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%I #32 Jan 10 2017 07:25:56
%S 1,2,3,10,11,12,20,30,100,110,200,300,1000,1100,2000,2100,3000,10000,
%T 20000,30000,100000,110000,120000,200000,300000,1000000,1100000,
%U 2000000,3000000,10000000,11000000,20000000,21000000,30000000,100000000,200000000,300000000
%N Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in base 10.
%H Colin Barker, <a href="/A278742/b278742.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10000).
%F a(n+17) = 10000*a(n) for any n>0.
%F a(17k+1) = 10^(4k), k >= 0. - _N. J. A. Sloane_, Jan 06 2017
%F G.f.: x*(1 +2*x +3*x^2 +10*x^3 +11*x^4 +12*x^5 +20*x^6 +30*x^7 +100*x^8 +110*x^9 +200*x^10 +300*x^11 +1000*x^12 +1100*x^13 +2000*x^14 +2100*x^15 +3000*x^16) / (1 -10000*x^17). - _Colin Barker_, Jan 10 2017
%e The first terms, alongside their partial sums, are:
%e n a(n) Partial sums (A280730)
%e -- ----- ----------------------
%e 1 1 1
%e 2 2 3
%e 3 3 6
%e 4 10 16
%e 5 11 27
%e 6 12 39
%e 7 20 59
%e 8 30 89
%e 9 100 189
%e 10 110 299
%e 11 200 499
%e 12 300 799
%e 13 1000 1799
%e 14 1100 2899
%e 15 2000 4899
%e 16 2100 6999
%e 17 3000 9999
%e -- ----- -----
%e 18 10000 19999
%e 19 20000 39999
%e 20 30000 69999
%t f[a_] := Function[w, Function[s, Total@ Map[PadLeft[#, s] &, w]]@ Max@ Map[Length, w]]@ Map[IntegerDigits, a]; g[n_] := FixedPoint[Function[k, If[Total@ Drop[RotateRight@ DigitCount@ k, 4] > 0, k + (6 * 10^(Position[#, 4][[1, 1]] - 1)) &@ Reverse@ IntegerDigits@ k, k]], n]; a = {1}; Do[If[n <= 17, k = g[Max@ a + 1]; While[Max@ f@ Join[a, {k}] > 9, k = g[k + 1]], k = 10^4 * a[[n - 17]]]; AppendTo[a, k], {n, 2, 37}]; a (* _Michael De Vlieger_, Dec 18 2016 *)
%o (PARI) a(n)=if(n>17, a(n-17)*10000, [1, 2, 3, 10, 11, 12, 20, 30, 100, 110, 200, 300, 1000, 1100, 2000, 2100, 3000][n]) \\ _Charles R Greathouse IV_, Nov 27 2016
%o (PARI) Vec(x*(1 +2*x +3*x^2 +10*x^3 +11*x^4 +12*x^5 +20*x^6 +30*x^7 +100*x^8 +110*x^9 +200*x^10 +300*x^11 +1000*x^12 +1100*x^13 +2000*x^14 +2100*x^15 +3000*x^16) / (1 -10000*x^17) + O(x^50)) \\ _Colin Barker_, Jan 10 2017
%Y Cf. A278743, A279732, A280730.
%K nonn,base,easy
%O 1,2
%A _Rémy Sigrist_, Nov 27 2016
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