A206373 -id:A206373 - 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

A199210 a(n) = (11*4^n + 1)/3.

Original entry on oeis.org

4, 15, 59, 235, 939, 3755, 15019, 60075, 240299, 961195, 3844779, 15379115, 61516459, 246065835, 984263339, 3937053355, 15748213419, 62992853675, 251971414699, 1007885658795, 4031542635179, 16126170540715, 64504682162859

Offset: 0

Vincenzo Librandi, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Sequences of the form (m*4^n + 1)/3: A007583 (m=2), A136412 (m=5), this sequence (m=11), A199210 (m=11), A206373 (m=14).

Magma
```
[(11*4^n+1)/3: n in [0..30]];
```

LinearRecurrence[{5,-4}, {4,15}, 31] (* G. C. Greubel, Jan 19 2023 *)

[(11*4^n+1)/3 for n in range(31)] # G. C. Greubel, Jan 19 2023

a(n) = 4*a(n-1) - 1.
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: (4-5*x)/((1-x)*(1-4*x)). - Bruno Berselli, Nov 04 2011
E.g.f.: (1/3)*(11*exp(4*x) + exp(x)). - G. C. Greubel, Jan 19 2023

cp's OEIS Frontend

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

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