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 A322403 #21 Jun 09 2025 14:40:34
%S A322403 0,0,0,0,1,0,0,2,2,0,0,3,2,3,0,0,4,12,12,4,0,0,5,4,15,4,5,0,0,6,42,48,
%T A322403 48,42,6,0,0,7,6,51,16,51,6,7,0,0,8,56,60,292,292,60,56,8,0,0,9,8,63,
%U A322403 12,5,12,63,8,9,0,0,10,150,192,448,438,438,448,192
%N A322403 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0: the lengths of runs in binary expansion of T(n, k) are obtained by multiplying those of n and of k (see Comments for precise definition).
%C A322403 For any n >= 0 and k >= 0:
%C A322403 - let r_n be the lengths of runs in binary expansion of n,
%C A322403 - for n = 0: we assume that r_0 = (),
%C A322403 - when n > 0: let R_n be the #r_n-periodic sequence whose first #r_n terms match r_n,
%C A322403 - r_{T(n, k)} has lcm(#r_n, #r_k) terms and r_{T(n, k)}(i) = R_n(i) * R_k(i) for i = 1..lcm(#r_n, #r_k).
%H A322403 <a href="http://oeis.org/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A322403 For any m >= 0, n >= 0 and k >= 0:
%F A322403 - T(n, k) = T(k, n) (T is commutative),
%F A322403 - T(m, T(n, k)) = T(T(m, n), k) (T is associative),
%F A322403 - T(m, A322404(n, k)) = A322404(T(m, n), T(m, k)) (T distributes over A322404),
%F A322403 - T(n, 0) = 0 (0 is an absorbing element for T),
%F A322403 - T(n, 1) = n (1 is an neutral element for T),
%F A322403 - T(n, 3) = A001196(n),
%F A322403 - T(n, 7) = A097254(n+1),
%F A322403 - T(n, 15) = A097262(n),
%F A322403 - T(n, n) = A322149(n),
%F A322403 - A005811(T(n, k)) = lcm(A005811(n), A005811(k)),
%F A322403 - T(2^n - 1, 2^k - 1) = 2^(n*k) - 1.
%F A322403 - T(2^n, 2^k) = 2^(n*k) when n > 0 and k > 0,
%F A322403 - T(n, k) is odd iff both n and k are odd.
%e A322403 Array T(n, k) begins (in decimal):
%e A322403 n\k| 0 1 2 3 4 5 6 7 8 9 10
%e A322403 ---+--------------------------------------------------------
%e A322403 0| 0 0 0 0 0 0 0 0 0 0 0
%e A322403 1| 0 1 2 3 4 5 6 7 8 9 10
%e A322403 2| 0 2 2 12 4 42 6 56 8 150 10
%e A322403 3| 0 3 12 15 48 51 60 63 192 195 204
%e A322403 4| 0 4 4 48 16 292 12 448 64 2124 36
%e A322403 5| 0 5 42 51 292 5 438 455 2184 9 2730
%e A322403 6| 0 6 6 60 12 438 30 504 24 3294 54
%e A322403 7| 0 7 56 63 448 455 504 511 3584 3591 3640
%e A322403 8| 0 8 8 192 64 2184 24 3584 512 33048 136
%e A322403 Array T(n, k) begins (in binary):
%e A322403 n\k| 0 1 10 11 100 101 110
%e A322403 ----+---------------------------------------------------------------
%e A322403 0| 0 0 0 0 0 0 0
%e A322403 1| 0 1 10 11 100 101 110
%e A322403 10| 0 10 10 1100 100 101010 110
%e A322403 11| 0 11 1100 1111 110000 110011 111100
%e A322403 100| 0 100 100 110000 10000 100100100 1100
%e A322403 101| 0 101 101010 110011 100100100 101 110110110
%e A322403 110| 0 110 110 111100 1100 110110110 11110
%e A322403 111| 0 111 111000 111111 111000000 111000111 111111000
%e A322403 1000| 0 1000 1000 11000000 1000000 100010001000 11000
%o A322403 (PARI) T(n,k) = my (v=0, p=1, rn=n, rk=k, b=if ((max(n,1)%2)&&(max(k,1)%2), 1, 0)); while (1, my (vn=if (rn==0, 0, valuation(rn+(rn%2), 2)), vk=if(rk==0, 0, valuation(rk+(rk%2), 2)), w=vn*vk); v+=b*p*(2^w-1); rn\=2^vn; rk\=2^vk; if (rn==0 && rk==0, return (v), rn==0, rn=n, rk==0, rk=k); p*=2^w; b=1-b)
%Y A322403 See A322404 for the additive variant.
%Y A322403 Cf. A001196, A097254, A097262, A322149.
%K A322403 nonn,base,tabl
%O A322403 0,8
%A A322403 _Rémy Sigrist_, Dec 06 2018