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A140660
a(n) = 3*4^n + 1.
8
4, 13, 49, 193, 769, 3073, 12289, 49153, 196609, 786433, 3145729, 12582913, 50331649, 201326593, 805306369, 3221225473, 12884901889, 51539607553, 206158430209, 824633720833, 3298534883329, 13194139533313, 52776558133249
OFFSET
0,1
COMMENTS
An Engel expansion of 4/3 to the base 4 as defined in A181565, with the associated series expansion 4/3 = 4/4 + 4^2/(4*13) + 4^3/(4*13*49) + 4^4/(4*13*49*193) + .... Cf. A199115. - Peter Bala, Oct 29 2013
LINKS
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
FORMULA
a(n) = A002001(n+1) + 1.
a(n) = 4*a(n-1) - 3.
First differences: a(n+1) - a(n) = A002063(n).
a(n+k) - a(n) = 3*(4^k - 1)*A000302(n) = 9*A002450(k)*A000302(n).
a(n) = A140529(n) - A096045(n).
O.g.f.: (7*x - 4)/((1 - x)*(4*x - 1)). - R. J. Mathar, Jul 14 2008
From G. C. Greubel, Sep 15 2017: (Start)
E.g.f.: 3*exp(4*x) + exp(x).
a(n) = 5*a(n-1) - 4*a(n-2). (End)
MATHEMATICA
LinearRecurrence[{5, -4}, {4, 13}, 50] (* or *) CoefficientList[Series[ (7*x-4)/((1-x)*(4*x-1)), {x, 0, 50}], x] (* G. C. Greubel, Sep 15 2017 *)
PROG
(Magma) [3*4^n+1: n in [0..30] ]; // Vincenzo Librandi, May 23 2011
(PARI) x='x+O('x^50); Vec((7*x-4)/((1-x)*(4*x-1))) \\ G. C. Greubel, Sep 15 2017
CROSSREFS
Sequence in context: A149451 A376803 A149452 * A149453 A149454 A101125
KEYWORD
nonn
AUTHOR
Paul Curtz, Jul 10 2008
EXTENSIONS
Edited and extended R. J. Mathar, Jul 14 2008
STATUS
approved