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A291660 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=2, a(1)=3, a(2)=5, a(3)=7, a sequence related to Lucas numbers.

%I A291660 #29 Dec 01 2019 15:25:11
%S A291660 2,3,5,7,11,18,30,49,79,127,205,332,538,871,1409,2279,3687,5966,9654,
%T A291660 15621,25275,40895,66169,107064,173234,280299,453533,733831,1187363,
%U A291660 1921194,3108558,5029753,8138311,13168063,21306373,34474436,55780810,90255247,146036057,236291303
%N A291660 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=2, a(1)=3, a(2)=5, a(3)=7, a sequence related to Lucas numbers.
%C A291660 The array of successive differences begins:
%C A291660 2,  3,  5,  7, 11, 18, 30, 49, 79, 127, ... = a(n)
%C A291660 1,  2,  2,  4,  7, 12, 19, 30, 48,  78, ... = b(n)
%C A291660 1,  0,  2,  3,  5,  7, 11, 18, 30,  49, ... = a(n-2)
%C A291660 -1, 2,  1,  2,  2,  4,  7, 12, 19,  30, ... = b(n-2)
%C A291660 3, -1,  1,  0,  2,  3,  5,  7, 11,  18, ... = a(n-4)
%C A291660 ...
%C A291660 Main diagonal is 2,2,2,... = A007395.
%C A291660 Adding a(n) and first column with alternating signs, one gets two autosequences: 2*Lucas numbers A000032 (2, 1, 3, 4, 7, 11, 18, ...) or 2*A286350 (0, 2, 2, 3, 4, 7, 12, ...) according to signs.
%H A291660 OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%H A291660 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 1).
%F A291660 G.f.: (2 - x + x^2)/(1 - 2*x + x^2 - x^4).
%F A291660 a(3n) = A097924(n).
%F A291660 a(3n) + a(3n+1) = a(3n+2).
%F A291660 a(n) = (1/15)*2^(-n-1)*((30-9*sqrt(5))*(1-sqrt(5))^n + (1+sqrt(5))^n*(30 + 9*sqrt(5)) + 5*2^(n+1)*sqrt(3)*sin(n*Pi/3)).
%t A291660 LinearRecurrence[{2, -1, 0, 1}, {2, 3, 5, 7}, 40]
%o A291660 (GAP)
%o A291660 L:=[2,3,5,7];; for i in [5..10^3] do L[i]:=2*L[i-1]-L[i-2]+L[i-4]; od; L;  #  _Muniru A Asiru_, Sep 02 2017
%Y A291660 Cf. A000032, A097924, A286350.
%K A291660 nonn
%O A291660 0,1
%A A291660 _Jean-François Alcover_ and _Paul Curtz_, Aug 31 2017