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A101544 Smallest permutation of the natural numbers with a(3*k-2) + a(3*k-1) = a(3*k), k > 0.

%I A101544 #40 Aug 14 2025 04:07:08
%S A101544 1,2,3,4,5,9,6,7,13,8,10,18,11,12,23,14,15,29,16,17,33,19,20,39,21,22,
%T A101544 43,24,25,49,26,27,53,28,30,58,31,32,63,34,35,69,36,37,73,38,40,78,41,
%U A101544 42,83,44,45,89,46,47,93,48,50,98,51,52,103,54,55,109,56,57,113,59,60
%N A101544 Smallest permutation of the natural numbers with a(3*k-2) + a(3*k-1) = a(3*k), k > 0.
%C A101544 Inverse: A101545; A101546(n) = a(a(n)).
%C A101544 From _Bernard Schott_, Jun 30 2019: (Start)
%C A101544 The terms can also be written simply following this array with 3 columns:
%C A101544    1st column       2nd column          3rd column
%C A101544         1      +        2          =         3
%C A101544         4      +        5          =         9
%C A101544         6      +        7          =        13
%C A101544         8      +       10          =        18
%C A101544        11      +       12          =        23
%C A101544        14      +       15          =        29
%C A101544        16      +       17          =        33
%C A101544        ...             ...                  ...
%C A101544 Question: in which column ends up the repdigit R_m(d) with m times the digit d?
%C A101544 Answer: R_m(d) will be in:
%C A101544 1) column 1 if d = 1, 4, 6, 8, or if d = 9 and m is even;
%C A101544 2) column 2 if d = 2, 5, 7;
%C A101544 3) column 3 if d = 3, or if d = 9 and m is odd.
%C A101544 Problem coming from Krusemeyer et al. (End)
%H A101544 Ivan Neretin, <a href="/A101544/b101544.txt">Table of n, a(n) for n = 1..10000</a>
%H A101544 Mark I. Krusemeyer, George T. Gilbert, Loren C. Larson, <a href="https://www.google.com/books/edition/A_Mathematical_Orchard/3Zeib5hjVLkC?gbpv=1&amp;pg=PA179">A Mathematical Orchard, Problems and Solutions</a>, MAA, 2012, Problem 99, pp. 179-181.
%H A101544 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A101544 From _Rémy Sigrist_, Apr 05 2020: (Start)
%F A101544 - a(3*n-2) = A249031(2*n-1),
%F A101544 - a(3*n-1) = A249031(2*n),
%F A101544 - a(3*n)   = A075326(n).
%F A101544 (End)
%F A101544 a(3*(4k + m) + c) = (10k + 3m - [m>1])*2^[c=3] + c - [m = 3 and c <> 2 and k+1 is in A036554], where 1 <= c <= 3, 0 <= m <= 3, and [.] is the Iverson bracket. - _M. F. Hasler_, Nov 26 2024
%p A101544 N:= 100: # to get a(1) .. a(N)
%p A101544 S:= {$1..N}:
%p A101544 for n from 1 to N do
%p A101544   if n mod 3 = 0 then A[n] := A[n-1]+A[n-2]
%p A101544   else A[n]:= min(S)
%p A101544   fi;
%p A101544   S:= S minus {A[n]};
%p A101544 od:
%p A101544 seq(A[i],i=1..N); # _Robert Israel_, Feb 07 2016
%t A101544 Fold[Append[#1, If[Divisible[#2, 3], #1[[-1]] + #1[[-2]], Min@Complement[Range[Max@#1 + 1], #1]]] &, {1}, Range[2, 71]] (* _Ivan Neretin_, Feb 05 2016 *)
%o A101544 (PARI) A101544_upto(N, U=[], T=0)=vector(N, n, if(n%=3, while(if(U, U[1])==T+=1, U=U[^1]); n>1 || N=T; T, U=concat(U, N+=T); N))
%o A101544 apply( {A101544(n, k=(n-=1)\12, m=n\3%4, c=n%3)=(10*k+3*m-(m>1))<<(c>1)+c+(m<3 || c==1 || valuation(k+1,2)%2)}, [1..99]) \\ _M. F. Hasler_, Nov 26 2024
%Y A101544 Cf. A075326, A249031.
%K A101544 nonn
%O A101544 1,2
%A A101544 _Reinhard Zumkeller_, Dec 06 2004