A393461 Positions of zeros in A393422.

1, 180, 203, 361, 406, 2786, 3357, 5375, 5494, 5573, 5727, 5794, 6493, 6560, 6714, 6793, 6912, 36844, 36848, 38880, 39800, 39853, 39857, 39878, 40298, 40344, 40355, 40508, 40545, 40707, 42750, 42948, 43516, 43868, 43879, 43925, 44251, 44261, 44345, 44366

Offset

1,2

Comments

The binary expansions of the terms of the sequence encode zero sums of the form +- 1^2 +- 2^2 +- 3^2 +- 4^2 +- ... +- k^2 for some choice of +- signs.

This sequence is infinite.

k is a term iff A054429(k) is a term.

For any k > 0, there are A158092(k) terms with k+1 binary digits.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Example

The first terms, alongside the corresponding zero sums, are:
  n  a(n)  Zero sum
  -  ----  ---------------------------------------------------
  1     1  0 (empty sum)
  2   180  +1^2+2^2-3^2+4^2-5^2-6^2+7^2
  3   203  -1^2-2^2+3^2-4^2+5^2+6^2-7^2
  4   361  -1^2+2^2+3^2-4^2+5^2-6^2-7^2+8^2
  5   406  +1^2-2^2-3^2+4^2-5^2+6^2+7^2-8^2
  6  2786  +1^2-2^2+3^2+4^2+5^2-6^2-7^2-8^2+9^2-10^2+11^2
  7  3357  -1^2+2^2-3^2-4^2-5^2+6^2+7^2+8^2-9^2+10^2-11^2
  8  5375  -1^2-2^2-3^2-4^2-5^2-6^2-7^2-8^2+9^2+10^2-11^2+12^2
  9  5494  +1^2-2^2-3^2+4^2-5^2-6^2-7^2+8^2-9^2+10^2-11^2+12^2

Mathematica

s = {0}; {1}~Join~Flatten@ Reap[Do[s = Join[s + n^2, s - n^2]; Sow[-1 + 2^n + Position[s, 0]], {n, 15}] ][[-1, 1]] (* Michael De Vlieger, Feb 16 2026 *)

Prog

(PARI) { for (n = 1, 44366, if (sum(e = 0, exponent(n)-1, (e+1)^2 * (-1)^bittest(n, e))==0, print1 (n", "); ); ); }

Crossrefs

Cf. A054429, A158092, A393422.

Sequence in context: A119542 A154360 A253393 * A260265 A117551 A383026

Adjacent sequences: A393458 A393459 A393460 * A393462 A393463 A393464

Keyword

nonn,base

Author

Rémy Sigrist, Feb 15 2026

Status

approved