A133600 -id:A133600 - cp's OEIS frontend

A133599 A097806 * A133080 * A007318.

1, 3, 1, 3, 3, 1, 3, 7, 5, 1, 3, 9, 10, 5, 1, 3, 13, 22, 18, 7, 1, 3, 15, 31, 34, 21, 7, 1, 3, 19, 51, 75, 65, 33, 9, 1, 3, 21, 64, 111, 120, 83, 36, 9, 1, 3, 25, 92, 196, 266, 238, 140, 52, 11, 1

Offset: 1

Gary W. Adamson, Sep 18 2007

Row sums = A133600: (1, 4, 7, 16, 28, 64, 112, ...).

			First few rows of the triangle:
  1;
  3,  1;
  3,  3,  1;
  3,  7,  5,  1;
  3,  9, 10,  5,  1;
  3, 13, 22, 18,  7,  1;
  ...

Cf. A097806, A133080, A133600.

A133599aux := proc(n,k)
    add(A097806(n,j)*A133080(j,k),j=k..n) ;
end proc:
A133599 := proc(n,k)
    add(A133599aux(n,j)*binomial(j-1,k-1),j=k..n) ;
end proc: # R. J. Mathar, Jun 20 2015

A097806 * A133080 * A007318 as infinite lower triangular matrices, where A097806 = the pairwise operator and A133080 = an interpolation operator.

A137480 a(n)=4a(n-2).

Original entry on oeis.org

12, 21, 48, 84, 192, 336, 768, 1344, 3072, 5376, 12288, 21504, 49152, 86016, 196608, 344064, 786432, 1376256, 3145728, 5505024, 12582912, 22020096, 50331648, 88080384, 201326592, 352321536, 805306368, 1409286144, 3221225472, 5637144576, 12884901888

Offset: 0

Paul Curtz, Apr 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4).

LinearRecurrence[{0, 4}, {12, 21}, 50] (* G. C. Greubel, Feb 23 2017 *)

x='x + O('x^50); Vec((21*x + 12)/(-4*x^2 + 1)) \\ G. C. Greubel, Feb 23 2017

G.f.: (21*x + 12)/(-4*x^2 + 1).
a(n) = -(9/4)*sqrt(2)*(1/2)^(-1/2*(-1)^n)*(-2)^n + (33/4)*2^n*sqrt(2)*(1/2)^(-(1/2)*(-1)^n) - Alexander R. Povolotsky, Apr 25 2008
a(n) = 3*A133600(n+2) - R. J. Mathar, Jun 08 2016

A171475 a(n) = 6a(n-1) - 8a(n-2), for n > 2, with a(0) = 1, a(1) = 6, a(2) = 27.

Original entry on oeis.org

1, 6, 27, 114, 468, 1896, 7632, 30624, 122688, 491136, 1965312, 7862784, 31454208, 125822976, 503304192, 2013241344, 8053014528, 32212156416, 128848822272, 515395682304, 2061583515648, 8246335635456, 32985345687552

Offset: 0

Klaus Brockhaus, Dec 09 2009

Binomial transform of A037480; second binomial transform of A133600.

First differences of A080960.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A037480 ((5*3^n +(-1)^n -6)/8), A133600 (row sums of triangle A133599), A080960 (third binomial transform of A010685).

I:=[6,27]; [1] cat [n le 2 select I[n] else 6*Self(n-1) - 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 02 2021

Table[If[n==0, 1, 3*(5*4^n - 2*2^n)/8],{n,0,30}] (* G. C. Greubel, Dec 02 2021 *)
LinearRecurrence[{6,-8},{1,6,27},30] (* Harvey P. Dale, Oct 25 2023 *)

{m=21; v=concat([1, 6, 27], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]); v}

[1]+[3*(5*4^n - 2*2^n)/8 for n in (1..30)] # G. C. Greubel, Dec 02 2021

a(n) = 3*(5*4^n - 2*2^n)/8 for n > 0.
G.f.: (1-x)*(1+x)/((1-2*x)*(1-4*x)).
E.g.f.: (1/8)*(-1 - 6*exp(2*x) + 15*exp(4*x)). - G. C. Greubel, Dec 02 2021

cp's OEIS Frontend

A133599 A097806 * A133080 * A007318.

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A137480 a(n)=4a(n-2).

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Formula

A171475 a(n) = 6a(n-1) - 8a(n-2), for n > 2, with a(0) = 1, a(1) = 6, a(2) = 27.

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cp's OEIS Frontend

A133599 A097806 * A133080 * A007318.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A137480 a(n)=4a(n-2).

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

A171475 a(n) = 6*a(n-1) - 8*a(n-2), for n > 2, with a(0) = 1, a(1) = 6, a(2) = 27.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A171475 a(n) = 6a(n-1) - 8a(n-2), for n > 2, with a(0) = 1, a(1) = 6, a(2) = 27.