A221175 -id:A221175 - cp's OEIS frontend

A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

-1, 2, 3, 8, 19, 46, 111, 268, 647, 1562, 3771, 9104, 21979, 53062, 128103, 309268, 746639, 1802546, 4351731, 10506008, 25363747, 61233502, 147830751, 356895004, 861620759, 2080136522, 5021893803, 12123924128, 29269742059, 70663408246, 170596558551, 411856525348

Offset: 0

Benoit Cloitre, Nov 22 2002

Inverse binomial transform of -1, 1, 6, 22, 76, 260, ... (see A111566). Binomial transform of -1, 3, -2, 6, -4, 12, -8, 24, -16, ... (see A162255). - R. J. Mathar, Oct 02 2012

			G.f. = -1 + 2*x + 3*x^2 + 8*x^3 + 19*x^4 + 46*x^5 + 111*x^6 + ... - _Michael Somos_, Jun 30 2022

H. S. M. Coxeter, 1998, Numerical distances among the circles in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
H. S. M. Coxeter, Numerical distances among the spheres in a loxodromic sequence, Math. Intell. 19(4) 1997 pp. 41-47. See page 41. See pp. 46-47.
Tanya Khovanova, Recursive Sequences.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv:1212.1368 [cs.DM], 2012.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A048654, A048655, A048746, A111566, A162255, A221172, A221173, A221174, A221175.

a078343 n = a078343_list !! n
a078343_list = -1 : 2 : zipWith (+)
                        (map (* 2) $ tail a078343_list) a078343_list
-- Reinhard Zumkeller, Jan 04 2013

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)/(-1+2*x+x^2))); // G. C. Greubel, Jul 26 2018

f:=proc(n) option remember; if n=0 then RETURN(-1); fi; if n=1 then RETURN(2); fi; 2*f(n-1)+f(n-2); end;

Table[4 Fibonacci[n, 2] - Fibonacci[n + 1, 2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)
LinearRecurrence[{2,1},{-1,2},40] (* Harvey P. Dale, Apr 15 2019 *)

a(n)=([0,1;1,2]^n*[-1;2])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

For the unsigned version: a(1)=1; a(2)=2; a(n) = Sum_{k=2..n-1} (a(k) + a(k-1)).
a(n) is asymptotic to (1/4)*(-2+3*sqrt(2))*(1+sqrt(2))^n.
a(n) = A048746(n-3) + 2, for n > 2. - Ralf Stephan, Oct 17 2003
a(n) = 2*A000129(n) - A000129(n-1) if n > 0; abs(a(n)) = Sum_{k=0..floor(n/2)} (C(n-k-1, k) - C(n-k-1, k-1))2^(n-2k). - Paul Barry, Dec 23 2004
O.g.f.: (1-4*x)/(-1 + 2*x + x^2). - R. J. Mathar, Feb 15 2008
a(n) = 4*Pell(n) - Pell(n+1), where Pell = A000129. - Vladimir Reshetnikov, Sep 27 2016
a(n) = -(-1)^n * A048654(-n) = ( (-2+3*sqrt(2))*(1+sqrt(2))^n + (-2-3*sqrt(2))*(1-sqrt(2))^n )/4 for all n in Z. - Michael Somos, Jun 30 2022
2*a(n+1)^2 = A048655(n)^2 + (-1)^n*7. - Philippe Deléham, Mar 07 2023
E.g.f.: 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2) - exp(x)*cosh(sqrt(2)*x). - Stefano Spezia, May 26 2024

Entry revised by N. J. A. Sloane, Apr 29 2004

A221174 a(0)=-4, a(1)=5; thereafter a(n) = 2*a(n-1) + a(n-2).

Original entry on oeis.org

-4, 5, 6, 17, 40, 97, 234, 565, 1364, 3293, 7950, 19193, 46336, 111865, 270066, 651997, 1574060, 3800117, 9174294, 22148705, 53471704, 129092113, 311655930, 752403973, 1816463876, 4385331725, 10587127326, 25559586377, 61706300080, 148972186537, 359650673154

Offset: 0

N. J. A. Sloane, Jan 04 2013

From Greg Dresden, May 08 2023: (Start)
For n >= 3, 2*a(n) is the number of ways to tile this figure of length n-1 with two colors of squares and one color of domino. For n=8, we have here the figure of length n-1=7, and it has 2*a(8) = 2728 different tilings.
. 
|||_   _ _
|||_|||_|_|
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A078343, A221172, A221173, A221175.

a221174 n = a221174_list !! n
a221174_list = -4 : 5 : zipWith (+)
                        (map (* 2) $ tail a221174_list) a221174_list
-- Reinhard Zumkeller, Jan 04 2013

LinearRecurrence[{2, 1}, {-4, 5}, 50] (* Paolo Xausa, Sep 02 2024 *)

Vec(-(13*x-4)/(x^2+2*x-1) + O(x^50)) \\ Colin Barker, Jul 10 2015

a(n) = 13*A000129(n) - 4*A000129(n+1). - R. J. Mathar, Jan 14 2013
G.f.: -(13*x-4) / (x^2+2*x-1). - Colin Barker, Jul 10 2015
a(n) is the numerator of the continued fraction [4, 2, ..., 2, 4] with n-3 2's in the middle. For denominators, see A048654. - Greg Dresden and Tongjia Rao, Sep 02 2021

A221172 a(0)=-2, a(1)=3; thereafter a(n) = 2*a(n-1) + a(n-2).

Original entry on oeis.org

-2, 3, 4, 11, 26, 63, 152, 367, 886, 2139, 5164, 12467, 30098, 72663, 175424, 423511, 1022446, 2468403, 5959252, 14386907, 34733066, 83853039, 202439144, 488731327, 1179901798, 2848534923, 6876971644, 16602478211, 40081928066, 96766334343, 233614596752, 563995527847

Offset: 0

N. J. A. Sloane, Jan 04 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A078343, A221173, A221174, A221175.

a221172 n = a221172_list !! n
a221172_list = -2 : 3 : zipWith (+)
                        (map (* 2) $ tail a221172_list) a221172_list
-- Reinhard Zumkeller, Jan 04 2013

LinearRecurrence[{2,1},{-2,3},40] (* Harvey P. Dale, May 30 2013 *)
Table[7 Fibonacci[n, 2] - 2 Fibonacci[n + 1, 2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)

G.f.: (2-7*x)/(-1+2*x+x^2). - R. J. Mathar, Jan 04 2013
a(n) = 7*Pell(n) - 2*Pell(n+1), where Pell = A000129. - Vladimir Reshetnikov, Sep 27 2016
E.g.f.: -2*exp(x)*cosh(sqrt(2)*x) + 5*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, May 26 2024

A221173 a(0)=-3, a(1)=4; thereafter a(n) = 2*a(n-1) + a(n-2).

Original entry on oeis.org

-3, 4, 5, 14, 33, 80, 193, 466, 1125, 2716, 6557, 15830, 38217, 92264, 222745, 537754, 1298253, 3134260, 7566773, 18267806, 44102385, 106472576, 257047537, 620567650, 1498182837, 3616933324, 8732049485, 21081032294, 50894114073, 122869260440, 296632634953

Offset: 0

N. J. A. Sloane, Jan 04 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A078343, A221172, A221174, A221175.

a221173 n = a221173_list !! n
a221173_list = -3 : 4 : zipWith (+)
                        (map (* 2) $ tail a221173_list) a221173_list
-- Reinhard Zumkeller, Jan 04 2013

LinearRecurrence[{2,1},{-3,4},50] (* Harvey P. Dale, Apr 09 2022 *)

Vec(-(10*x-3)/(x^2+2*x-1) + O(x^100)) \\ Colin Barker, Jul 10 2015

a(n) = 10*A000129(n)-3*A000129(n+1). - R. J. Mathar, Jan 14 2013

G.f.: -(10*x-3) / (x^2+2*x-1). - Colin Barker, Jul 10 2015

cp's OEIS Frontend

A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

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A221174 a(0)=-4, a(1)=5; thereafter a(n) = 2*a(n-1) + a(n-2).

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A221172 a(0)=-2, a(1)=3; thereafter a(n) = 2*a(n-1) + a(n-2).

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Haskell

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A221173 a(0)=-3, a(1)=4; thereafter a(n) = 2*a(n-1) + a(n-2).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula