A278742 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.
1, 2, 3, 10, 11, 12, 20, 30, 100, 110, 200, 300, 1000, 1100, 2000, 2100, 3000, 10000, 20000, 30000, 100000, 110000, 120000, 200000, 300000, 1000000, 1100000, 2000000, 3000000, 10000000, 11000000, 20000000, 21000000, 30000000, 100000000, 200000000, 300000000
Offset: 1
Examples
The first terms, alongside their partial sums, are: n a(n) Partial sums (A280730) -- ----- ---------------------- 1 1 1 2 2 3 3 3 6 4 10 16 5 11 27 6 12 39 7 20 59 8 30 89 9 100 189 10 110 299 11 200 499 12 300 799 13 1000 1799 14 1100 2899 15 2000 4899 16 2100 6999 17 3000 9999 -- ----- ----- 18 10000 19999 19 20000 39999 20 30000 69999
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10000).
Programs
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Mathematica
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 *) -
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
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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
Formula
a(n+17) = 10000*a(n) for any n>0.
a(17k+1) = 10^(4k), k >= 0. - N. J. A. Sloane, Jan 06 2017
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