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

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, 43, 24, 25, 49, 26, 27, 53, 28, 30, 58, 31, 32, 63, 34, 35, 69, 36, 37, 73, 38, 40, 78, 41, 42, 83, 44, 45, 89, 46, 47, 93, 48, 50, 98, 51, 52, 103, 54, 55, 109, 56, 57, 113, 59, 60

Offset: 1

Author

Reinhard Zumkeller, Dec 06 2004

Keywords

• nonn

Comments

Inverse: A101545; A101546(n) = a(a(n)).

From Bernard Schott, Jun 30 2019: (Start)

The terms can also be written simply following this array with 3 columns:

1st column 2nd column 3rd column

1 + 2 = 3

4 + 5 = 9

6 + 7 = 13

8 + 10 = 18

11 + 12 = 23

14 + 15 = 29

16 + 17 = 33

... ... ...

Question: in which column ends up the repdigit R_m(d) with m times the digit d?

Answer: R_m(d) will be in:

1) column 1 if d = 1, 4, 6, 8, or if d = 9 and m is even;

2) column 2 if d = 2, 5, 7;

3) column 3 if d = 3, or if d = 9 and m is odd.

Problem coming from Krusemeyer et al. (End)

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Mark I. Krusemeyer, George T. Gilbert, Loren C. Larson, A Mathematical Orchard, Problems and Solutions, MAA, 2012, Problem 99, pp. 179-181.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A075326, A249031.

Programs

Maple

N:= 100: # to get a(1) .. a(N)
S:= {$1..N}:
for n from 1 to N do
  if n mod 3 = 0 then A[n] := A[n-1]+A[n-2]
  else A[n]:= min(S)
  fi;
  S:= S minus {A[n]};
od:
seq(A[i],i=1..N); # Robert Israel, Feb 07 2016

Mathematica

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 *)

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))
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

Formula

From Rémy Sigrist, Apr 05 2020: (Start)

- a(3*n-2) = A249031(2*n-1),

- a(3*n-1) = A249031(2*n),

- a(3*n) = A075326(n).

(End)

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