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A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4]. 2
1, 7, 10, 25, 46, 97, 190, 385, 766, 1537, 3070, 6145, 12286, 24577, 49150, 98305, 196606, 393217, 786430, 1572865, 3145726, 6291457, 12582910, 25165825, 50331646, 100663297, 201326590, 402653185, 805306366, 1610612737, 3221225470, 6442450945, 12884901886 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) mod 9 = period 2: repeat [1, 7].
The last digit from 7 is of period 4: repeat [7, 0, 5, 6].
The bisection A096045 = 1, 10, 46, ... is based on Bernoulli numbers.
a(n) is a companion to A051049(n).
With an initial 0, A051049(n) is an autosequence of the first kind.
With an initial 2, this sequence is an autosequence of the second kind.
See the reference.
Difference table:
1, 7, 10, 25, 46, 97, ... = this sequence.
6, 3, 15, 21, 51, 93, ... = 3*A014551(n)
-3, 12, 6, 30, 42, 102, ... = -3 followed by 6*A014551(n).
The main diagonal of the difference table gives A003945: 1, 3, 6, 12, 24, ...
LINKS
Wikipedia, Autosuite de nombres, (in French).
FORMULA
a(2n) = 3*4^n - 2, a(2n+1) = 6*4^n + 1.
a(n+2) = a(n) + 9*2^n, a(0) = 1, a(1) = 7.
a(n) = 2*A051049(n+1) - A051049(n).
From Colin Barker, Dec 28 2016: (Start)
a(n) = 3*2^n - 2 for n even.
a(n) = 3*2^n + 1 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2.
G.f.: (1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)
EXAMPLE
a(0) = 1, a(1) = 2*1 + 5 = 7, a(2) = 2*7 - 4 = 10, a(3) = 2*10 + 5 = 25.
MAPLE
seq(3*2^n-(-1)^n*(1+irem(n+1, 2)), n=0..32); # Peter Luschny, Dec 29 2016
MATHEMATICA
LinearRecurrence[{2, 1, -2}, {1, 7, 10}, 50] (* Paolo Xausa, Nov 13 2023 *)
PROG
(PARI) Vec((1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 28 2016
CROSSREFS
Sequence in context: A134329 A175492 A241052 * A229310 A064948 A064950
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Dec 28 2016
STATUS
approved

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Last modified March 28 07:33 EDT 2024. Contains 371235 sequences. (Running on oeis4.)