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A206373
a(n) = (14*4^n + 1)/3.
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
0,1
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
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.
Sequence in context: A212403 A098913 A126392 * A149767 A149768 A149769
KEYWORD
nonn,easy
AUTHOR
Brad Clardy, Feb 07 2012
STATUS
approved