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%I #36 Mar 11 2024 01:46:30
%S 1,7,10,25,46,97,190,385,766,1537,3070,6145,12286,24577,49150,98305,
%T 196606,393217,786430,1572865,3145726,6291457,12582910,25165825,
%U 50331646,100663297,201326590,402653185,805306366,1610612737,3221225470,6442450945,12884901886
%N a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].
%C a(n) mod 9 = period 2: repeat [1, 7].
%C The last digit from 7 is of period 4: repeat [7, 0, 5, 6].
%C The bisection A096045 = 1, 10, 46, ... is based on Bernoulli numbers.
%C a(n) is a companion to A051049(n).
%C With an initial 0, A051049(n) is an autosequence of the first kind.
%C With an initial 2, this sequence is an autosequence of the second kind.
%C See the reference.
%C Difference table:
%C 1, 7, 10, 25, 46, 97, ... = this sequence.
%C 6, 3, 15, 21, 51, 93, ... = 3*A014551(n)
%C -3, 12, 6, 30, 42, 102, ... = -3 followed by 6*A014551(n).
%C The main diagonal of the difference table gives A003945: 1, 3, 6, 12, 24, ...
%H Colin Barker, <a href="/A280173/b280173.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="http://fr.wikipedia.org/wiki/Autosuite_de_nombres">Autosuite de nombres</a>, (in French).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F a(2n) = 3*4^n - 2, a(2n+1) = 6*4^n + 1.
%F a(n+2) = a(n) + 9*2^n, a(0) = 1, a(1) = 7.
%F a(n) = 2*A051049(n+1) - A051049(n).
%F From _Colin Barker_, Dec 28 2016: (Start)
%F a(n) = 3*2^n - 2 for n even.
%F a(n) = 3*2^n + 1 for n odd.
%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2.
%F G.f.: (1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
%F (End)
%e a(0) = 1, a(1) = 2*1 + 5 = 7, a(2) = 2*7 - 4 = 10, a(3) = 2*10 + 5 = 25.
%p seq(3*2^n-(-1)^n*(1+irem(n+1,2)),n=0..32); # _Peter Luschny_, Dec 29 2016
%t LinearRecurrence[{2,1,-2},{1,7,10},50] (* _Paolo Xausa_, Nov 13 2023 *)
%o (PARI) Vec((1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, Dec 28 2016
%Y Cf. A003945, A005010, A010688, A010710, A014551, A051049, A096045, A199116.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Dec 28 2016
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