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A005255 Atkinson-Negro-Santoro sequence: a(n+1) = 2*a(n) - a(n-floor(n/2+1)).

%I A005255 M1076 #38 Sep 03 2022 23:02:56
%S A005255 0,1,2,4,7,13,24,46,88,172,337,667,1321,2629,5234,10444,20842,41638,
%T A005255 83188,166288,332404,664636,1328935,2657533,5314399,10628131,21254941,
%U A005255 42508561,85014493,170026357,340047480,680089726,1360169008,2720327572
%N A005255 Atkinson-Negro-Santoro sequence: a(n+1) = 2*a(n) - a(n-floor(n/2+1)).
%C A005255 For each n, the n-term sequence (b(k) = a(n) - a(n-k), 1 <= k <= n), has the property that all 2^n sums of subsets of the terms are distinct.
%C A005255 a(n) = A062178(n+1) - 1; see also A002083. - _Reinhard Zumkeller_, Nov 18 2012
%D A005255 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.28.
%D A005255 T. V. Narayana, Recent progress and unsolved problems in dominance theory, pp. 68-78 of Combinatorial mathematics (Canberra 1977), Lect. Notes Math. Vol. 686, 1978.
%D A005255 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
%D A005255 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005255 T. D. Noe, <a href="/A005255/b005255.txt">Table of n, a(n) for n=0..300</a>
%H A005255 M. D. Atkinson et al., <a href="http://dx.doi.org/10.1016/0012-365X(90)90112-U">Sums of lexicographically ordered sets</a>, Discrete Math., 80 (1990), 115-122.
%H A005255 W. F. Lunnon, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0917837-5">Integer sets with distinct subset-sums</a>, Math. Comp. 50 (1988), 297-320.
%H A005255 B. E. Wynne & N. J. A. Sloane, <a href="/A002083/a002083_1.pdf">Correspondence, 1976-84</a>
%H A005255 B. E. Wynne & T. V. Narayana, <a href="/A002083/a002083_2.pdf">Tournament configuration, weighted voting, and partitioned catalans</a>, Preprint.
%H A005255 Bayard Edmund Wynne, and T. V. Narayana, <a href="http://www.numdam.org/item?id=BURO_1981__36__75_0">Tournament configuration and weighted voting</a>, Cahiers du bureau universitaire de recherche opérationnelle, 36 (1981): 75-78.
%e A005255 For n = 4, the sequence b is 7-4,7-2,7-1,7-0 = 3,5,6,7, which has subset sums (grouped by number of terms) 0, 3,5,6,7, 8,9,10,11,12,13, 14,15,16,18, 21.
%t A005255 a[ 0 ] := 0; a[ 1 ] := 1; a[ n_ ] := 2*a[ n - 1 ] - a[(n - 1) - Floor[ (n - 1)/2 + 1 ] ]; For[ n = 1, n <= 100, n++, Print[ a[ n ] ] ];
%o A005255 (Haskell)
%o A005255 a005255 n = a005255_list !! (n-1)
%o A005255 a005255_list = scanl (+) 0 $ tail a002083_list
%o A005255 -- _Reinhard Zumkeller_, Nov 18 2012
%Y A005255 Cf. A002083, A005318, A062178.
%K A005255 nonn,easy,nice
%O A005255 0,3
%A A005255 _N. J. A. Sloane_, _Simon Plouffe_
%E A005255 More terms from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 26 2000
%E A005255 Edited by _Franklin T. Adams-Watters_, Apr 11 2009