A199115 -id:A199115 - cp's OEIS frontend

A136412 a(n) = (5*4^n + 1)/3.

2, 7, 27, 107, 427, 1707, 6827, 27307, 109227, 436907, 1747627, 6990507, 27962027, 111848107, 447392427, 1789569707, 7158278827, 28633115307, 114532461227, 458129844907, 1832519379627, 7330077518507, 29320310074027

Offset: 0

Paul Curtz, Mar 31 2008

An Engel expansion of 4/5 to the base b := 4/3 as defined in A181565, with the associated series expansion 4/5 = b/2 + b^2/(2*7) + b^3/(2*7*27) + b^4/(2*7*27*107) + .... Cf. A199115 and A140660. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Sequences of the form (m*4^n + 1)/3: A007583 (m=2), this sequence (m=5), A199210 (m=11), A199210 (m=11), A206373 (m=14).

Cf. A007302, A140660, A181565, A199115.

a136412 = (`div` 3) . (+ 1) . (* 5) . (4 ^)
-- Reinhard Zumkeller, Jun 17 2012

[(5*4^n+1)/3: n in [0..30]]; // Vincenzo Librandi, Nov 04 2011

LinearRecurrence[{5,-4}, {2,7}, 31] (* G. C. Greubel, Jan 19 2023 *)

a(n)=(5*4^n+1)/3 \\ Charles R Greathouse IV, Oct 07 2015

[(5*4^n+1)/3 for n in range(31)] # G. C. Greubel, Jan 19 2023

a(n) = 4*a(n-1) - 1.
a(n) = A199115(n)/3.
O.g.f.: (2-3*x)/((1-x)*(1-4*x)). - R. J. Mathar, Apr 04 2008
a(n) = 5*a(n-1) - 4*a(n-2). - Vincenzo Librandi, Nov 04 2011
E.g.f.: (1/3)*(5*exp(4*x) + exp(x)). - G. C. Greubel, Jan 19 2023

Formula in definition and more terms from R. J. Mathar, Apr 04 2008

A140660 a(n) = 3*4^n + 1.

Original entry on oeis.org

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

Paul Curtz, Jul 10 2008

nonn

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

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A181565, A199115.

[3*4^n+1: n in [0..30] ]; // Vincenzo Librandi, May 23 2011

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 *)

x='x+O('x^50); Vec((7*x-4)/((1-x)*(4*x-1))) \\ G. C. Greubel, Sep 15 2017

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)

Edited and extended R. J. Mathar, Jul 14 2008

cp's OEIS Frontend

A136412 a(n) = (5*4^n + 1)/3.

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