A092184 -id:A092184 - cp's OEIS frontend

A102206 a(0) = 3, a(1) = 8, a(n+2) = 4*a(n+1) - a(n) - 2.

3, 8, 27, 98, 363, 1352, 5043, 18818, 70227, 262088, 978123, 3650402, 13623483, 50843528, 189750627, 708158978, 2642885283, 9863382152, 36810643323, 137379191138, 512706121227, 1913445293768, 7141075053843, 26650854921602, 99462344632563, 371198523608648

Offset: 0

Creighton Dement, Dec 30 2004

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A092184, A001834, A001353, A102207.

a[0] = 3; a[1] = 8; a[n_] := a[n] = 4a[n - 1] - a[n - 2] - 2; Table[a[n], {n, 0, 23}] (* Or *)
CoefficientList[ Series[(2x - 1)(x - 3)/((1 - x)(x^2 - 4x + 1)), {x, 0, 22}], x] (* Robert G. Wilson v, Jan 12 2005 *)
LinearRecurrence[{5,-5,1},{3,8,27},30] (* Harvey P. Dale, Jul 25 2012 *)

Vec((2*x-1)*(x-3)/((1-x)*(x^2-4*x+1)) + O(x^30)) \\ Colin Barker, Nov 03 2016

G.f.: (2x-1)(x-3)/((1-x)(x^2-4x+1)).
a(n) = A092184(n+1) + 2; a(n+1) - a(n) = A001834(n+1) (see comment).
a(0)=3, a(1)=8, a(2)=27, a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). - Harvey P. Dale, Jul 25 2012
a(n) = (2+(2-sqrt(3))^(1+n)+(2+sqrt(3))^(1+n))/2. - Colin Barker, Nov 03 2016

More terms from Robert G. Wilson v, Jan 12 2005

Recurrence in the definition corrected by R. J. Mathar, Aug 07 2008

A095004 a(n) = 9a(n-1) - 9a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.

Original entry on oeis.org

1, 10, 81, 640, 5041, 39690, 312481, 2460160, 19368801, 152490250, 1200553201, 9451935360, 74414929681, 585867502090, 4612525087041, 36314333194240, 285902140466881, 2250902790540810, 17721320183859601, 139519658680336000, 1098435949258828401, 8647967935390291210

Offset: 1

Gary W. Adamson, May 27 2004

A sequence derived from A076765, with a(n)/a(n-1) tending to 4 + sqrt(15).
a(n)/a(n-1) tends to C = 4 + sqrt(15) = 7.87298334... (C having the property that C + 1/C = 8). Eigenvalues of M (1, C, 1/C) are roots to x^3 - 9x^2 + 9x - 1.
This is the r=10 member of the r-family of sequences S_r(n), n>=1, defined in A092184, where more information can be found.

			a(4) = 640 = 568 + 72 = A076765(3) + A076765(2).
a(4) = 640 = 9*81 - 9*10 + 1.
a(4) = 640, rightmost term in M^4 * [1 0 0]: [145 352 640] = [A095002(4) A095003(4) A095004(4)].

Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Cf. A095002, A095003, A076765.

a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1, 3]:
seq(a(n), n=1..23);  # Alois P. Heinz, Jun 06 2021

a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)

a(n) = A076765(n-1) + A076765(n-2).
Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]; then M^n * [1 0 0] = [A095002(n) A095003(n) a(n)].
a(n)= (T(n, 4)-1)/3 with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n). a(0):=0. - Wolfdieter Lang, Oct 18 2004
G.f.: x*(1+x)/((1-x)*(1-8*x+x^2)) = x*(1+x)/(1-9*x+9*x^2-x^3).

Edited and extended by Robert G. Wilson v, May 29 2004

Definition aligned with A095002, A095003 by Georg Fischer, Jun 06 2021

A129743 a(n) = -(u^n-1)*(v^n-1) with u = 2+sqrt(3), v = 2-sqrt(3).

Original entry on oeis.org

2, 12, 50, 192, 722, 2700, 10082, 37632, 140450, 524172, 1956242, 7300800, 27246962, 101687052, 379501250, 1416317952, 5285770562, 19726764300, 73621286642, 274758382272, 1025412242450, 3826890587532, 14282150107682, 53301709843200, 198924689265122, 742397047217292

Offset: 1

N. J. A. Sloane, May 13 2007

Each term of this sequence beyond the sixth has a primitive prime divisor. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007

a(n) is also the number of spanning trees for the n-gear graph. - Eric W. Weisstein, Jul 16 2011

Stefano Spezia, Table of n, a(n) for n = 1..1700
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
Anthony Flatters, Primitive Divisors of some Lehmer-Pierce Sequences, arXiv:0708.2190 [math.NT], 2007.
Eric Weisstein's World of Mathematics, Gear Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001353, A001834, A092184.

u:=2+sqrt(3): v:=2-sqrt(3): a:=n->expand(-(u^n-1)*(v^n-1)): seq(a(n),n=1..28); # Emeric Deutsch, May 13 2007

Table[-((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1), {n, 30}] // Expand (* Stefan Steinerberger, May 15 2007 *)
LinearRecurrence[{5, -5, 1}, {2, 12, 50}, 30]
LucasL[2 Range[20], Sqrt[2]] - 2 // Round (* Eric W. Weisstein, Mar 28 2018 *)

my(x='x+O('x^30)); Vec(2*x*(1+x)/((1-x)*(1-4*x+x^2))) \\ Altug Alkan, Mar 28 2018

a(2*n) = 12*A001353(n)^2, a(2*n+1) = 2*A001834(n)^2. - Vladeta Jovovic, May 30 2007
a(n) = 2*A092184(n). - Robert G. Wilson v, Jul 04 2007
O.g.f.: 2*x*(1+x)/((1-x)*(1-4*x+x^2)). - R. J. Mathar, Dec 05 2007
a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). - Eric W. Weisstein, Jul 15 2011
E.g.f.: 2*exp(x)*(exp(x)*cosh(sqrt(3)*x) - 1). - Stefano Spezia, May 05 2024

More terms from Emeric Deutsch and Stefan Steinerberger, May 13 2007

More terms from Vladeta Jovovic, May 30 2007

A102207 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(0) = 4, a(1) = 17, a(2) = 65.

Original entry on oeis.org

4, 17, 65, 244, 912, 3405, 12709, 47432, 177020, 660649, 2465577, 9201660, 34341064, 128162597, 478309325, 1785074704, 6661989492, 24862883265, 92789543569, 346295291012, 1292391620480, 4823271190909, 18000693143157

Offset: 0

Creighton Dement, Dec 30 2004

nonn

Index entries for linear recurrences with constant coefficients, signature (5, -5, 1).

Cf. A061278, A092184, A001834, A001353, A102206.

a[0] = 4; a[1] = 17; a[2] = 65; a[n_] := a[n] = 5a[n - 1] - 5a[n - 2] + a[n - 3]; Table[ a[n], {n, 0, 22}] (* Or *)
CoefficientList[ Series[(3x - 4)/((x - 1)(x^2 - 4x + 1)), {x, 0, 22}], x] (* Robert G. Wilson v, Jan 12 2005 *)
LinearRecurrence[{5,-5,1},{4,17,65},30] (* or *) With[{c=Sqrt[3]},Table[ Simplify[ ((3-7c)(2-c)^x+(2+c)^x (3+7c)-6)/12],{x,30}]] (* Harvey P. Dale, Mar 15 2013 *)

G.f.: (3x-4)/((x-1)(x^2-4x+1))
(1/2) [A001353(n+1) + 5*A001353(n) - 1 ]. - Ralf Stephan, May 17 2007
a(n)=1/12*((3-7*Sqrt[3])*(2-Sqrt[3])^n+(3+7*Sqrt[3])*(2+Sqrt[3])^n-6). - Harvey P. Dale, Mar 15 2013

More terms from Robert G. Wilson v, Jan 12 2005

A140824 Expansion of (x-x^3)/(1-3x+2x^2-3*x^3+x^4).

Original entry on oeis.org

0, 1, 3, 6, 15, 41, 108, 281, 735, 1926, 5043, 13201, 34560, 90481, 236883, 620166, 1623615, 4250681, 11128428, 29134601, 76275375, 199691526, 522799203, 1368706081, 3583319040, 9381251041, 24560434083, 64300051206, 168339719535, 440719107401, 1153817602668

Offset: 0

N. J. A. Sloane, Sep 07 2009, based on email from R. K. Guy, Mar 09 2009

Case P1 = 3, P2 = 0, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Cf. A006238, A005248, A054493, A078070, A092184, A098306, A100047, A100048, A108196, A138573, A152090, A218134.

LinearRecurrence[{3, -2, 3, -1}, {0, 1, 3, 6}, 50] (* G. C. Greubel, Aug 08 2017 *)

x='x+O('x^50); concat([0], Vec((x-x^3)/(1-3*x+2*x^2-3*x^3+x^4))) \\ G. C. Greubel, Aug 08 2017

a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 6, a(n) - 3 a(n + 1) + 2 a(n + 2) - 3 a(n + 3) + a(n + 4) = 0.
From Peter Bala, Mar 25 2014: (Start)
a(n) = 2/3*( T(n,3/2) - T(n,0) ), where T(n,x) is a Chebyshev polynomial of the first kind.
a(n) = 1/3 * (A005248(n) - (i^n + (-i)^n)) = 1/3 * (Fibonacci(2*n-1) + Fibonacci(2*n+1) - (i^n + (-i)^n)).
a(n) = bottom left entry of the 2 X 2 matrix 2*T(n, 1/2*M), where M is the 2 X 2 matrix [0, 0; 1, 3].
The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/sqrt(2)*(sqrt(5) + i)*x + x^2) and x/(1 - 1/sqrt(2)*(sqrt(5) - i)*x + x^2). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = A099483(n) - A099483(n-2). - R. J. Mathar, Feb 10 2016

A335649 a(n) is the frequency of multi-pairs in a sequence of multi-set designs with 2 blocks.

Original entry on oeis.org

0, 10, 200, 3040, 43320, 607050, 8468880, 118007680, 1643826800, 22896269770, 318906570840, 4441805503200, 61866406977960, 861688028423050, 12001766499380000, 167163044860403200, 2328280868627854560, 32428769142358413450, 451674487223023755240, 6291014052348080593120

Offset: 1

John P. McSorley, Jun 15 2020

			For V={x,y} the design for n=2 are the blocks {xxxxxy,xyyyyy}. Pair frequencies of the multi-pairs xx, yy, and xy in these 2 blocks are all a(2)=10.
A092184(3)=6, and the above example has blocks of size 6.

A. Assaf, A. Hartman, E. Mendelsohn, Multi-set Designs-Designs having blocks with repeated elements, Congressus Numerantium, 48 (1985), 7-24.

Index entries for linear recurrences with constant coefficients, signature (19,-76,76,-19,1).

A092184(n+1) is the block size of the n-th design in the sequence.

LinearRecurrence[{19, -76, 76, -19, 1}, {0, 10, 200, 3040, 43320, 607050}, 20] (* Amiram Eldar, Jun 16 2020 *)

concat(0, Vec(10*x^2*(1 + x) / ((1 - x)*(1 - 14*x + x^2)*(1 - 4*x + x^2)) + O(x^20))) \\ Colin Barker, Jun 16 2020

a(n) = (1/12)*((2+sqrt(3))^(2*n) + (2-sqrt(3))^(2*n) - 6*(2+sqrt(3))^n - 6*(2-sqrt(3))^n + 10).
a(n) = (1/3)*(A092184(n+1)*(A092184(n+1)-1)).
From Colin Barker, Jun 16 2020: (Start)
G.f.: 10*x^2*(1 + x) / ((1 - x)*(1 - 14*x + x^2)*(1 - 4*x + x^2)).
a(n) = 19*a(n-1) - 76*a(n-2) + 76*a(n-3) - 19*a(n-4) + a(n-5) for n>5.
(End)

More terms from Jinyuan Wang, Jun 15 2020

cp's OEIS Frontend

A102206 a(0) = 3, a(1) = 8, a(n+2) = 4*a(n+1) - a(n) - 2.

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A095004 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.

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A129743 a(n) = -(u^n-1)*(v^n-1) with u = 2+sqrt(3), v = 2-sqrt(3).

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A102207 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(0) = 4, a(1) = 17, a(2) = 65.

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A140824 Expansion of (x-x^3)/(1-3*x+2*x^2-3*x^3+x^4).

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Programs

Mathematica

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A335649 a(n) is the frequency of multi-pairs in a sequence of multi-set designs with 2 blocks.

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Mathematica

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Formula

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A095004 a(n) = 9a(n-1) - 9a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.

A140824 Expansion of (x-x^3)/(1-3x+2x^2-3*x^3+x^4).