A199210 -id:A199210

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A136412

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-4).

FORMULA

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

MATHEMATICA

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

PROG

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

(Haskell)

a136412 = (`div` 3) . (+ 1) . (* 5) . (4 ^)

-- Reinhard Zumkeller, Jun 17 2012

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

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

CROSSREFS

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.

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Mar 31 2008

EXTENSIONS

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

STATUS

approved

A206373

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-4).

FORMULA

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

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

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.