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A336109 First column of dispersion array A120861.

%I A336109 #19 Jul 29 2020 02:57:30
%S A336109 1,2,3,4,5,6,8,9,10,11,13,14,15,16,17,18,20,21,22,23,25,26,27,28,29,
%T A336109 30,32,33,34,35,37,38,39,40,42,43,44,45,46,47,49,50,51,52,54,55,56,57,
%U A336109 58,59,61,62,63,64,66,67,68,69,71,72,73,74,75,76,78,79,80,81,83,84,85,86,87,88,90,91,92,93,95,96
%N A336109 First column of dispersion array A120861.
%C A336109 For n = 1 to 34, we have a(n) = A039216(n), but a(35) = 42 <>  = 41 = A039216(35).
%C A336109 Is this the same as A188040? - _R. J. Mathar_, Jul 22 2020
%C A336109 Indeed, it seems that A336109(1) = A188040(1) and A336109(n) = A188040(n-1) for n >= 3, but A336109(2) = 2 does not appear in A188040. Because of the way A120861 is constructed (cf. _Michel Marcus_'s PARI program below), a(2) = 2 cannot be omitted. - _Petros Hadjicostas_, Jul 22 2020
%H A336109 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling2/kimberling45.html">The equation (j+k+1)^2 - 4*k = Q*n^2 and related dispersions</a>, Journal of Integer Sequences, 10 (2007), Article #07.2.7; see Section 5.
%o A336109 (PARI) /* This is _Michel Marcus_'s program from A120861. */
%o A336109 f(n) = 3*n + 2*sqrtint(2*n^2) + 2;
%o A336109 unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };
%o A336109 D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 6*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; };
%o A336109 firstcol(nb) = my(m=D(nb)); vector(nb, k, m[k, 1]);
%Y A336109 Cf. A039216, A120861, A188040.
%K A336109 nonn
%O A336109 1,2
%A A336109 _Petros Hadjicostas_, Jul 08 2020