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A171441 Expansion of g.f. (1+x)^6/(1-x).

%I A171441 #27 Mar 19 2023 09:44:49
%S A171441 1,7,22,42,57,63,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,
%T A171441 64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,
%U A171441 64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64
%N A171441 Expansion of g.f. (1+x)^6/(1-x).
%C A171441 a(n)=2^6=64 for n>=6. We observe that this sequence is the transform of A171440 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).
%C A171441 Also continued fraction expansion of 1+(1233212607598+5*sqrt(41))/8688482797079. - _Bruno Berselli_, Sep 23 2011
%H A171441 Richard Choulet, <a href="https://mp.sbpm.be/MP157.PDF">Une nouvelle formule de combinatoire?</a>, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
%H A171441 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A171441 With m=7, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).
%e A171441 a(4) = C(7,4-0) + C(7,4-2) + C(7,4-4) = 35+21+1 = 57.
%p A171441 m:=7:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);
%Y A171441 Cf. A040000, A113311, A115291, A171418, A171440, A171442, A171443.
%K A171441 nonn,easy
%O A171441 0,2
%A A171441 _Richard Choulet_, Dec 09 2009
%E A171441 Definition rewritten by _Bruno Berselli_, Sep 23 2011