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%I #26 Sep 08 2022 08:45:13
%S 1,2,4,11,29,76,199,521,1364,3571,9349,24476,64079,167761,439204,
%T 1149851,3010349,7881196,20633239,54018521,141422324,370248451,
%U 969323029,2537720636,6643838879,17393796001,45537549124,119218851371,312119004989,817138163596
%N a(1) = 1, a(2) = 2, a(n+1) = n*a(1) + (n-1)*a(2) + ... + (n-r)*a(r+1) + ... + a(n).
%C a(1) = a(2) = 1 gives A088305, i.e., Fibonacci numbers with even indices. This can be called 'fake Fibonacci sequence'. 4 = 3+1, 11 = 8+3, 29 = 21+8, 76 = 55+21, etc. a(n) = F(2n-2) + F(2n-4).
%C Except for the initial terms, this is the same as the bisection of the Lucas sequence (A002878). - _Franklin T. Adams-Watters_, Jul 17 2006
%H Colin Barker, <a href="/A093960/b093960.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F a(n) = F(2*n-2) + F(2*n-4), where F(k) is k-th Fibonacci number, n > 2.
%F a(n) = 3*a(n-1) - a(n-2) for n>4. - _Colin Barker_, Mar 26 2015
%F G.f.: x*(1-x)^2*(1+x) / (1-3*x+x^2). - _Colin Barker_, Mar 26 2015
%F a(n) = 2^(2-n)*[n<3] + LucasL(2*n-3). - _G. C. Greubel_, Dec 30 2021
%p a[1]:=1: a[2]:=2: for n from 2 to 33 do a[n+1]:=sum((n-r)*a[r+1],r=0..n-1) od: seq(a[n],n=1..33); # _Emeric Deutsch_, Aug 01 2005
%p A093960List := proc(m) local A, P, n; A := [1,2]; P := [1];
%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(A), P[-1]]);
%p A := [op(A), P[-1]] od; A end: A093960List(30); # _Peter Luschny_, Mar 24 2022
%t Print[1]; Print[2]; Do[Print[Fibonacci[2*n - 2] + Fibonacci[2*n - 4]], {n, 3, 20}] (* _Ryan Propper_, Jun 19 2005 *)
%t LinearRecurrence[{3,-1},{1,2,4,11},30] (* _Harvey P. Dale_, Nov 17 2018 *)
%o (PARI) Vec(x*(x-1)^2*(x+1)/(x^2-3*x+1) + O(x^100)) \\ _Colin Barker_, Mar 26 2015
%o (Magma) [1,2] cat [Lucas(2*n-3): n in [3..30]]; // _G. C. Greubel_, Dec 30 2021
%o (Sage) [2^(2-n)*bool(n<3) + lucas_number2(2*n-3, 1, -1) for n in (1..30)] # _G. C. Greubel_, Dec 30 2021
%Y Cf. A000045, A002878, A088305.
%K nonn,easy
%O 1,2
%A _Amarnath Murthy_, May 22 2004
%E More terms from _Ryan Propper_, Jun 19 2005
%E More terms from _Emeric Deutsch_, Aug 01 2005
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