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A308663 Partial sums of A097805.

%I A308663 #50 Sep 15 2019 21:06:29
%S A308663 1,1,2,2,3,4,4,5,7,8,8,9,12,15,16,16,17,21,27,31,32,32,33,38,48,58,63,
%T A308663 64,64,65,71,86,106,121,127,128,128,129,136,157,192,227,248,255,256,
%U A308663 256,257,265,293,349,419,475,503,511,512
%N A308663 Partial sums of A097805.
%C A308663 Curtz (1965), page 15, from right to left, gives (F1):
%C A308663   1/2;
%C A308663   1/4,  3/4;
%C A308663   1/8,  4/8,   7/8;
%C A308663   1/16, 5/16, 11/16, 15/16;
%C A308663   ... .
%C A308663 Numerators + Denominators = (C) =
%C A308663    3;
%C A308663    5,  7;
%C A308663    9, 12, 15;
%C A308663   17, 21, 27, 31;
%C A308663   ... .
%C A308663 This is the current sequence without powers of 2.
%C A308663 The triangle (P) for a(n) is
%C A308663   1;
%C A308663   1, 2;
%C A308663   2, 3,  4;
%C A308663   4, 5,  7,  8;
%C A308663   8, 9, 12, 15, 16;
%C A308663   ... .
%C A308663 (C) is the core of (P).
%C A308663 Extension of (F1). (F2) =
%C A308663   0/1;
%C A308663   0/1, 1/1;
%C A308663   0/2, 1/2, 2/2;
%C A308663   0/4, 1/4, 3/4, 4/4;
%C A308663   0/8, 1/8, 4/8, 7/8, 8/8;
%C A308663   ... .
%C A308663 (Mentioned, without 0's, op. cit., page 16.)
%C A308663 a(n) = Numerators + Denominators.
%C A308663 Row sums of triangle (P): A084858(n).
%C A308663 From right to left, with alternating signs: 1, 1, 3, 2, 12, 8, 48, 32, ..., see A098646.
%C A308663 For triangle (C), row sums give A167667(n+1).
%C A308663 From right to left, with alternating signs: A098646(n).
%C A308663 Rank of A016116(n): 0 together with A117142.
%H A308663 Paul Curtz, <a href="http://paul.curtz.free.fr/wp-content/uploads/These_Curtz_1965.pdf">Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation</a>, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.
%F A308663 T(n,k) = ceiling(2^(n-1)) + Sum_{j=0..k-1} binomial(n-1,j). - _Alois P. Heinz_, Jun 15 2019
%F A308663 a(n+1) = a(n) + A097805(n+1) for n >= 0.
%Y A308663 Cf. A097805.
%K A308663 nonn,tabl
%O A308663 0,3
%A A308663 _Paul Curtz_, Jun 15 2019
%E A308663 Edited by _N. J. A. Sloane_, Sep 15 2019