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A244749 0-additive sequence: a(n) is the smallest number larger than a(n-1) that is not the sum of any subset of earlier terms, starting with initial values {2, 5}.

%I A244749 #30 Dec 21 2024 02:00:45
%S A244749 2,5,6,9,10,28,29,85,86,256,257,769,770,2308,2309,6925,6926,20776,
%T A244749 20777,62329,62330,186988,186989,560965,560966,1682896,1682897,
%U A244749 5048689,5048690,15146068,15146069,45438205,45438206,136314616,136314617,408943849,408943850,1226831548,1226831549
%N A244749 0-additive sequence: a(n) is the smallest number larger than a(n-1) that is not the sum of any subset of earlier terms, starting with initial values {2, 5}.
%C A244749 This sequence differs from A003664.
%D A244749 R. K. Guy, "s-Additive sequences," preprint, 1994.
%H A244749 Steven R. Finch, <a href="http://dx.doi.org/10.2307/2325001">Are 0-additive sequences always regular?</a>, Amer. Math. Monthly, 99 (1992), 671-673.
%H A244749 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,3).
%F A244749 a(2n) = 4a(2n - 2) - 3a(2n - 4) and a(2n +1) = a(2n) +1, for n>2.
%F A244749 a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) for n>6. - _Colin Barker_, Jul 11 2014
%F A244749 G.f.: x*(7*x^5+14*x^4+6*x^3-5*x^2-7*x-2) / ((x+1)*(3*x^2-1)). - _Colin Barker_, Jul 11 2014
%e A244749 The numbers 11-27 are not in the sequence since some combination of the previous terms add to it. example 17=2+5+10.
%e A244749 The number 28 however is a term since no combination of the previous terms cannot be found which sum to 28.
%t A244749 f[s_List] := f[n] = Block[{k = s[[-1]] + 1, ss = Union[ Plus @@@ Subsets[s]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#] &, {2, 5}, 20] (* or *)
%t A244749 b = LinearRecurrence[{4, -3}, {9, 28}, 18]; Join[{2, 5, 6}, Riffle[b, b + 1]]
%t A244749 Join[{2, 5, 6},LinearRecurrence[{-1, 3, 3},{9, 10, 28},36]] (* _Ray Chandler_, Aug 03 2015 *)
%o A244749 (PARI) Vec(x*(7*x^5+14*x^4+6*x^3-5*x^2-7*x-2)/((x+1)*(3*x^2-1)) + O(x^100)) \\ _Colin Barker_, Jul 11 2014
%Y A244749 Cf. A003662, A003663, A005408, A026471, A026474, A033627, A051039, A051040, A244151, A244750.
%Y A244749 Cf. A060469 - A060472.
%K A244749 nonn,easy
%O A244749 1,1
%A A244749 _N. J. A. Sloane_ and _Robert G. Wilson v_, Jul 05 2014