A199210 -id:A199210 - 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

A206373 a(n) = (14*4^n + 1)/3.

Original entry on oeis.org

5, 19, 75, 299, 1195, 4779, 19115, 76459, 305835, 1223339, 4893355, 19573419, 78293675, 313174699, 1252698795, 5010795179, 20043180715, 80172722859, 320690891435, 1282763565739, 5131054262955, 20524217051819, 82096868207275, 328387472829099, 1313549891316395

Offset: 0

Brad Clardy, Feb 07 2012

A generalized Engel expansion of 2/7 to the base b := 4/3 as defined in A181565 with associated series expansion 2/7 = b/5 + b^2/(5*19) + b^3/(5*19*75) + b^4/(5*19*75*299) + .... - Peter Bala, Oct 30 2013

G. C. Greubel, 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), A136412 (m=5), A199210 (m=11), A199210 (m=11), this sequence (m=14).

Cf. A181565.

Magma
```
[(14*4^n+1)/3 : n in [0..30]];
```

(14*4^Range[0,30]+1)/3 (* or *) LinearRecurrence[{5,-4},{5,19},30] (* Harvey P. Dale, Jan 13 2023 *)

a(n)=(14*4^n + 1)/3 \\ Charles R Greathouse IV, Jun 01 2015

[(7*2^(2*n+1)+1)/3 for n in range(31)] # G. C. Greubel, Jan 19 2023

a(n) = (14*4^n + 1)/3.
From Peter Bala, Oct 30 2013: (Start)
a(n+1) = 4*a(n) - 1 with a(0) = 5.
a(n) = 5*a(n-1) - 4*a(n-2) with a(0) = 5 and a(1) = 19.
O.g.f. (5 - 6*x)/((1 - x)*(1 - 4*x)). (End)
E.g.f.: (1/3)*(14*exp(4*x) + exp(x)). - G. C. Greubel, Jan 19 2023

A199211 a(n) = 11*4^n + 1.

Original entry on oeis.org

12, 45, 177, 705, 2817, 11265, 45057, 180225, 720897, 2883585, 11534337, 46137345, 184549377, 738197505, 2952790017, 11811160065, 47244640257, 188978561025, 755914244097, 3023656976385, 12094627905537, 48378511622145, 193514046488577, 774056185954305, 3096224743817217

Offset: 0

Vincenzo Librandi, Nov 04 2011

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

Cf. A199210.

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

11*4^Range[0,30]+1 (* or *) LinearRecurrence[{5,-4},{12,45},30] (* Harvey P. Dale, Oct 10 2012 *)

a(n) = 4*a(n-1) - 3.
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: 3*(4-5*x)/((1-x)*(1-4*x)). - Bruno Berselli, Nov 04 2011
From Elmo R. Oliveira, Mar 06 2025: (Start)
E.g.f.: exp(x)*(1 + 11*exp(3*x)).
a(n) = 3*A199210(n). (End)

A383953 a(0) = 4, a(n) = 2*a(n-1) + (-1)^n.

Original entry on oeis.org

4, 7, 15, 29, 59, 117, 235, 469, 939, 1877, 3755, 7509, 15019, 30037, 60075, 120149, 240299, 480597, 961195, 1922389, 3844779, 7689557, 15379115, 30758229, 61516459, 123032917, 246065835, 492131669, 984263339, 1968526677, 3937053355, 7874106709, 15748213419, 31496426837

Offset: 0

Paul Curtz, Aug 19 2025

Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A005015, A052997, A340627.

Bisections give A199210 and A072261.

a[n_] := (11*2^n + (-1)^n)/3; Array[a, 34, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n) = (11*2^n + (-1)^n)/3.
a(n) = A340627(n+1)/2.
a(n) = 2*A052997(n) + 1 for n >= 1.
a(n) = a(n-4) + 55*2^(n-4) for n >= 4.
G.f.: (3*x + 4)/((x + 1)*(1 - 2*x)).
E.g.f: (11*exp(2*x) + exp(-x))/3.

cp's OEIS Frontend

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

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A206373 a(n) = (14*4^n + 1)/3.

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A199211 a(n) = 11*4^n + 1.

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A383953 a(0) = 4, a(n) = 2*a(n-1) + (-1)^n.

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