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%I #35 May 29 2024 19:57:38
%S 1,4,32,256,2048,16384,131072,1048576,8388608,67108864,536870912,
%T 4294967296,34359738368,274877906944,2199023255552,17592186044416,
%U 140737488355328,1125899906842624,9007199254740992,72057594037927936,576460752303423488,4611686018427387904
%N Expansion of g.f. (1-4*x)/(1-8*x).
%C 4th binomial transform of (1,0,16,0,256,...).
%C Number of compositions of even natural numbers into n parts <= 7. - _Adi Dani_, May 28 2011
%H Vincenzo Librandi, <a href="/A092811/b092811.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (8).
%F a(n) = 8^n/2 + 0^n/2.
%F a(n) = A001045(3n+1) - A001045(3n-1) + 0^n/2.
%F a(n) = A013731(n-1), n > 0. - _R. J. Mathar_, Sep 08 2008
%F a(n) = 4 * 8^(n-1), a(0)=1. - _Vincenzo Librandi_, Jun 16 2011
%F a(n) = Sum_{k=0..n} A134309(n,k)*4^k = Sum_{k=0..n} A055372(n,k)*3^k. - _Philippe Deléham_, Feb 04 2012
%F E.g.f.: (1 + exp(8*x))/2. - _Stefano Spezia_, May 29 2024
%e From _Adi Dani_, May 28 2011: (Start)
%e a(2)=32: there are 32 compositions of even natural numbers into 2 parts <= 7:
%e (0,0);
%e (0,2),(2,0),(1,1);
%e (0,4),(4,0),(1,3),(3,1),(2,2);
%e (0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
%e (1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
%e (3,7),(7,3),(4,6),(6,4),(5,5);
%e (5,7),(7,5),(6,6);
%e (7,7). (End)
%t Table[EulerPhi[8^n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 10 2009 *)
%o (Magma) [8^n/2+0^n/2: n in [0..20]]; // _Vincenzo Librandi_, Jun 16 2011
%o (PARI) a(n)=max(1,8^n/2) \\ _Charles R Greathouse IV_, Apr 09 2012
%Y Cf. A001045, A013731 (same sequence omitting initial 1), A055372, A134309.
%K easy,nonn,changed
%O 0,2
%A _Paul Barry_, Mar 10 2004
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