A154383 -id:A154383

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A154407

a(n) = 5*2^(n-1) + 3*6^n/2.

+10
3

4, 14, 64, 344, 1984, 11744, 70144, 420224, 2520064, 15117824, 90701824, 544200704, 3265183744, 19591061504, 117546287104, 705277558784, 4231665025024, 25389989494784, 152339935657984, 914039611326464, 5484237662715904

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

One of the diagonals of the n-th differences of A154383.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..150

Index entries for linear recurrences with constant coefficients, signature (8, -12).

FORMULA

a(n+1) = 6*a(n) - 10*2^n.

a(n) = 6*a(n) - 5*A020714(n+1).

G.f.: 2*(2 - 9*x)/((6*x-1)*(2*x-1)). - R. J. Mathar, May 21 2009

E.g.f.: (1/2)*( 5*exp(2*x) + 3*exp(6*x) ). - G. C. Greubel, Sep 16 2016

EXAMPLE

Sequence A154383 and its k-th iterated difference in the k-th row are

...1.....0.....4.....2.....16......8.....64.....32....256....128...1024.

..-1.....4....-2....14.....-8.....56....-32....224...-128....896...-512.

...5....-6....16...-22.....64....-88....256...-352...1024..-1408...4096.

.-11....22...-38....86...-152....344...-608...1376..-2432...5504..-9728.

..33...-60...124..-238....496...-952...1984..-3808...7936.-15232..31744.

.-93...184..-362...734..-1448...2936..-5792..11744.-23168..46976.-92672.

.277..-546..1096.-2182...4384..-8728..17536.-34912..70144.-139648.280576.

The sequence is the diagonal T(k,k+2) in this array.

MAPLE

A154407:=n->5*2^(n-1)+3*6^n/2; seq(A154407(n), n=0..50); # Wesley Ivan Hurt, Nov 13 2013

MATHEMATICA

Table[5*2^(n-1)+3*6^n/2, {n, 0, 50}] (* Wesley Ivan Hurt, Nov 13 2013 *)

PROG

(Magma) [5*2^(n-1)+3*6^n/2: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jan 09 2009

EXTENSIONS

Edited by R. J. Mathar, May 21 2009

STATUS

approved

A167784

a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

+10
3

1, 0, 4, 2, 16, 14, 64, 74, 256, 350, 1024, 1562, 4096, 6734, 16384, 28394, 65536, 117950, 262144, 484922, 1048576, 1979054, 4194304, 8034314, 16777216, 32491550, 67108864, 131029082, 268435456, 527304974, 1073741824, 2118785834, 4294967296, 8503841150

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Binomial transform of A077917, the signed variant of A127864.

LINKS

Robert Israel, Table of n, a(n) for n = 0..3318

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

FORMULA

a(n) = A167936(n+1) - A167936(n).

a(2n) = A000302(n). a(2n+1) = 2*A005061(n).

a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).

G.f.: (x-1)^2/((2*x-1)*(3*x^2-1)).

a(n+4) mod 9 = A153130(n+4) = A146501(n+2), n>=0.

E.g.f.: exp(2*x) - (2/sqrt(3))*sinh(sqrt(3)*x). - G. C. Greubel, Jun 27 2016

MAPLE

seq(2^n - (1 - (-1)^n)*3^((n-1)/2), n=0..100); # Robert Israel, Apr 11 2019

MATHEMATICA

LinearRecurrence[{2, 3, -6}, {1, 0, 4}, 40] (* Harvey P. Dale, Nov 29 2011 *)

CROSSREFS

Cf. A154383.

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Nov 12 2009

EXTENSIONS

Edited and extended by R. J. Mathar, Feb 27 2010

Incorrect b-file corrected by Robert Israel, Apr 11 2019

STATUS

approved

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