A393461 - OEIS

%I #11 Feb 18 2026 15:49:20

%S 1,180,203,361,406,2786,3357,5375,5494,5573,5727,5794,6493,6560,6714,

%T 6793,6912,36844,36848,38880,39800,39853,39857,39878,40298,40344,

%U 40355,40508,40545,40707,42750,42948,43516,43868,43879,43925,44251,44261,44345,44366

%N Positions of zeros in A393422.

%C 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.

%C This sequence is infinite.

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

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

%H Rémy Sigrist, <a href="/A393461/b393461.txt">Table of n, a(n) for n = 1..10000</a>

%e The first terms, alongside the corresponding zero sums, are:

%e   n  a(n)  Zero sum

%e   -  ----  ---------------------------------------------------

%e   1     1  0 (empty sum)

%e   2   180  +1^2+2^2-3^2+4^2-5^2-6^2+7^2

%e   3   203  -1^2-2^2+3^2-4^2+5^2+6^2-7^2

%e   4   361  -1^2+2^2+3^2-4^2+5^2-6^2-7^2+8^2

%e   5   406  +1^2-2^2-3^2+4^2-5^2+6^2+7^2-8^2

%e   6  2786  +1^2-2^2+3^2+4^2+5^2-6^2-7^2-8^2+9^2-10^2+11^2

%e   7  3357  -1^2+2^2-3^2-4^2-5^2+6^2+7^2+8^2-9^2+10^2-11^2

%e   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

%e   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

%t 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 *)

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

%Y Cf. A054429, A158092, A393422.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Feb 15 2026