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a(n) = (11*2^n - 2*(-1)^n)/3 for n >= 0.
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%I #47 May 07 2021 09:39:25

%S 3,8,14,30,58,118,234,470,938,1878,3754,7510,15018,30038,60074,120150,

%T 240298,480598,961194,1922390,3844778,7689558,15379114,30758230,

%U 61516458,123032918,246065834,492131670,984263338,1968526678,3937053354,7874106710,15748213418,31496426838

%N a(n) = (11*2^n - 2*(-1)^n)/3 for n >= 0.

%C Based on A112387.

%C Prepended with 0, 1, its difference table is

%C 0, 1, 1, 2, 1, 4, 3, 8, ... = mix A001045(n), 2^n.

%C 1, 0, 1, -1, 3, -1, 5, -3, ... = mix A001045(n+1), -A001045(n).

%C -1, 1, -2, 4, -4, 6, -8, 14, ... = mix -2^n, A084214(n+1).

%C 2, -3, 6, -8, 10, -14, 22, -30, ... = mix 2*A001045(n+2), -a(n).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).

%F a(n) = 2^(n+2) - A078008(n), n>=0.

%F a(n) = (A062510(n) = 3*A001045(n)) + A001045(n+3), n>=0.

%F a(0)=3, a(2*n+1) = 2*a(2*n) + 2, a(2*n+2) = 2*a(2*n+1) - 2, n>=0.

%F a(n) = 4*A052997(n-1) + 2, n>=2. - _Hugo Pfoertner_, Apr 25 2021

%F a(n+1) = 11*2^n - a(n) for n>=0.

%F a(n+3) = 33*2^n - a(n) for n>=0.

%F a(n+5) = 121*2^n - a(n) for n>=0.

%F etc.

%F a(n+2) = a(n) + 11*2^n for n>=0.

%F a(n+4) = a(n) + 55*2^n for n>=0.

%F a(n+6) = a(n) + 231*2^n for n>=0.

%F etc.

%F G.f.: (3 + 5*x)/(1 - x - 2*x^2). - _Stefano Spezia_, Apr 26 2021

%F E.g.f: (11*exp(2*x) - 2*exp(-x))/3. - _Jianing Song_, Apr 26 2021

%t LinearRecurrence[{1, 2}, {3, 8}, 35] (* _Amiram Eldar_, Apr 25 2021 *)

%o (PARI) a(n) = (11*2^n - 2*(-1)^n)/3 \\ _Felix Fröhlich_, Apr 25 2021

%Y Cf. A000079, A001045, A062510, A078008, A112387.

%Y Cf. also A052997.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Apr 25 2021

%E More terms from _Michel Marcus_, Apr 25 2021

%E New name from _Jianing Song_, Apr 25 2021