A100088 -id:A100088 - cp's OEIS frontend

A137470 Inverse binomial transform of 1, 2, 2, 4, 10, 20, ... = A100088.

1, 1, -1, 3, -1, -1, 7, -9, 7, 7, -25, 39, -25, -25, 103, -153, 103, 103, -409, 615, -409, -409, 1639, -2457, 1639, 1639, -6553, 9831, -6553, -6553, 26215, -39321, 26215, 26215, -104857, 157287, -104857, -104857, 419431

Offset: 0

Paul Curtz, Apr 20 2008

Index entries for linear recurrences with constant coefficients, signature (-1,0,2)

a(n)=[3+(1-2i)(i-1)^n+(1+2i)(-1-i)^n]/5 where i=sqrt(-1). - R. J. Mathar, Apr 25 2008
O.g.f.: -(1+2x)/((1+2x+2x^2)(-1+x)). - R. J. Mathar, Apr 25 2008
a(n+1)-a(n)=A090132(n+1). - R. J. Mathar, Apr 25 2008
G.f.: Q(0)*(1+2*x)/(2- 2*x), where Q(k) = 1 + 1/(1 - x*(4*k+2 +2*x)/(x*(4*k+4 +2*x) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 01 2014

More terms from R. J. Mathar, Apr 25 2008

A122117 a(n) = 3a(n-1) + 4a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 10, 38, 154, 614, 2458, 9830, 39322, 157286, 629146, 2516582, 10066330, 40265318, 161061274, 644245094, 2576980378, 10307921510, 41231686042, 164926744166, 659706976666, 2638827906662, 10555311626650, 42221246506598

Offset: 0

Philippe Deléham, Oct 19 2006

Inverse binomial transform of A005053. Binomial transform of [1, 1, 7, 13, 55, ...] = A015441(n+1).
Convolved with [1, 2, 2, 2, ...] = powers of 4: [1, 4, 16, 64, ...]. - Gary W. Adamson, Jun 02 2009
a(n) is the number of compositions of n when there are 2 types of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A055380, A100088, A108981, A122016, A201455.

a:=[1,2];; for n in [3..30] do a[n]:=3*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, May 18 2019

I:=[1, 2]; [n le 2 select I[n] else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 06 2012

CoefficientList[Series[(1-x)/(1-3*x-4*x^2),{x,0,30}],x] (* Vincenzo Librandi, Jul 06 2012 *)

Vec((1-x)/(1-3*x-4*x^2)+O(x^30)) \\ Charles R Greathouse IV, Jan 11 2012

def A122117(n): return ((4<<(m:=n<<1))|2)//5-((1<Chai Wah Wu, Apr 22 2025

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,3,4, lambda n: 0); [next(it) for i in range(24)] # Zerinvary Lajos, Jul 03 2008

((1-x)/(1-3*x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 18 2019

a(n) = 2*A108981(n-1) for n > 0, with a(0) = 1.
a(2*n) = 4*a(2*n-1) + 2, a(2*n+1) = 4*a(2*n) - 2.
a(n) = Sum_{k=0..n} 2^(n-k)*A055380(n,k).
G.f.: (1-x)/(1-3*x-4*x^2).
Lim_{n->infinity} a(n+1)/a(n) = 4.
a(n) = Sum_{k=0..n} A122016(n,k)*2^k. - Philippe Deléham, Nov 05 2008
a(n) = A100088(2*n). - Chai Wah Wu, Apr 22 2025

Corrected by T. D. Noe, Nov 07 2006

A100087 Expansion of x/(sqrt(1-4*x^2) + x - 1).

Original entry on oeis.org

1, 2, 4, 10, 24, 60, 148, 370, 920, 2300, 5736, 14340, 35808, 89520, 223668, 559170, 1397496, 3493740, 8732920, 21832300, 54575888, 136439720, 341082504, 852706260, 2131706864, 5329267160, 13322959888, 33307399720, 83267756400, 208169391000, 520420803060, 1301052007650

Offset: 0

Paul Barry, Nov 03 2004

Inverse Chebyshev transform of (1-x^2)/((1-2*x)*(1+x^2)), the g.f. of A100088, under the mapping g(x) -> (1/sqrt(1-4*x^2))*g(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. Equivalently, its image under the Chebyshev map A(x) -> ((1-x^2)/(1+x^2))*A(x/(1+x^2)) is A100088.

Transform of 1/(1-2*x) under the mapping g(x) -> g(x*c(x^2)). - Paul Barry, Jan 17 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A100067, A100088.

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( x/(Sqrt(1-4*x^2) +x-1) )); // G. C. Greubel, Jul 08 2022

CoefficientList[Series[x/(Sqrt[1-4*x^2]+x-1), {x, 0, 50}], x] (* Vaclav Kotesovec, Dec 06 2012 *)

my(x='x+O('x^66)); Vec(x/(sqrt(1-4*x^2)+x-1)) \\ Joerg Arndt, May 12 2013

@CachedFunction
def A100067(n): return sum( binomial(n,k)*2^(n-2*k) for k in (0..(n//2)) )
def A100087(n): return (3/5)*A100067(n) + (1/5)*((1+(-1)^n) -2*I*(1-(-1)^n))*I^n*(-1)^floor(n/2)*binomial(n-1, floor(n/2))
[A100087(n) for n in (0..60)] # G. C. Greubel, Jul 08 2022

a(n) = Sum_{k=0..floor(n/2)} C(n, k)*(3*2^(n-2*k) + 2*cos(Pi*(n-2*k)/2) + 4*sin(Pi*(n-2*k)/2))/5.
a(n) = Sum_{k=0..floor(n/2)} C(n, k)*A100088(n-2*k).
a(n) = Sum_{k=0..n} k*C(n-1,(n-k)/2)*(1 + (-1)^(n-k))*2^k/(n+k). - Paul Barry, Jan 17 2005
D-finite with recurrence: 4*n*a(n) + 2*(2*n-7)*a(n-1) - (51*n-83)*a(n-2) - 8*(2*n-13)*a(n-3) + 140*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 22 2012
a(n) ~ 3*5^(n-1)/2^n. - Vaclav Kotesovec, Dec 06 2012

cp's OEIS Frontend

A137470 Inverse binomial transform of 1, 2, 2, 4, 10, 20, ... = A100088.

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A122117 a(n) = 3a(n-1) + 4a(n-2), with a(0)=1, a(1)=2.

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A100087 Expansion of x/(sqrt(1-4*x^2) + x - 1).

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cp's OEIS Frontend

A137470 Inverse binomial transform of 1, 2, 2, 4, 10, 20, ... = A100088.

Views

Author

Keywords

Links

Formula

Extensions

A122117 a(n) = 3*a(n-1) + 4*a(n-2), with a(0)=1, a(1)=2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Python

Sage

Sage

Formula

Extensions

A100087 Expansion of x/(sqrt(1-4*x^2) + x - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A122117 a(n) = 3a(n-1) + 4a(n-2), with a(0)=1, a(1)=2.