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A232896 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4), where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 8.

%I A232896 #47 Sep 08 2022 08:46:06
%S A232896 1,3,5,8,12,18,27,41,63,98,154,244,389,623,1001,1612,2600,4198,6783,
%T A232896 10965,17731,28678,46390,75048,121417,196443,317837,514256,832068,
%U A232896 1346298,2178339,3524609,5702919,9227498,14930386,24157852,39088205,63246023,102334193
%N A232896 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4), where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 8.
%C A232896 Conjecture: a(n) is the position of 2*n-1, for n >= 1, in the sequence S = A232895 of positive integers generated by these rules:  1 and 2 are in S; if x is in S then x + 2 and 2*x are in S, where duplicates are deleted as they occur.
%H A232896 Clark Kimberling, <a href="/A232896/b232896.txt">Table of n, a(n) for n = 1..1000</a>
%H A232896 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A232896 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4), where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 8.
%F A232896 a(n) = n-1 + A000045(n+1). - _Tom Edgar_, Mar 09 2015
%F A232896 G.f.: x*(1-2*x^2)/((1-x)^2*(1-x-x^2)). - _Vincenzo Librandi_, Mar 18 2015
%F A232896 a(n) = -1 + (2^(-1-n)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / sqrt(5) + n. - _Colin Barker_, Mar 11 2017
%F A232896 a(n) = Sum_{k=1..n} A001611(k-1). - _Ehren Metcalfe_, Apr 15 2019
%e A232896 a(5) = 3*a(4) - 2*a(3) - a(4) + a(5) = 3*8 - 2*5 - 3 + 1 = 12.
%t A232896 a[1] = 1; a[2] = 3; a[3] = 5; a[4] = 8; a[n_] := a[n] = 3*a[n - 1] - 2*a[n - 2] - a[n - 3] + a[n - 4]; t = Table[a[n], {n, 1, 100}]
%t A232896 CoefficientList[Series[(1 - 2 x^2) / ((1 - x)^2 (1 - x - x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 18 2015 *)
%t A232896 LinearRecurrence[{3, -2, -1, 1}, {1, 3, 5, 8}, 39] (* _Robert G. Wilson v_, Jul 23 2018 *)
%o A232896 (Magma) I:=[1,3,5,8]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Mar 18 2015
%o A232896 (PARI) Vec(x*(1-2*x^2)/((1-x)^2*(1-x-x^2)) + O(x^50)) \\ _Michel Marcus_, Mar 18 2015
%Y A232896 Cf. A232895.
%K A232896 nonn,easy
%O A232896 1,2
%A A232896 _Clark Kimberling_, Dec 02 2013