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A237714 Expansion of (1 + x)/(1 - x^2 - 2*x^5).

%I A237714 #32 Nov 24 2024 12:37:12
%S A237714 1,1,1,1,1,3,3,5,5,7,11,13,21,23,35,45,61,87,107,157,197,279,371,493,
%T A237714 685,887,1243,1629,2229,2999,4003,5485,7261,9943,13259,17949,24229,
%U A237714 32471,44115,58989,80013,107447,144955,195677,262933,355703,477827,645613,869181,1171479,1580587
%N A237714 Expansion of (1 + x)/(1 - x^2 - 2*x^5).
%H A237714 Vincenzo Librandi, <a href="/A237714/b237714.txt">Table of n, a(n) for n = 0..1000</a>
%H A237714 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,0,2).
%F A237714 a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(n) = 2*a(n-5) + a(n-2) for n>=5.
%F A237714 a(2n) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-3)/5} C(n-2-3j,2j+1)*2^(2j+1).
%F A237714 a(2n+1) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-2)/5} C(n-1-3j,2j+1)*2^(2j+1).
%t A237714 For[j = 0, j < 5, j++, a[j] = 1]
%t A237714 For[j = 5, j < 51, j++, a[j] = 2 a[j - 5] + a[j - 2]]
%t A237714 Table[a[j], {j, 0, 50}]
%t A237714 CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^5), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 26 2014 *)
%t A237714 LinearRecurrence[{0,1,0,0,2},{1,1,1,1,1},70] (* _Harvey P. Dale_, Nov 24 2024 *)
%o A237714 (PARI) Vec( (1 + x)/(1 - x^2 - 2*x^5) + O(x^66) ) \\ _Joerg Arndt_, Feb 24 2014
%Y A237714 Cf. A000129, A122522, A159284.
%K A237714 nonn,easy
%O A237714 0,6
%A A237714 _Sergio Falcon_, Feb 12 2014