This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348299 #16 Oct 12 2021 11:28:09
%S A348299 0,4,24,28,140,144,164,168,816,820,840,844,956,960,980,984,4756,4760,
%T A348299 4780,4784,4896,4900,4920,4924,5572,5576,5596,5600,5712,5716,5736,
%U A348299 5740,27720,27724,27744,27748,27860,27864,27884,27888,28536,28540,28560,28564,28676,28680,28700,28704
%N A348299 Indices of 0 in A348295: numbers m such that Sum_{k=1..m} (-1)^(floor(k*(sqrt(2)-1))) = Sum_{k=1..m} (-1)^A097508(k) = 0.
%C A348299 Conjecture: sequence is infinite.
%C A348299 Conjecture: all terms are divisible by 4. There are no counterexamples <= 10^9.
%H A348299 Jianing Song, <a href="/A348299/b348299.txt">Table of n, a(n) for n = 1..2048</a> (all terms <= 10^9)
%F A348299 Conjecture: for n >= 2, a(2n-1) = ceiling(a(n) * (3+2*sqrt(2))), a(2n) = a(2n-1) + 4. This is correct for the first 2048 terms (all terms <= 10^9).
%F A348299 Conjectured explicit formula: if the binary expansion of n-1 is n-1 = 2^(e_0) + 2^(e_1) + ... + 2^(e_k), then a(n) = 4*(A001109(1+(e_0)) + A001109(1+(e_1)) + ... + A001109(1+(e_k))). For example, since 28-1 = 27 = 2^0 + 2^1 + 2^3 + 2^4, a(28) = 4*(A001109(1) + A001109(2) + A001109(4) + A001109(5)) = 5600.
%e A348299 24 is a term: A097508(k) is even for k = 1, 2, 5, 6, 7, 10, 11, 12, 15, 16, 20, 21 and odd for k = 3, 4, 8, 9, 13, 14, 17, 18, 19, 22, 23, 24, so Sum_{k=1..24} (-1)^A097508(k) = 0.
%t A348299 Join[{0}, Position[Accumulate@ Table[(-1)^Floor[k*(Sqrt[2] - 1)], {k, 1, 30000}], 0] // Flatten] (* _Amiram Eldar_, Oct 11 2021 *)
%o A348299 (PARI) list(lim) = my(Sum=-1, v=[]); for(k=0, lim, Sum+=(-1)^(sqrtint(2*k^2)-k); if(Sum==0, v=concat(v, k))); v
%Y A348299 Cf. A348295, A097508, A001109.
%K A348299 nonn
%O A348299 1,2
%A A348299 _Jianing Song_, Oct 10 2021