A307695 -id:A307695 - cp's OEIS frontend

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

1, 20, 336, 5440, 87296, 1397760, 22368256, 357908480, 5726601216, 91625881600, 1466015154176, 23456246661120, 375299963355136, 6004799480791040, 96076791961092096, 1537228672451215360, 24595658763514413056, 393530540233410478080, 6296488643803287126016

Offset: 0

Klaus Brockhaus, Oct 26 2009

Partial sums of A166965.

First differences of A006105. - Klaus Purath, Oct 15 2020

Seiichi Manyama, Table of n, a(n) for n = 0..830 (terms 0..200 from Vincenzo Librandi)
E. Saltürk and I. Siap, Generalized Gaussian Numbers Related to Linear Codes over Galois Rings, European Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 2012, 250-259; ISSN 1307-5543. - From _N. J. A. Sloane_, Oct 23 2012
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A000302, A002450, A002452, A002453, A006105, A079598, A083584.

Cf. A115490, A166914, A166917, A166927, A166965, A307695.

[n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];

LinearRecurrence[{20,-64},{1,20},30] (* Harvey P. Dale, Jul 04 2012 *)

a(n) = (4*16^n - 4^n)/3 \\ Charles R Greathouse IV, Jun 21 2022

A166984=BinaryRecurrenceSequence(20,-64,1,20)
[A166984(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

a(n) = (4*16^n - 4^n)/3.
G.f.: 1/((1-4*x)*(1-16*x)).
Limit_{n -> oo} a(n)/a(n-1) = 16.
a(n) = A115490(n+1)/3.
Sum_{n>=0} a(n) x^(2*n+4)/(2*n+4)! = ( sinh(x) )^4/4!. - Robert A. Russell, Apr 03 2013
From Klaus Purath, Oct 15 2020: (Start)
a(n) = A002450(n+1)*(A002450(n+2) - A002450(n))/5.
a(n) = (A083584(n+1)^2 - A083584(n)^2)/80. (End)
a(n) = (A079598(n) - A000302(n))/24. - César Aguilera, Jun 21 2022
a(n) = 16*a(n-1) + 4^n with a(0) = 1. - Nadia Lafreniere, Aug 08 2022
E.g.f.: (4/3)*exp(10*x)*sinh(6*x + log(2)). - G. C. Greubel, Oct 02 2024

A307810 Expansion of 1/AGM(1-64x, sqrt((1-16x)(1-256x))).

Original entry on oeis.org

1, 100, 13924, 2371600, 453093796, 92598490000, 19745403216400, 4333667896360000, 971177275449892900, 221106619001508490000, 50967394891692703241104, 11866732390447357481358400, 2785834789480617203561744656, 658549235163074008904405646400

Offset: 0

Seiichi Manyama, Apr 30 2019

nonn

See A246923.

Also the squares of coefficients in g.f. 1/sqrt((1-4*x)*(1-16*x)).

Seiichi Manyama, Table of n, a(n) for n = 0..416

Cf. A307695.

(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), A307811 (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 14, 0] // Flatten (* Amiram Eldar, May 13 2021 *)

N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))

{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}

{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}

a(n) = A307695(n)^2 = (Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k))^2.

a(n) ~ 2^(8*n+2) / (3*Pi*n). - Vaclav Kotesovec, Sep 27 2019

cp's OEIS Frontend

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

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A307810 Expansion of 1/AGM(1-64x, sqrt((1-16x)(1-256x))).

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cp's OEIS Frontend

A166984 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 20.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A307810 Expansion of 1/AGM(1-64*x, sqrt((1-16*x)*(1-256*x))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

A307810 Expansion of 1/AGM(1-64x, sqrt((1-16x)(1-256x))).