp(7*x/2)*sinh(sqrt(37)*x/2). - G. C. Greubel, Dec 18 2015

A190974 a(n) = 7a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 7, 44, 273, 1691, 10472, 64849, 401583, 2486836, 15399937, 95365379, 590557968, 3657078881, 22646762327, 140241941884, 868459781553, 5378008761451, 33303762422392, 206236293149489, 1277135239934463, 7908765213793796, 48975680296884257

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-5).

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 7*Self(n-1) - 5*Self(n-2): n in [1..51]]; // G. C. Greubel, Jun 11 2022

Mathematica
```
LinearRecurrence[{7,-5}, {0,1}, 50]
```

[lucas_number1(n,7,5) for n in (0..50)] # G. C. Greubel, Jun 11 2022

a(n) = ((7/2 + 1/2*sqrt(29))^n - (7/2 - 1/2*sqrt(29))^n)/sqrt(29). - Giorgio Balzarotti, May 28 2011
G.f.: x/(1 - 7*x + 5*x^2). - Philippe Deléham, Oct 12 2011
From G. C. Greubel, Jun 11 2022: (Start)
a(n) = 5^((n-1)/2)*ChebyshevU(n-1, 7/(2*sqrt(5))).
E.g.f.: (2/sqrt(29))*exp(7*x/2)*sinh(sqrt(29)*x/2). (End)

A190978 a(n) = 8a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 8, 58, 416, 2980, 21344, 152872, 1094912, 7842064, 56167040, 402283936, 2881269248, 20636450368, 147803987456, 1058613197440, 7582081654784, 54304974053632, 388947302500352, 2785748575681024, 19952304790446080, 142903946869482496, 1023517746213183488

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (8,-6).

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 8*Self(n-1) -6*Self(n-2): n in [1..41]]; // G. C. Greubel, Jun 17 2022

LinearRecurrence[{8,-6}, {0,1}, 50]
CoefficientList[Series[x/(1-8x+6x^2),{x,0,30}],x] (* Harvey P. Dale, Aug 03 2021 *)

[sum( (-1)^k*binomial(n-k-1, k)*6^k*8^(n-2*k-1) for k in (0..((n-1)//2))) for n in (0..40)] # G. C. Greubel, Jun 17 2022

a(n) = ((4 + sqrt(10))^n - (4 - sqrt(10))^n)/(2*sqrt(10)). - Giorgio Balzarotti, May 28 2011
G.f.: x/(1 - 8*x + 6*x^2). - Philippe Deléham, Oct 12 2011
From G. C. Greubel, Jun 17 2022: (Start)
a(n) = 6^((n-1)/2)*ChebyshevU(n-1, 4/sqrt(6)).
E.g.f.: (1/sqrt(10))*exp(4*x)*sinh(sqrt(10)*x). (End)

A190990 a(n) = 10a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 92, 840, 7664, 69920, 637888, 5819520, 53092096, 484364800, 4418911232, 40314193920, 367790649344, 3355392942080, 30611604226048, 279272898723840, 2547836153430016, 23244178344509440, 212059094217654272, 1934637515420467200, 17649902400463437824

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-8).

Cf. A190958 (index to generalized Fibonacci sequences)

[Round(2^(3*(n-1)/2)*Evaluate(ChebyshevU(n), 5/(2*Sqrt(2)))): n in [0..30]]; // G. C. Greubel, Sep 15 2022

Mathematica
```
LinearRecurrence[{10,-8}, {0,1}, 50]
```

A190990 = BinaryRecurrenceSequence(10, -8, 0, 1)
[A190990(n) for n in (0..30)] # G. C. Greubel, Sep 15 2022

G.f.: x / ( 1-10*x+8*x^2 ). - R. J. Mathar, May 26 2011

E.g.f.: (1/sqrt(17))*exp(5*x)*sinh(sqrt(17)*x). - G. C. Greubel, Sep 15 2022

A190960 a(n) = 3a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 3, -9, -45, -81, 27, 567, 1539, 1215, -5589, -24057, -38637, 28431, 317115, 780759, 439587, -3365793, -12734901, -18009945, 22379571, 175198383, 391317723, 122762871, -1979617725, -6675430401, -8148584853, 15606827847, 95711992659, 193495010895

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 16.
Index entries for linear recurrences with constant coefficients, signature (3,-6).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

Mathematica
```
LinearRecurrence[{3,-6}, {0,1}, 50]
```

x='x+O('x^30); concat([0], Vec(x/(1-3*x+6*x^2))) \\ G. C. Greubel, Jan 25 2018

G.f.: x/(1-3*x+6*x^2). - Philippe Deléham, Oct 11 2011
a(n) = (i/sqrt(15))*((3/2 - i*sqrt(15)/2)^n - (3/2 + i*sqrt(15)/2)^n), where i=sqrt(-1). - Taras Goy, Jan 04 2025
E.g.f.: 2*exp(3*x/2)*sin(sqrt(15)*x/2)/sqrt(15). - Stefano Spezia, Jan 05 2025

A190965 a(n) = 4a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 4, 10, 16, 4, -80, -344, -896, -1520, -704, 6304, 29440, 79936, 143104, 92800, -487424, -2506496, -7101440, -13366784, -10858496, 36766720, 212217856, 628271104, 1239777280, 1189482496, -2680733696, -17859829760, -55354916864, -114260688896

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the quaternion Q = 2+j+k, Q^n = r(n) + a(n)*(j+k). The sequence of real-parts r(n) is A266046. - Stanislav Sykora, Dec 20 2015

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (4,-6).

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A088137 (Inv. Bin. Trans.), A168175, A213421, A266046.

[n le 2 select n-1 else 4*Self(n-1) -6*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 10 2024

w := I*sqrt(2): a := n -> (w/4)*((2 - w)^n - (2 + w)^n):
seq(simplify(a(n)), n = 0..20);  # (after Taras Goy), Peter Luschny, Jan 03 2025

Mathematica
```
LinearRecurrence[{4,-6}, {0,1}, 50]
```

a(n)=([0,1;0,0]*[0,-6;1,4]^n)[1,1] \\ Charles R Greathouse IV, May 31 2011

A190965=BinaryRecurrenceSequence(4,-6,0,1)
[A190965(n) for n in range(41)] # G. C. Greubel, Jan 10 2024

G.f.: x/(1-4*x+6*x^2). - Philippe Deléham, Oct 12 2011
2*a(n)^2 + A266046(n)^2 = 6^n. - Stanislav Sykora, Dec 20 2015
From G. C. Greubel, Jan 10 2024: (Start)
a(n) = 6^((n-1)/2)*ChebyshevU(n-1, sqrt(2/3)).
E.g.f.: (1/sqrt(2))*exp(2*x)*sin(sqrt(2)*x). (End)
a(n) = (i*sqrt(2)/4)*((2 - i*sqrt(2))^n - (2 + i*sqrt(2))^n), where i = sqrt(-1). - Taras Goy, Jan 03 2025

cp's OEIS Frontend

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

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A138395 a(n) = 6*a(n-1) - 3*a(n-2), a(1) = 1, a(2) = 6.

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A190984 a(n) = 9*a(n-1) - 7*a(n-2), with a(0)=0, a(1)=1.

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A190970 a(n) = 5*a(n-1) - 9*a(n-2), with a(0)=0, a(1)=1.

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A190972 a(n) = 7*a(n-1) - 3*a(n-2), with a(0)=0, a(1)=1.

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Magma

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A190974 a(n) = 7*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

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Magma

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A190978 a(n) = 8*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.

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A138395 a(n) = 6a(n-1) - 3a(n-2), a(1) = 1, a(2) = 6.

A190984 a(n) = 9a(n-1) - 7a(n-2), with a(0)=0, a(1)=1.

A190970 a(n) = 5a(n-1) - 9a(n-2), with a(0)=0, a(1)=1.

A190972 a(n) = 7a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

A190974 a(n) = 7a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

A190978 a(n) = 8a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

A190990 a(n) = 10a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.

A190960 a(n) = 3a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

A190965 a(n) = 4a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.