A221174 -id:A221174 - cp's OEIS frontend

A048654 a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=4.

1, 4, 9, 22, 53, 128, 309, 746, 1801, 4348, 10497, 25342, 61181, 147704, 356589, 860882, 2078353, 5017588, 12113529, 29244646, 70602821, 170450288, 411503397, 993457082, 2398417561, 5790292204

Offset: 0

Barry E. Williams

Generalized Pellian with second term equal to 4.

The generalized Pellian with second term equal to s has the terms a(n) = A000129(n)*s + A000129(n-1). The generating function is -(1+s*x-2*x)/(-1+2*x+x^2). - R. J. Mathar, Nov 22 2007

T. D. Noe, Table of n, a(n) for n = 0..300
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A001333, A048655, A038761, A084214, A100525.

a048654 n = a048654_list !! n
a048654_list =
   1 : 4 : zipWith (+) a048654_list (map (* 2) $ tail a048654_list)
-- Reinhard Zumkeller, Aug 01 2011

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!((1+2*x)/(1-2*x-x^2))); // G. C. Greubel, Jul 26 2018

LinearRecurrence[{2,1},{1,4},30] (* Harvey P. Dale, Jul 27 2011 *)

a[0]:1$
a[1]:4$
a[n]:=2*a[n-1]+a[n-2]$
A048654(n):=a[n]$
makelist(A048654(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=(([0, 1; 1,2]^n)*[1,4]~)[1] \\ Charles R Greathouse IV, May 18 2015

[lucas_number1(n+1,2,-1) +2*lucas_number1(n,2,-1) for n in (0..40)] # G. C. Greubel, Aug 09 2022

a(n) = ((3+sqrt(2))*(1+sqrt(2))^n - (3-sqrt(2))*(1-sqrt(2))^n)/2*sqrt(2).
a(n) = 2*A000129(n+2) - 3*A000129(n+1). - Creighton Dement, Oct 27 2004
G.f.: (1+2*x)/(1-2*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = binomial transform of 1, 3, 2, 6, 4, 12, ... . - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
E.g.f.: exp(x)*cosh(sqrt(2)*x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Vaclav Kotesovec, Feb 16 2015
a(n) is the denominator of the continued fraction [4, 2, ..., 2, 4] with n-1 2's in the middle. For the numerators, see A221174. - Greg Dresden and Tongjia Rao, Sep 02 2021
a(n) = A001333(n) + A000129(n). - G. C. Greubel, Aug 09 2022

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

Original entry on oeis.org

-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

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

A221175 a(0)=-5, a(1)=6; thereafter a(n) = 2*a(n-1) + a(n-2).

Original entry on oeis.org

-5, 6, 7, 20, 47, 114, 275, 664, 1603, 3870, 9343, 22556, 54455, 131466, 317387, 766240, 1849867, 4465974, 10781815, 26029604, 62841023, 151711650, 366264323, 884240296, 2134744915, 5153730126, 12442205167, 30038140460, 72518486087, 175075112634

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, A221173, A221174.

a221175 n = a221175_list !! n
a221175_list = -5 : 6 : zipWith (+)
                        (map (* 2) $ tail a221175_list) a221175_list
-- Reinhard Zumkeller, Jan 04 2013

Vec(-(16*x-5)/(x^2+2*x-1) + O(x^50)) \\ Colin Barker, Jul 10 2015

a(n) = 16*A000129(n)-5*A000129(n+1). - R. J. Mathar, Jan 14 2013

G.f.: -(16*x-5) / (x^2+2*x-1). - Colin Barker, Jul 10 2015

A335747 Number of ways to tile vertically-fault-free 3 X n strip with squares and dominoes.

Original entry on oeis.org

1, 3, 13, 26, 66, 154, 380, 904, 2204, 5286, 12818, 30854, 74636, 179948, 434820, 1049122, 2533818, 6115538, 14766868, 35646080, 86064196, 207766110, 501609946, 1210964110, 2923573588, 7058053972, 17039774268

Offset: 0

Greg Dresden and Oluwatobi Jemima Alabi, Jun 20 2020

nonn

By "vertically-fault-free" we mean that the tilings of the 3 X n strip do not split along any interior vertical line. Here are two of the 66 possible vertically-fault-free tilings of a 3 X 4 strip with squares and dominoes:
.  _       _  
| |_ |   | _|_| |
|| _| |   | | |_|
|| _||   ||_| |

			a(2) = 13 thanks to these thirteen vertically-fault-free tilings of a 3 X 2 strip:
._ _     _ _     _ _     _ _     _ _     _ _     _ _
|_ _|   |_|_|   |_|_|   |_ _|   |_|_|   |_ _|   |_ _|
|_|_|   |_ _|   |_|_|   |_ _|   |_ _|   |_|_|   |_ _|
|_|_|   |_|_|   |_ _|   |_|_|   |_ _|   |_ _|   |_ _|
._ _     _ _     _ _     _ _     _ _     _ _
|_ _|   |_ _|   |_ _|   | |_|   |_| |   | | |
| |_|   |_| |   | | |   |_|_|   |_|_|   |_|_|
|_|_|   |_|_|   |_|_|   |_ _|   |_ _|   |_ _|

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-1).

Cf. A033506 (which gives all tilings of 3 X n strip), A112577, A134438, A291227.

Cf. A000045, A000129.

I:=[26, 66, 154, 380]; [1,3,13] cat [n le 4 select I[n] else Self(n-1) +4*Self(n-2) -Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jan 15 2022

CoefficientList[Series[(1+2x+6x^2+2x^3-8x^4+x^6)/((1+x-x^2)(1-2x-x^2)), {x, 0, 26}], x] (* Michael De Vlieger, Jul 03 2020 *)
LinearRecurrence[{1,4,-1,-1}, {1,3,13,26,66,154,380}, 40] (* G. C. Greubel, Jan 15 2022 *)

def P(n): return lucas_number1(n,2,-1)
def A335747(n): return (1/3)*(-9*bool(n==0) - 3*bool(n==1) + 3*bool(n==2) + 2*(3*P(n+1) + 2*P(n-1)) + 2*(-1)^n*fibonacci(n-1))
[A335747(n) for n in (0..40)] # G. C. Greubel, Jan 15 2022

a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4) for n >= 7.
a(n) = 2*A291227(n) - 8*A112577(n-2) + 2*A112577(n-4) for n >= 4.
a(n) = (2/3)*(A221174(n+1) + (-1)^n*A000045(n-1)) for n >= 3. - Greg Dresden, Jul 03 2020
G.f.: (1 + 2*x + 6*x^2 + 2*x^3 - 8*x^4 + x^6) / ((1 + x - x^2)*(1 - 2*x - x^2)). - Colin Barker, Jun 21 2020
a(n) = (1/3)*(-9*[n=0] - 3*[n=1] + 3*[n=2] + 2*(3*A000129(n+1) + 2*A000129(n-1)) + 2*(-1)^n*Fibonacci(n-1)). - G. C. Greubel, Jan 15 2022

cp's OEIS Frontend

A048654 a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=4.

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A078343 a(0) = -1, a(1) = 2; 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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A221173 a(0)=-3, a(1)=4; thereafter a(n) = 2*a(n-1) + a(n-2).

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A221175 a(0)=-5, a(1)=6; thereafter a(n) = 2*a(n-1) + a(n-2).

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A335747 Number of ways to tile vertically-fault-free 3 X n strip with squares and dominoes.

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