A100058 -id:A100058 - cp's OEIS frontend

A019481 a(n) = 3a(n-1) + a(n-2) - 2a(n-3) (agrees with A019480 for n <= 19 only).

4, 12, 37, 115, 358, 1115, 3473, 10818, 33697, 104963, 326950, 1018419, 3172281, 9881362, 30779529, 95875387, 298642966, 930245227, 2897627873, 9025842914, 28114666161, 87574585651, 272786737286, 849705465187, 2646753961545, 8244393875250

Offset: 0

N. J. A. Sloane

R. K. Guy, personal communication.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-2)

I:=[4, 12, 37]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 18 2017

LinearRecurrence[{3,1, -2}, {4, 12, 37}, 30] (* Harvey P. Dale, Jan 17 2017 *)

From R. J. Mathar, Aug 22 2008: (Start)
O.g.f.: (4 - 3*x^2)/(1 - 3*x - x^2 + 2*x^3).
a(n) = 4*A100058(n) - 3*A100058(n-2). (End)

A052911 Expansion of (1-x)/(1 - 3x - x^2 + 2x^3).

Original entry on oeis.org

1, 2, 7, 21, 66, 205, 639, 1990, 6199, 19309, 60146, 187349, 583575, 1817782, 5662223, 17637301, 54938562, 171128541, 533049583, 1660400166, 5171992999, 16110279997, 50182032658, 156312391973, 486898648583, 1516644272406

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 891
Index entries for linear recurrences with constant coefficients, signature (3,1,-2).

Cf. A100058, A058071, A100059.

a:=[1,2,7];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Oct 15 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x-x^2+2*x^3) )); // G. C. Greubel, Oct 15 2019

spec := [S,{S=Sequence(Union(Z,Prod(Union(Sequence(Z),Z,Z),Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

LinearRecurrence[{3,1,-2}, {1,2,7}, 30] (* G. C. Greubel, Oct 15 2019 *)

my(x='x+O('x^30)); Vec((1-x)/(1-3*x-x^2+2*x^3)) \\ G. C. Greubel, Oct 15 2019

def A052911_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-3*x-x^2+2*x^3)).list()
A052911_list(30) # G. C. Greubel, Oct 15 2019

G.f.: (1-x)/(1 - 3*x - x^2 + 2*x^3)
a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = Sum_{alpha=RootOf(1 - 3*z - z^2 + 2*z^3)} (1/229)*(43 + 41*alpha - 46*alpha^2)*alpha^(-1-n).
a(n) = center term in M^n * [1 1 1] where M = Hosoya's triangle considered as an upper triangular 3 X 3 matrix: [2 1 2 / 1 1 0 / 1 0 0]. E.g., a(4) = 66 since M^4 * [1 1 1] = [139 66 45]. The analogous procedure using M^n * [1 0 0] generates A100058. - Gary W. Adamson, Oct 31 2004
a(n) = A100058(n) - A100058(n-1). - R. J. Mathar, May 04 2018

More terms from James Sellers, Jun 06 2000

A100059 First differences of A052911.

Original entry on oeis.org

1, 5, 14, 45, 139, 434, 1351, 4209, 13110, 40837, 127203, 396226, 1234207, 3844441, 11975078, 37301261, 116189979, 361921042, 1127350583, 3511592833, 10938286998, 34071752661, 106130359315, 330586256610

Offset: 1

Gary W. Adamson, Oct 31 2004

nonn

a(n)/a(n-1) tends to 3.11490754148...an eigenvalue of M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.

			a(5) = 139 = rightmost term in M^5 * [1 1 1] which is [434 205 139]. 434 = a(6), while 205 = A052911(5).
a(6) = 434 = 3*a(5) + a(4) - 2*a(3) = 3*139 + 45 - 2*14.

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Index entries for linear recurrences with constant coefficients, signature (3,1,-2).
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 27.

Cf. A019481, A052550, A052939, A100058, A058071.

LinearRecurrence[{3,1,-2},{1,5,14},30] (* Harvey P. Dale, Apr 21 2016 *)

G.f.: (2*x^2-2*x-1)*x / (-2*x^3+x^2+3*x-1).
Recurrence: a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = rightmost term in M^5 * [1 1 1], where M = the 3 X 3 upper triangular matrix [2 1 2 / 1 1 0 / 1 0 0].
INVERT transform of (1, 4, 5, 6, 7, 8, 9, ...) with offset 0.

Edited by Ralf Stephan, Nov 02 2004

cp's OEIS Frontend

A019481 a(n) = 3a(n-1) + a(n-2) - 2a(n-3) (agrees with A019480 for n <= 19 only).

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A052911 Expansion of (1-x)/(1 - 3x - x^2 + 2x^3).

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A100059 First differences of A052911.

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cp's OEIS Frontend

A019481 a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3) (agrees with A019480 for n <= 19 only).

Views

Author

Keywords

References

Links

Programs

Magma

Mathematica

Formula

A052911 Expansion of (1-x)/(1 - 3*x - x^2 + 2*x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A100059 First differences of A052911.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A019481 a(n) = 3a(n-1) + a(n-2) - 2a(n-3) (agrees with A019480 for n <= 19 only).

A052911 Expansion of (1-x)/(1 - 3x - x^2 + 2x^3).