%I #21 Sep 17 2024 14:01:30
%S 0,1,0,2,1,3,1,4,2,5,2,6,3,7,3,8,4,9,4,10,5,11,5,12,6,13,6,14,7,15,7,
%T 16,8,17,8,18,9,19,9,20,10,21,10,22,11,23,11,24,12,25,12,26
%N First differences of A129819.
%C A129819 and its repeated differences are
%C 0.0.1..1..3..4..7...8..12..14.19..21.27....
%C ..0.1..0..2..1..3...1...4...2..5...2..6....
%C ....1.-1..2.-1..2..-2...3..-2..3..-3..4....
%C ......-2..3.-3..3..-4...5..-5..5..-6..7....
%C ..........5.-6..6..-7...9.-10.10.-11.13...
%C ...........-11.12.-13..16.-19.20.-21.24.-27
%C ...............23.-25..29.-35.39.-41.45.-51
%C The left edge is A130668.
%C I discovered the array 1 1 -2 1 -3 2 in studying the singular points of planar polynomial differential systems (inspired by the reference).
%H G. C. Greubel, <a href="/A124072/b124072.txt">Table of n, a(n) for n = 0..5000</a>
%H Paul Curtz, <a href="http://www.numdam.org/item?id=ASENS_1980_4_13_3_293_0">Stabilite locale des systemes quadratiques</a>, Ann. sc. Ecole Norm. Sup. vol 13 no 3 (1980) pp 293-302.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1,0,-1).
%F a(2n) = A004526(n).
%F a(2n+1) = A000027(n+1) .
%F G.f.: x*(1+x^2+x^3)/((1-x)^2*(1+x)^2*(1+x^2)). - _R. J. Mathar_, Feb 25 2009
%F From _G. C. Greubel_, Sep 17 2024: (Start)
%F a(n) = (1/8)*(3*n + 1 - (-1)^n*(n + 3) + i^n*(1 + (-1)^n)).
%F E.g.f.: (1/4)*( cos(x) - (1-2*x)*cosh(x) + (2+x)*sinh(x) ). (End)
%t a[n_?OddQ] := (n+1)/2; a[n_?EvenQ] := Floor[n^2/16] - Floor[(n-2)^2/16]; Table[a[n], {n, 0, 51}] (* _Jean-François Alcover_, Aug 13 2012 *)
%t LinearRecurrence[{0,1,0,1,0,-1}, {0,1,0,2,1,3}, 61] (* _G. C. Greubel_, Sep 17 2024 *)
%o (Magma)
%o R<x>:=PowerSeriesRing(Integers(), 61);
%o [0] cat Coefficients(R!( x*(1+x^2+x^3)/((1-x^2)*(1-x^4)) )); // _G. C. Greubel_, Sep 17 2024
%o (SageMath)
%o def A124072_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x*(1+x^2+x^3)/((1-x^2)*(1-x^4)) ).list()
%o A124072_list(60) # _G. C. Greubel_, Sep 17 2024
%Y Cf. A000027, A004526, A129819, A130668.
%K nonn,easy,changed
%O 0,4
%A _Paul Curtz_, Jun 26 2007
%E Partially edited by _R. J. Mathar_, Jul 07 2008