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a(n) = 2^(2n+3) - 2^n*(n+3).
6

%I #37 Apr 10 2023 12:50:24

%S 1,5,24,108,464,1936,7936,32192,129792,521472,2091008,8375296,

%T 33525760,134156288,536739840,2147205120,8589344768,34358493184,

%U 137436332032,549750308864,2199011721216,8796068904960,35184321757184,140737383497728,562949735317504

%N a(n) = 2^(2n+3) - 2^n*(n+3).

%C a(n-2) = 2^(2n-1) - (n+1)*2^(n-2) is number of compositions of 2n where each part <= n: e.g., for n=2, 4 can be written 5 ways with no part greater than 2 as 1+1+1+1 = 1+1+2 = 1+2+1 = 2+1+1 = 2+2. - _Henry Bottomley_, Feb 03 2003

%H Delbert L. Johnson, <a href="/A008464/b008464.txt">Table of n, a(n) for n = -1..1659</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-20,16).

%F G.f.: 8/(1-4*x) - 3/(1-2*x) - 2*x/(1-2*x)^2 + 1/x. - _R. J. Mathar_, Nov 21 2007

%F a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3). - _Wesley Ivan Hurt_, Sep 04 2022

%p A008464:=n->2^(2*n+3)-2^n*(n+3); seq(A008464(n), n=-1..22);

%t Table[Coefficient[Series[(1 - x)/(1 - 2 x + x^(n + 1)), {x, 0, 2 n}], x^(2 n)], {n, 1, 22}] (* _Geoffrey Critzer_, May 29 2013 *)

%t LinearRecurrence[{8,-20,16},{1,5,24},30] (* _Harvey P. Dale_, May 15 2022 *)

%o (PARI) a(n)=2^(2*n+3)-2^n*(n+3) \\ _Charles R Greathouse IV_, Oct 07 2015

%K nonn,easy

%O -1,2

%A _N. J. A. Sloane_