cp's OEIS Frontend

A115291 Expansion of (1+x)^3/(1-x).

%I A115291 #37 Aug 16 2025 06:19:32
%S A115291 1,4,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%T A115291 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%U A115291 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8
%N A115291 Expansion of (1+x)^3/(1-x).
%C A115291 Partial sums are A086570. Partial sums of squares are A115295. Correlation triangle is A115292.
%C A115291 Let m=4. We observe that a(n) = Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A113311 and A040000: it is the same formula with respectively m=3 and m=2. We can generalize this result with the sequence whose G.f is given by (1+z)^(m-1)/(1-z). - _Richard Choulet_, Dec 08 2009
%C A115291 Also continued fraction expansion of (132-sqrt(17))/103. - _Bruno Berselli_, Sep 23 2011
%C A115291 Also decimal expansion of 1331/9000. - _Vincenzo Librandi_, Sep 23 2011
%H A115291 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A115291 a(n) = 8 - C(2, n) - 2*C(1, n) - 4*C(0, n).
%F A115291 a(n) = Sum_{k=0..n} C(3, k).
%F A115291 a(n) = A004070(n, 3).
%F A115291 From _Elmo R. Oliveira_, Aug 09 2024: (Start)
%F A115291 E.g.f.: 8*exp(x) - 7 - 4*x - x^2/2.
%F A115291 a(n) = 8, n > 2. (End)
%t A115291 CoefficientList[Series[(1+x)^3/(1-x),{x,0,100}],x] (* or *) PadRight[ {1,4,7},120,{8}] (* _Harvey P. Dale_, May 23 2016 *)
%Y A115291 Cf. A040000, A113311, A171418, A171440, A171441, A171442, A171443.
%K A115291 nonn,easy
%O A115291 0,2
%A A115291 _Paul Barry_, Jan 19 2006