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 A341288 #18 Dec 10 2023 17:24:12
%S A341288 0,0,0,0,1,0,0,2,2,0,0,3,1,3,0,0,4,3,3,4,0,0,5,8,0,8,5,0,0,6,10,12,12,
%T A341288 10,6,0,0,7,9,15,1,15,9,7,0,0,8,11,15,5,5,15,11,8,0,0,9,4,12,9,0,9,12,
%U A341288 4,9,0,0,10,6,12,13,15,15,13,12,6,10,0
%N A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.
%C A341288 For any x >= 0, the function n -> T(n, 2^x) is a self-inverse permutation of the nonnegative integers.
%C A341288 The set of nonnegative integers equipped with T forms a commutative monoid; its invertible elements are the odious numbers (A000069).
%C A341288 Hence A000069 equipped with T forms a group.
%H A341288 Rémy Sigrist, <a href="/A341288/b341288.txt">Table of n, a(n) for n = 0..10010</a> (rows 0..140)
%H A341288 Rémy Sigrist, <a href="/A341288/a341288.png">Colored representation of the table for n, k < 2^10</a>
%H A341288 Rémy Sigrist, <a href="/A341288/a341288_1.png">Colored representation of the table over the first 128 odious numbers</a>
%H A341288 Rémy Sigrist, <a href="/A341288/a341288_2.png">Colored representation of the table over the first 1024 evil numbers</a> (white pixels correspond to 0's)
%F A341288 T(n, k) = T(k, n) (T is commutative).
%F A341288 T(m, T(n, k)) = T(T(m, n), k) (T is associative).
%F A341288 T(n, 0) = 0 (0 is an absorbing element for T).
%F A341288 T(n, 1) = n (1 is the neutral element for T).
%F A341288 T(n, 2) = A057300(n).
%F A341288 T(n, 4) = A126006(n).
%F A341288 T(n, n) = A010060(n).
%F A341288 A010060(T(n, k)) = A010060(n) * A010060(k).
%e A341288 Array T(n, k) begins:
%e A341288 n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e A341288 ---+---------------------------------------------------------------
%e A341288 0| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e A341288 1| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e A341288 2| 0 2 1 3 8 10 9 11 4 6 5 7 12 14 13 15 -> A057300
%e A341288 3| 0 3 3 0 12 15 15 12 12 15 15 12 0 3 3 0
%e A341288 4| 0 4 8 12 1 5 9 13 2 6 10 14 3 7 11 15 -> A126006
%e A341288 5| 0 5 10 15 5 0 15 10 10 15 0 5 15 10 5 0
%e A341288 6| 0 6 9 15 9 15 0 6 6 0 15 9 15 9 6 0
%e A341288 7| 0 7 11 12 13 10 6 1 14 9 5 2 3 4 8 15
%e A341288 8| 0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15
%e A341288 9| 0 9 6 15 6 15 0 9 9 0 15 6 15 6 9 0
%e A341288 10| 0 10 5 15 10 0 15 5 5 15 0 10 15 5 10 0
%e A341288 11| 0 11 7 12 14 5 9 2 13 6 10 1 3 8 4 15
%e A341288 12| 0 12 12 0 3 15 15 3 3 15 15 3 0 12 12 0
%e A341288 13| 0 13 14 3 7 10 9 4 11 6 5 8 12 1 2 15
%e A341288 14| 0 14 13 3 11 5 6 8 7 9 10 4 12 2 1 15
%e A341288 15| 0 15 15 0 15 0 0 15 15 0 0 15 0 15 15 0
%e A341288 \
%e A341288 v
%e A341288 A010060
%o A341288 (PARI) B(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n -= 2^(b[k] = valuation(n, 2))); b }
%o A341288 T(n,k) = { my (nn=B(n), kk=B(k), v=0); for (i=1, #nn, for (j=1, #kk, v=bitxor(v, 2^bitxor(nn[i], kk[j])))); v }
%Y A341288 Cf. A000069, A010060, A057300, A126006, A133457.
%K A341288 nonn,tabl,look,base
%O A341288 0,8
%A A341288 _Rémy Sigrist_, Feb 08 2021