A028860 -id:A028860 - cp's OEIS frontend

A293007 Expansion of 2x^2 / (1-2x-2*x^2).

0, 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896, 13376, 36544, 99840, 272768, 745216, 2035968, 5562368, 15196672, 41518080, 113429504, 309895168, 846649344, 2313089024, 6319476736, 17265131520, 47169216512, 128868696064, 352075825152, 961889042432

Offset: 0

J. Devillet, Sep 28 2017

Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n} that have neutral and annihilator elements.

Colin Barker, Table of n, a(n) for n = 0..1000
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA] (2017).
Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A002605, A293005, A293006.

Essentially the same as A028860 and A152035.

concat(vector(2), Vec(2*x^2 / (1-2*x-2*x^2) + O(x^50))) \\ Colin Barker, Sep 28 2017

a(n) = 2*A002605(n-1), a(0) = 0.
a(n) = A028860(n+1), a(0) = 0.
From Colin Barker, Sep 28 2017: (Start)
a(n) = ((1-sqrt(3))^n*(1+sqrt(3)) + (-1+sqrt(3))*(1+sqrt(3))^n) / (2*sqrt(3)) for n>0.
a(n) = 2*a(n-1) + 2*a(n-2) for n>2.
(End)

A152035 Expansion of g.f. (1-2x^2)/(1-2x-2*x^2).

Original entry on oeis.org

1, 2, 4, 12, 32, 88, 240, 656, 1792, 4896, 13376, 36544, 99840, 272768, 745216, 2035968, 5562368, 15196672, 41518080, 113429504, 309895168, 846649344, 2313089024, 6319476736, 17265131520, 47169216512, 128868696064, 352075825152, 961889042432, 2627929735168, 7179637555200, 19615134580736

Offset: 0

Roger L. Bagula, Nov 20 2008

Essentially same as A028860. - Philippe Deléham, Sep 21 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A028860. Row sums of A322942.

[1] cat [n le 2 select 2^n else 2*(Self(n-1) +Self(n-2)): n in [1..30]]; // G. C. Greubel, Sep 20 2023

a := proc(n) option remember;
`if`(n < 3, [1, 2, 4][n+1], 2*(a(n-1) + a(n-2))) end:
seq(a(n), n=0..31); # Peter Luschny, Jan 03 2019

f[n_] = 2^n*Product[(1 + 2*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 15}]
CoefficientList[Series[(1-2x^2)/(1-2x-2x^2),{x,0,40}],x] (* Harvey P. Dale, Sep 23 2014 *)
LinearRecurrence[{2,2},{1,2,4},40] (* Harvey P. Dale, May 12 2023 *)

@CachedFunction
def a(n): # a = A152035
    if n<3: return (1,2,4)[n]
    else: return 2*(a(n-1) + a(n-2))
[a(n) for n in range(31)] # G. C. Greubel, Sep 20 2023

a(n) = 2*(a(n-1) + a(n-2)) for n >= 3. - Peter Luschny, Jan 03 2019

Edited by N. J. A. Sloane, Apr 11 2009, based on comments from Philippe Deléham and R. J. Mathar

More terms from Philippe Deléham, Sep 21 2009

A106434 The (1,1)-entry of the matrix A^n, where A = [0,1;2,3].

Original entry on oeis.org

0, 2, 6, 22, 78, 278, 990, 3526, 12558, 44726, 159294, 567334, 2020590, 7196438, 25630494, 91284358, 325114062, 1157910902, 4123960830, 14687704294, 52311034542, 186308512214, 663547605726, 2363259841606, 8416874736270, 29977143892022, 106765181148606

Offset: 1

Roger L. Bagula, May 29 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,2).

Cf. A028860, A100638.

a[1]:=0: a[2]:=2: for n from 3 to 25 do a[n]:=3*a[n-1]+2*a[n-2] od: seq(a[n],n=1..25);

LinearRecurrence[{3, 2}, {0, 2}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

A106434(n)=([0,1;2,3]^n)[1,1] /* M. F. Hasler, Dec 01 2008 */

a(n) = 3*a(n-1) + 2*a(n-2) for n>=3; a(1)=0, a(2)=2.
O.g.f.: 2*x^2/(1-3*x-2*x^2). - R. J. Mathar, Dec 05 2007
a(n) = 2 * A007482(n-2) for n >= 2.

Simplified definition and added cross reference. - M. F. Hasler, Dec 01 2008

Edited by N. J. A. Sloane, May 20 2006 and Dec 04 2008

A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).

Original entry on oeis.org

0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, 915814, 3504600, 13419898, 51371996, 196683278, 752970528, 2882724002, 11036241700, 42251551414, 161756794728, 619274449354, 2370846461804, 9076614069086, 34749153370800

Offset: 0

Peter Luschny, Jun 02 2018

			Array ((1+y)^n - (1-y)^n)/y with y = sqrt(k).
[k\n]
[1]   1, 2, 4,  8, 16, 32,   64,  128,    256,   512,   1024, ...
[2]   0, 2, 4, 10, 24, 58,  140,  338,    816,  1970,   4756, ...
[3]   0, 2, 4, 12, 32, 88,  240,  656,   1792,  4896,  13376, ...
[4]   0, 2, 4, 14, 40, 122, 364,  1094,  3280,  9842,  29524, ...
[5]   0, 2, 4, 16, 48, 160, 512,  1664,  5376, 17408,  56320, ...
[6]   0, 2, 4, 18, 56, 202, 684,  2378,  8176, 28242,  97364, ...
[7]   0, 2, 4, 20, 64, 248, 880,  3248, 11776, 43040, 156736, ...
[8]   0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, ...
[9]   0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, ...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7).

Let f(n, y) = ((1 + y)^n - (1 - y)^n)/y.
f(n,      1 ) = A000079(n);
f(n, sqrt(2)) = A163271(n+1);
f(n, sqrt(3)) = A028860(n+2);
f(n,      2 ) = A152011(n) for n>0;
f(n, sqrt(5)) = A103435(n);
f(n, sqrt(6)) = A083694(n);
f(n, sqrt(7)) = A274520(n);
f(n, sqrt(8)) = a(n);
f(n,      3 ) = A192382(n+1);
Cf. A305491.
Equals 2 * A015519.

egf :=  (n,x) -> 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n):
ser := series(egf(8,x), x, 26):
seq(n!*coeff(ser,x, n), n=0..24);

Table[Simplify[((1 + Sqrt[8])^n - (1 - Sqrt[8])^n)/ Sqrt[8]], {n, 0, 24}]

concat(0, Vec(2*x / (1 - 2*x - 7*x^2) + O(x^40))) \\ Colin Barker, Jun 05 2018

E.g.f.: 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n) for n = 8.
From Colin Barker, Jun 02 2018: (Start)
G.f.: 2*x / (1 - 2*x - 7*x^2).
a(n) = 2*a(n-1) + 7*a(n-2) for n>1.
(End)

A106568 Expansion of 4x/(1 - 4x - 4*x^2).

Original entry on oeis.org

0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360, 782005818097664, 3775858103812096, 18231455687639040

Offset: 0

Roger L. Bagula, May 30 2005

This sequence is part of a class of sequences with the properties: a(n) = m*(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = m, g.f.: m*x/(1 - m*x - m*x^2), and have the Binet form m*(alpha^n - beta^n)/(alpha - beta) where 2*alpha = m + sqrt(m^2 + 4*m) and 2*beta = p - sqrt(m^2 + 4*m). - G. C. Greubel, Sep 06 2021

G. C. Greubel, 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 (4,4).

Cf. A000129, A057087, A009013.

Sequences of the form a(n) = m*(a(n-1) + a(n-2)): A000045 (m=1), A028860 (m=2), A106435 (m=3), A094013 (m=4), A106565 (m=5), A221461 (m=6), A221462 (m=7).

[n le 2 select 4*(n-1) else 4*(Self(n-1) +Self(n-2)): n in [1..41]]; // G. C. Greubel, Sep 06 2021

A106568 := n -> ifelse(n=0, 0, 4^(n)*hypergeom([(1-n)/2, 1-n/2], [1-n], -1)):
seq(simplify(A106568(n)), n = 0..24);  # Peter Luschny, Mar 30 2025

LinearRecurrence[{4,4}, {0,4}, 40] (* G. C. Greubel, Sep 06 2021 *)

[2^(n+1)*lucas_number1(n,2,-1) for n in (0..40)] # G. C. Greubel, Sep 06 2021

a(n) = 4 * A057087(n).
a(n) = A094013(n+1). - R. J. Mathar, Aug 24 2008
From Philippe Deléham, Sep 19 2009: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) for n > 2; a(0) = 0, a(1)=4.
G.f.: 4*x/(1 - 4*x - 4*x^2). (End)
G.f.: Q(0) - 1, where Q(k) = 1 + 2*(1+2*x)*x + 2*(2*k+3)*x - 2*x*(2*k+1 +2*x+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013
a(n) = 2^(n+1)*A000129(n). - G. C. Greubel, Sep 06 2021
a(n) = 4^n*hypergeom([(1-n)/2, 1-n/2], [1-n], -1) for n > 0. - Peter Luschny, Mar 30 2025

Edited by N. J. A. Sloane, Apr 30 2006

Simpler name using o.g.f. by Joerg Arndt, Oct 05 2013

cp's OEIS Frontend

A293007 Expansion of 2*x^2 / (1-2*x-2*x^2).

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A152035 Expansion of g.f. (1-2*x^2)/(1-2*x-2*x^2).

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A106434 The (1,1)-entry of the matrix A^n, where A = [0,1;2,3].

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A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).

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A106568 Expansion of 4*x/(1 - 4*x - 4*x^2).

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A293007 Expansion of 2x^2 / (1-2x-2*x^2).

A152035 Expansion of g.f. (1-2x^2)/(1-2x-2*x^2).

A106568 Expansion of 4x/(1 - 4x - 4*x^2).