A282886 Sum_{k=0..n} (-1)^issquare(p(k))*p(k) where p=A282840 is the lexicographic-first permutation of the nonnegative integers such that these sums always remain nonnegative.

0, 2, 1, 4, 0, 5, 11, 2, 9, 17, 1, 11, 22, 34, 9, 22, 36, 0, 15, 32, 50, 1, 20, 40, 61, 83, 19, 42, 66, 92, 11, 38, 66, 95, 125, 25, 56, 88, 121, 0, 34, 69, 106, 144, 0, 39, 79, 120, 162, 205, 36, 80, 125, 171, 218, 22, 70, 120, 171, 223, 276, 51, 105, 160, 216, 273, 17, 75, 134, 194, 255, 317, 28, 91, 156, 222, 289, 357, 33, 102, 172, 243, 315

Offset

0,2

Comments

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

A variant of A282846 (corresponding to permutation A282864) and A282865: In those variants, "square" is replaced by "prime".

The graph yields a nice moirée pattern.

Links

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

Example

Starting from a(0)=0 we cannot subtract the square 1, so we add 2 to get a(1)=2, then we can subtract 1, a(2)=1. Now we must add nonsquare 3 to get a(3)=4 before subtracting the square 4, to yield zero sum a(4)=0. Now we have to add nonsquares 5, a(5)=5, and 6, a(6)=11, before subtracting the next square 9, a(7)=2. And so on.

Maple

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

Prog

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

Crossrefs

Sequence in context: A324055 A087664 A158032 * A282846 A326129 A336566

Adjacent sequences: A282883 A282884 A282885 * A282887 A282888 A282889

Keyword

nonn,look

Author

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

Status

approved