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

Offset: 0

Author

Brad Clardy, Feb 07 2012

Keywords

• nonn
• easy

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).

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.

Programs

Magma

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

Mathematica

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

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

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