A111282 -id:A111282 - cp's OEIS frontend

A025169 a(n) = 2Fibonacci(2n+2).

2, 6, 16, 42, 110, 288, 754, 1974, 5168, 13530, 35422, 92736, 242786, 635622, 1664080, 4356618, 11405774, 29860704, 78176338, 204668310, 535828592, 1402817466, 3672623806, 9615053952, 25172538050, 65902560198, 172535142544

Offset: 0

Wouter Meeussen

The pairs (x, y) = (a(n), a(n+1)) satisfy x^2 + y^2 = 3*x*y + 4. - Michel Lagneau, Feb 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 32).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A000045, A001906, A002878, A122367.

List([0..30], n-> 2*Fibonacci(2*n+2) ); # G. C. Greubel, Jan 16 2020

a025169 n = a025169_list !! n
a025169_list = 2 : 6 : zipWith (-) (map (* 3) $ tail a025169_list) a025169_list
-- Reinhard Zumkeller, Apr 08 2012

[2*Fibonacci(2*n+2): n in [0..30]]; // Vincenzo Librandi, Jul 11 2011

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 2/(1-3*x + x^2) )); // Marius A. Burtea, Jan 16 2020

seq( 2*fibonacci(2*n+2), n=0..30); # G. C. Greubel, Jan 16 2020

Table[2Fibonacci[2n+2], {n,0,30}] (* or *)
CoefficientList[Series[2/(1-3x+x^2), {x,0,30}], x] (* Michael De Vlieger, Mar 09 2016 *)
LinearRecurrence[{3, -1}, {2, 6}, 30] (* Jean-François Alcover, Sep 27 2017 *)

PARI
```
a(n)=2*fibonacci(2*n+2)
```

[2*fibonacci(2*n+2) for n in (0..30)] # G. C. Greubel, Jan 16 2020

G.f.: 2/(1 - 3*x + x^2).
a(n) = 3*a(n-1) - a(n-2).
a(n) = 2*A001906(n+1).
a(n) = A111282(n+2). - Reinhard Zumkeller, Apr 08 2012
a(n) = Fibonacci(2*n+1) + Lucas(2*n+1). - Bruno Berselli, Oct 13 2017

Better description from Michael Somos

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

cp's OEIS Frontend

A025169 a(n) = 2Fibonacci(2n+2).

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A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

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cp's OEIS Frontend

A025169 a(n) = 2*Fibonacci(2*n+2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A025169 a(n) = 2Fibonacci(2n+2).