A282840 Lexicographic-first permutation of the nonnegative integers such that the partial sums (-1)^issquare(a(n))*a(n) always remain nonnegative.

0, 2, 1, 3, 4, 5, 6, 9, 7, 8, 16, 10, 11, 12, 25, 13, 14, 36, 15, 17, 18, 49, 19, 20, 21, 22, 64, 23, 24, 26, 81, 27, 28, 29, 30, 100, 31, 32, 33, 121, 34, 35, 37, 38, 144, 39, 40, 41, 42, 43, 169, 44, 45, 46, 47, 196, 48, 50, 51, 52, 53, 225, 54, 55, 56, 57, 256, 58, 59, 60, 61, 62, 289, 63, 65, 66, 67, 68, 324, 69, 70, 71, 72, 73, 361, 74, 75, 76

Offset

0,2

Comments

In short: subtract squares when you can, else add nonsquares.

A variant of A282864 (with partial sums A282846) and A282865: In those variants, "square" is replaced by "prime".

The graph is a "superposition" of the nonsquares (on a roughly straight line of slope 1) and the squares in slowly increasingly spaced positions (at indices 0, 2, 4, 7, 10, 14, 17, 21,...: distances 2, 3, 4, 5, 6, 7,... appear 2, 3, 4, 5, 7, 9,... times, respectively).

The graph of the partial (signed) sums yields a nice "moirée" pattern, see A282886.

The variant with opposite signs (subtract nonsquares, add squares) is A282888, with sums A282887.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Example

Starting from 0 we cannot subtract the square 1, so we add a(1)=2, then we can subtract a(2)=1. Now we must add nonsquare a(3)=3 before subtracting the square  a(4)=4 (to yield zero sum). Now we have to add nonsquares a(5)=5 and a(6)=6 before subtracting the next square, a(7)=9. And so on.

Maple

a:= 1: b:= 2:
A[0]:= 0: S:= 0:
for n from 1 to 100 do
  if S - a^2 >= 0 then
    A[n]:= a^2; S:= S - a^2; a:= a+1;
  else A[n]:= b; S:= S + b; b:= b+1;
    if issqr(b) then b:= b+1 fi
  fi
od:
seq(A[i], i=0..100); # Robert Israel, Apr 15 2019

Prog

(PARI) {print1(a=0); c=1; p=2; for(n=1, 199, print1(", "abs(a+0*if(a<c^2, a+=p; while(issquare(p++), ), a-=c^2; c++)-a)))}

Crossrefs

Sequence in context: A345903 A346786 A239469 * A370655 A082750 A048212

Adjacent sequences: A282837 A282838 A282839 * A282841 A282842 A282843

Keyword

nonn,look

Author

M. F. Hasler and Eric Angelini, Feb 24 2017

Status

approved