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A332057 Partial sums (and absolute value of first differences) of A332056: if odd (resp. even) add (resp. subtract) the partial sum to get the next term.

%I A332057 #22 Dec 23 2024 14:53:46
%S A332057 1,3,2,3,7,4,5,11,6,7,15,8,9,19,10,11,23,12,13,27,14,15,31,16,17,35,
%T A332057 18,19,39,20,21,43,22,23,47,24,25,51,26,27,55,28,29,59,30,31,63,32,33,
%U A332057 67,34,35,71,36,37,75,38,39,79,40
%N A332057 Partial sums (and absolute value of first differences) of A332056: if odd (resp. even) add (resp. subtract) the partial sum to get the next term.
%C A332057 The terms show a 3-quasiperiodic pattern (2m-1, 4m-1, 2m), m = 1, 2, 3, ...
%C A332057 Or: group positive integers by pairs, then insert the sum of the pair between the two terms.
%H A332057 Antti Karttunen, <a href="/A332057/b332057.txt">Table of n, a(n) for n = 1..19683</a>
%H A332057 Eric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2020-February/020509.html">Re: Add or subtract my cumulative sum of terms</a>, SeqFan list, Feb 24 2020.
%H A332057 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A332057 a(3k-2) = 2k - 1, a(3k-1) = 4k - 1, a(3k) = 2k, for all k >= 1.
%F A332057 From _Colin Barker_, Feb 25 2020: (Start)
%F A332057 G.f.: x*(1 + x)*(1 + 2*x + x^3) / ((1 - x)^2*(1 + x + x^2)^2).
%F A332057 a(n) = 2*a(n-3) - a(n-6) for n>6.
%F A332057 (End)
%o A332057 (PARI) apply( {A332057(n)=n<<max(n%3,1)\/3}, [1..99])
%o A332057 (PARI) Vec(x*(1 + x)*(1 + 2*x + x^3) / ((1 - x)^2*(1 + x + x^2)^2) + O(x^60)) \\ _Colin Barker_, Feb 26 2020
%Y A332057 Cf. A332056.
%K A332057 nonn,easy
%O A332057 1,2
%A A332057 _Eric Angelini_ and _M. F. Hasler_, Feb 24 2020