A002605 -id:A002605 - cp's OEIS frontend

A111006 Another version of Fibonacci-Pascal triangle A037027.

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144

Offset: 0

Philippe Deléham, Oct 02 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).
Triangle read by rows: T(n,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)). - Philippe Deléham, Feb 17 2014
Diagonal sums are A013979(n). - Philippe Deléham, Feb 17 2014
T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 5,  5;
  0, 0, 0, 3, 10,  8;
  0, 0, 0, 1,  9, 20, 13;
  0, 0, 0, 0,  4, 22, 38,  21;
  0, 0, 0, 0,  1, 14, 51,  71,  34;
  0, 0, 0, 0,  0,  5, 40, 111, 130,  55;
  0, 0, 0, 0,  0,  1, 20, 105, 233, 235,  89;
  0, 0, 0, 0,  0,  0,  6,  65, 256, 474, 420, 144;

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045, A000129, A001045, A037027, A038112, A038149, A084938, A128100 (reversed version).

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A114197, A162741, A228074.

a111006 n k = a111006_tabl !! n !! k
a111006_row n = a111006_tabl !! n
a111006_tabl =  map fst $ iterate (\(us, vs) ->
   (vs, zipWith (+) (zipWith (+) ([0] ++ us ++ [0]) ([0,0] ++ us))
                    ([0] ++ vs))) ([1], [0,1])
-- Reinhard Zumkeller, Aug 15 2013

T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
T(n, k) = A037027(k, n-k). T(n, n) = A000045(n+1). T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).
G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006
Sum_{k=0..n} T(n,k)*x^(n-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) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014

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

Original entry on oeis.org

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)

A180222 a(n) = 4a(n-1) + 8a(n-2), with a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 1, 4, 24, 128, 704, 3840, 20992, 114688, 626688, 3424256, 18710528, 102236160, 558628864, 3052404736, 16678649856, 91133837312, 497964548096, 2720928890880, 14867431948288, 81237158920192, 443888091267072, 2425449636429824, 13252903275855872

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 16 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,8).

I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1) + 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Join[{a=0,b=1},Table[c=4*b+8*a;a=b;b=c,{n,100}]]
LinearRecurrence[{4,8}, {0,1}, 30] (* G. C. Greubel, Jan 16 2018 *)

concat(0,Vec(1/(1-4*x-8*x^2)+O(x^98))) \\ Charles R Greathouse IV, Dec 07 2011

a(n) = 2^(n-3)*((1+sqrt(3))^(n-1)-(1-sqrt(3))^(n-1))/sqrt(3). - Rolf Pleisch, May 14 2011
a(n) = (-1)^n*A174443(n-1). - Nathaniel Johnston, May 14 2011
G.f.: x^2/(1-4*x-8*x^2).
a(n+2) = Sum_{k=0..n} A201947(n,k)*3^(n-k). - Philippe Deléham, Dec 07 2011
a(n+2) = 2^n*A002605(n+1). - R. J. Mathar, May 07 2019

A239537 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no element equal to all horizontal neighbors or unequal to all vertical neighbors, and new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 2, 1, 6, 2, 6, 16, 6, 24, 13, 44, 16, 216, 68, 47, 120, 44, 1536, 1014, 406, 128, 328, 120, 11616, 11108, 13254, 1584, 405, 896, 328, 86400, 131988, 293741, 98304, 7790, 1181, 2448, 896, 645504, 1533792, 7199001, 3785280, 984150, 33630, 3598, 6688, 2448

Offset: 1

R. H. Hardin, Mar 21 2014

Table starts
.....1......2.........6...........16..............44...............120
.....1......2.........6...........16..............44...............120
.....6.....24.......216.........1536...........11616.............86400
....13.....68......1014........11108..........131988...........1533792
....47....406.....13254.......293741.........7199001.........171712936
...128...1584.....98304......3785280.......165336096........6976042240
...405...7790....984150.....71388971......5988724293......482858009716
..1181..33630...8368566...1093269814....169350916116....25030781490504
..3598.156032..77673624..18727824939...5466853947751..1514104924173186
.10705.695344.687582150.301846514891.164296850179323.84271877225127052

			Some solutions for n=4 k=4
..0..1..0..0..1....0..1..2..1..2....0..1..1..2..1....0..1..2..0..2
..0..1..0..0..1....0..1..2..1..2....0..1..1..2..1....0..1..2..0..2
..0..2..0..2..1....1..2..2..1..2....0..1..1..0..1....0..1..2..0..2
..0..2..0..2..1....1..2..2..1..0....2..0..2..0..2....1..0..2..1..2
..0..2..0..2..1....1..2..2..1..0....2..0..2..0..2....1..0..2..1..2

R. H. Hardin, Table of n, a(n) for n = 1..128

Row 1 and 2 are A002605

Row 3 is A231317

Empirical for column k:
k=1: a(n) = 2*a(n-1) +4*a(n-2) -3*a(n-3)
k=2: [order 10]
k=3: a(n) = 8*a(n-1) +26*a(n-2) -166*a(n-3) +96*a(n-4) +198*a(n-5) -81*a(n-6)
Empirical for row n:
n=1: a(n) = 2*a(n-1) +2*a(n-2)
n=2: a(n) = 2*a(n-1) +2*a(n-2)
n=3: a(n) = 6*a(n-1) +12*a(n-2) -8*a(n-3)
n=4: [order 10]
n=5: [order 37]
n=6: [order 85]

A084609 Coefficients of 1/sqrt(1-4x-8x^2); also, a(n) is the central coefficient of (1+2x+3x^2)^n.

Original entry on oeis.org

1, 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424, 78210099718, 414862155980, 2204273582236, 11729283976136, 62496686731924, 333400654676168

Offset: 0

Paul D. Hanna, Jun 01 2003

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U can have 3 colors and H can have 2 colors. - N-E. Fahssi, Mar 30 2008
Self-convolution of a(n)/2^n gives A002605(n+1). - Vladimir Reshetnikov, Oct 10 2016
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1358 (terms 0..200 from Vincenzo Librandi)
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A002426, A084600 - A084608, A084610 - A084615.

Row sums of A328347.

A084609:= func< n | (&+[Binomial(n,j)*Binomial(2*(n-j),n)*2^j: j in [0..Floor(n/2)]]) >;
[A084609(n): n in [0..50]]; // G. C. Greubel, Mar 26 2023

(* Programs from Robert G. Wilson v, Mar 02 2011 *)
a[n_]:= Sum[Binomial[n, k] Binomial[2(n-k), n] 2^k, {k, 0, n/2}]; Array[a, 30, 0]
a[n_]:= CoefficientList[Expand[(1 +2x +3x^2)^n], x][[n+1]]; Array[a, 30, 0]
CoefficientList[Series[1/Sqrt[1 -4x -8x^2], {x,0,30}], x]
Range[0, 30]! CoefficientList[ Series[ Exp[ 2x] BesselI[0, Sqrt[12] x], {x, 0, 30}], x] (* End *)
Table[2^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 3], {n,0,30}] (* Vladimir Reshetnikov, Oct 10 2016 *)

a(n):=coeff(expand((1+2*x+3*x^2)^n),x,n);
makelist(a(n),n,0,12);

for(n=0,30,t=polcoeff((1+2*x+3*x^2)^n,n,x); print1(t","))

def A084609(n): return sum(binomial(n,j)*binomial(2*(n-j),n)*2^j for j in range(n//2+1))
[A084609(n) for n in range(51)] # G. C. Greubel, Mar 26 2023

a(n) = Sum_{k = 0..floor(n/2)} C(n,k)*C(2*(n-k),n)*2^k. - Paul Barry, Sep 08 2004
a(n) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*3^k*2^(n-2*k); a(n) = Sum_{k = 0..floor(n/2)} C(n,k)*C(n-k,k)*3^k*2^(n-2k). - Paul Barry, Sep 19 2006
E.g.f.: exp(2*x) * Bessel_I(0,2*sqrt(3)*x)
a(n) = ( 2*(2*n-1)*a(n-1) + 8*(n-1)*a(n-2) )/n, a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 20 2012
a(n) ~ sqrt(18+6*sqrt(3))*(2+2*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - 2*x*(1+2*x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(1+2*x)/(k+1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: G(0), where G(k)= 1 + x*(2+4*x)*(4*k+1)/(2*k+1 - x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(1+2*x)*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 18 2013
a(n) = 2^n * hypergeom([(1-n)/2,-n/2], [1], 3) = binomial(2*n, n) * hypergeom([(1-n)/2,-n/2], [1/2-n], -2). - Vladimir Reshetnikov, Oct 10 2016
a(n) = (-2*sqrt(-2))^n * P(n, sqrt(-1/2)), where P(n,x) denotes the n-th Legendre polynomial. - Peter Bala, Feb 07 2022

A106435 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 9, 36, 135, 513, 1944, 7371, 27945, 105948, 401679, 1522881, 5773680, 21889683, 82990089, 314639316, 1192888215, 4522582593, 17146412424, 65006985051, 246460192425, 934401532428, 3542585174559, 13430960120961

Offset: 0

Roger L. Bagula, May 29 2005

The first entry of the vector v[n] = M*v[n-1], where M is the 2 x 2 matrix [[0,3],[1,3]] and v[1] is the column vector [0,1]. The characteristic polynomial of the matrix M is x^2-3x-3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,3).

Equals 3*A030195(n).
Cf. A028860.
Cf. A002605, A026150, A028859, A080040, A083337, A108898, A125145.

a106435 n = a106435_list !! n
a106435_list = 0 : 3 : map (* 3) (zipWith (+) a106435_list (tail
a106435_list))
-- Reinhard Zumkeller, Oct 15 2011

a:=[0,3]; [n le 2 select a[n] else    3*Self(n-1) + 3*Self(n-2) : n in [1..24]]; // Marius A. Burtea, Jan 21 2020

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!(3*x/(1-3*x-3*x^2))); // Marius A. Burtea, Jan 21 2020

seq(coeff(series(3*x/(1-3*x-3*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Mar 12 2020

LinearRecurrence[{3,3}, {0,3}, 30] (* G. C. Greubel, Mar 12 2020 *)

PARI
```
a(n)=([0,3;1,3]^n)[1,2]
```

[3^((n+1)/2)*i^(1-n)*chebyshev_U(n-1, i*sqrt(3)/2) for n in (0..30)] # G. C. Greubel, Mar 12 2020

G.f.: 3*x/(1-3*x-3*x^2). - Philippe Deléham, Nov 19 2008
From G. C. Greubel, Mar 12 2020: (Start)
a(n) = 3^((n+1)/2) * Fibonacci(n, sqrt(3)), where F(n, x) is the Fibonacci polynomial.
a(n) = 3^((n+1)/2)*i^(1-n)*ChebyshevU(n-1, i*sqrt(3)/2). (End)

Edited by N. J. A. Sloane, May 20 2006 and May 29 2006

Offset corrected by Reinhard Zumkeller, Oct 15 2011

A162517 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x + d)^n - (x - d)^n)/(2*d), where d = sqrt(x+4).

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 4, 4, 16, 0, 5, 10, 41, 8, 16, 6, 20, 86, 48, 96, 0, 7, 35, 161, 169, 348, 48, 64, 8, 56, 280, 456, 992, 384, 512, 0, 9, 84, 462, 1044, 2449, 1744, 2400, 256, 256, 10, 120, 732, 2136, 5482, 5920, 8640, 2560, 2560, 0, 11, 165, 1122, 4026, 11407, 16721, 26420, 14240, 14720, 1280, 1024

Offset: 1

Clark Kimberling, Jul 05 2009

			First six rows:
  0
  1
  2...0
  3...1...4
  4...4...16...0
  5...10..41...8...16

G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

Cf. A162514, A162515, A162516.

Cf. A002605, A063727.

m:=12;
Q:= func< n,x | ((x+Sqrt(x+4))^n - (x-Sqrt(x+4))^n)/(2*Sqrt(x+4)) >;
R:=PowerSeriesRing(Rationals(), m+1);
T:= func< n,k | Coefficient(R!( Q(n, x) ), n-k) >;
[0] cat [T(n,k): k in [1..n], n in [1..m]]; // G. C. Greubel, Jul 09 2023

Q[n_, x_]:= Q[n, x]= ((x+Sqrt[x+4])^n -(x-Sqrt[x+4])^n)/(2*Sqrt[x+4]);
T[n_, k_]:= Coefficient[Series[P[n,x], {x,0,n-k+1}], x, n-k];
Join[{0}, Table[T[n,k], {n,12}, {k,n}]//Flatten] (* G. C. Greubel, Jul 09 2023 *)

def Q(n,x): return ((x+sqrt(x+4))^n - (x-sqrt(x+4))^n)/(2*sqrt(x+4))
def T(n,k):
    P. = PowerSeriesRing(QQ)
    return P( Q(n,x) ).list()[n-k]
[0]+flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 09 2023

Q(n,x) = (P(n+1, x) - x*P(n,x))/(x+4), where P(n, x) is the n-th polynomial of A162516.
Q(n, x) also has the recurrence Q(n, x) = 2*x*Q(n-1, x) - (x^2 - x - 4)*Q(n-2, x).
From G. C. Greubel, Jul 09 2023: (Start)
T(n, k) = [x^(n-k)](((x+sqrt(x+4))^n -(x-sqrt(x+4))^n)/(2*sqrt(x+4))).
Sum_{k=1..n-1} T(n, k) = A063727(n-2), n >= 2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A002605(n-1). (End)

A368518 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 3x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 7, 3, 10, 18, 20, 5, 20, 51, 68, 61, 8, 40, 118, 220, 251, 182, 13, 76, 264, 584, 905, 888, 547, 21, 142, 558, 1452, 2678, 3540, 3076, 1640, 34, 260, 1145, 3380, 7279, 11536, 13418, 10456, 4921, 55, 470, 2286, 7548, 18391, 33990, 47600, 49552

Offset: 1

Clark Kimberling, Jan 22 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1    2
   2    4    7
   3   10   18    20
   5   20   51    68    61
   8   40  118   220   251   182
  13   76  264   584   905   888   547
  21  142  558  1452  2678  3540  3076  1640
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 18*x^2 + 20*x^3, so (T(4,k)) = (3,10,18,20), k=0..3.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 3x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + 32*x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 16*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A057093 Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.

Original entry on oeis.org

1, 10, 110, 1200, 13100, 143000, 1561000, 17040000, 186010000, 2030500000, 22165100000, 241956000000, 2641211000000, 28831670000000, 314728810000000, 3435604800000000, 37503336100000000, 409389409000000000, 4468927451000000000, 48783168600000000000

Offset: 0

Wolfdieter Lang, Aug 11 2000

This is the m=10 member of the m-family of sequences a(m,n)= S(n,i*sqrt(m))*(-i*sqrt(m))^n, with S(n,x) given in Formula and g.f.: 1/(1-m*x-m*x^2). The instances m=1..9 are A000045 (Fibonacci), A002605, A030195, A057087, A057088, A057089, A057090, A057091, A057092.
With the roots rp(m) := (m+sqrt(m*(m+4)))/2 and rm(m) := (m-sqrt(m*(m+4)))/2 the Binet form of these m-sequences is a(n,m)= (rp(m)^(n+1)-rm(m)^(n+1))/(rp(m)-rm(m)).
a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^10, 1->(1^10)0, starting from 0. The number of 1's and 0's of this word is 10*a(n-1) and 10*a(n-2), resp.

Colin Barker, Table of n, a(n) for n = 0..963
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
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=10, q=10.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=10.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,10).

Join[{a=0,b=1},Table[c=10*b+10*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{10,10},{1,10},30] (* Harvey P. Dale, Jan 18 2025 *)

Vec(1/(1-10*x-10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,10,-10) for n in range(1, 19)] # Zerinvary Lajos, Apr 26 2009

a(n) = 10*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, i*sqrt(10))*(-i*sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1 - 10*x - 10*x^2).
a(n) = Sum_{k=0..n} 9^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

Extended by T. D. Noe, May 23 2011

A083337 a(n) = 2a(n-1) + 2a(n-2); a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 20064, 54816, 149760, 409152, 1117824, 3053952, 8343552, 22795008, 62277120, 170144256, 464842752, 1269974016, 3469633536, 9479215104, 25897697280, 70753824768, 193303044096, 528113737728, 1442833563648, 3941894602752, 10769456332800

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,2).

Equals 3 * A002605.
Cf. A026150.
Cf. A028859, A028860, A030195, A080040, A106435, A108898, A125145.

a083337 n = a083337_list !! n
a083337_list =
   0 : 3 : map (* 2) (zipWith (+) a083337_list (tail a083337_list))
-- Reinhard Zumkeller, Oct 15 2011

CoefficientList[Series[3x/(1-2x-2x^2), {x, 0, 25}], x]
s = Sqrt[3]; a[n_] := Simplify[s*((1 + s)^n - (1 - s)^n)/2]; Array[a, 30, 0] (* or *)
LinearRecurrence[{2, 2}, {0, 3}, 31] (* Robert G. Wilson v, Aug 07 2018 *)

apply( a(n)=([1,1;3,1]^n)[2,1], [0..30]) \\ or: ([2,2;1,0]^n)[2,1]*3. - M. F. Hasler, Aug 06 2018

G.f.: 3x/(1 - 2x - 2x^2).
a(n) = a(n-1) + 3*A026150(n-1). a(n)/A026150(n) converges to sqrt(3).
a(n) = lower left term of [1,1; 3,1]^n. - Gary W. Adamson, Mar 12 2008

Edited and definition completed by M. F. Hasler, Aug 06 2018

cp's OEIS Frontend

A111006 Another version of Fibonacci-Pascal triangle A037027.

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

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A180222 a(n) = 4*a(n-1) + 8*a(n-2), with a(1)=0 and a(2)=1.

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A239537 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no element equal to all horizontal neighbors or unequal to all vertical neighbors, and new values 0..2 introduced in row major order.

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A084609 Coefficients of 1/sqrt(1-4*x-8*x^2); also, a(n) is the central coefficient of (1+2*x+3*x^2)^n.

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A106435 a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=3.

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A162517 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x + d)^n - (x - d)^n)/(2*d), where d = sqrt(x+4).

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A180222 a(n) = 4a(n-1) + 8a(n-2), with a(1)=0 and a(2)=1.

A084609 Coefficients of 1/sqrt(1-4x-8x^2); also, a(n) is the central coefficient of (1+2x+3x^2)^n.

A106435 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=3.

A368518 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 3x^2.

A083337 a(n) = 2a(n-1) + 2a(n-2); a(0)=0, a(1)=3.