A134581 -id:A134581 - cp's OEIS frontend

A140343 a(n)=4a(n-1)-7a(n-2)+6a(n-3)-3a(n-4), n>4.

0, 0, 0, 0, 1, 4, 9, 14, 14, 0, -41, -122, -243, -364, -364, 0, 1093, 3280, 6561, 9842, 9842, 0, -29525, -88574, -177147, -265720, -265720, 0, 797161, 2391484, 4782969, 7174454, 7174454, 0, -21523361, -64570082, -129140163, -193710244, -193710244, 0, 581130733

Offset: 0

Paul Curtz, May 29 2008

This is the main sequence representing the degenerate case of sequences which equal their sixth differences, where, besides the generic a(n)=6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+6a(n-5), cf. A135356, there is also a shorter recurrence. Another sequence of this kind is A134581.

Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

Cf. A135356.

Join[{0},LinearRecurrence[{4,-7,6,-3},{0,0,0,1},40]] (* Harvey P. Dale, Mar 11 2015 *)

O.g.f.: x^4/((x^2-x+1)(3x^2-3x+1)). - R. J. Mathar, Jul 10 2008

Edited and corrected by R. J. Mathar, Jul 10 2008

A106233 An inverse Catalan transform of A003462.

Original entry on oeis.org

0, 1, 3, 5, 5, 0, -14, -41, -81, -121, -121, 0, 364, 1093, 2187, 3281, 3281, 0, -9842, -29525, -59049, -88573, -88573, 0, 265720, 797161, 1594323, 2391485, 2391485, 0, -7174454, -21523361, -43046721, -64570081, -64570081, 0, 193710244, 581130733, 1162261467

Offset: 0

Paul Barry, Apr 26 2005

The g.f. is obtained from that of A003462 through the mapping g(x)->g(x(1-x)). A003462 may be retrieved through the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. Binomial transform of x(1+x)/(1+x^2+x^4).

The sequence is identical to its sixth differences. See A140344. - Paul Curtz, Nov 09 2012

			From _Paul Curtz_, Nov 09 2012: (Start)
The sequence and its higher-order differences (periodic after 6 rows):
   0,  1,  3,  5,  5,   0, -14, ...
   1,  2,  2,  0, -5, -14, -27, ...
   1,  0, -2, -5, -9, -13, -13, ...
  -1, -2, -3, -4, -4,   0,  13, ...   = -A134581(n+1)
  -1, -1, -1,  0,  4,  13,  27, ...
   0,  0,  1,  4,  9,  14,  14, ...   = A140343(n+2)
   0,  1,  3,  5,  5,   0, -14, ...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

Cf. A103368.

I:=[0,1,3,5]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+ 6*Self(n-3)-3*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 24 2018

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* Vincenzo Librandi, Dec 24 2018 *)

G.f.: x(1-x)/((1-x+x^2)*(1-3*x+3*x^2));
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(3^(n-k)-1)/2.
a(n) = Sum_{k=0..n} A109466(n,k)*A003462(k). - Philippe Deléham, Oct 30 2008
a(n) = (1/2)*[A057083(n) - [1,1,0,0,-1,-1]6 ]. - _Ralf Stephan, Nov 15 2010
a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4) = A140343(n+2) - A140343(n+1). - Paul Curtz, Nov 09 2012
a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - Paul Curtz, Nov 09 2012

cp's OEIS Frontend

A140343 a(n)=4a(n-1)-7a(n-2)+6a(n-3)-3a(n-4), n>4.

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