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 A357743 #52 Dec 05 2022 20:46:31
%S A357743 0,1,1,1,2,1,2,3,3,2,1,3,2,3,1,3,4,5,5,4,3,2,5,3,6,3,5,2,3,5,6,5,5,6,
%T A357743 5,3,1,4,3,5,2,5,3,4,1,4,5,7,8,7,7,8,7,5,4,3,7,4,9,5,10,5,9,4,7,3,5,8,
%U A357743 9,7,8,11,11,8,7,9,8,5,2,7,5,8,3,9,6,9,3,8,5,7,2
%N A357743 Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, for n, k >= 0, A(2*n, 2*k) = A(n, k), A(2*n, 2*k+1) = A(n, k) + A(n, k+1), A(2*n+1, 2*k) = A(n, k) + A(n+1, k), A(2*n+1, 2*k+1) = A(n, k+1) + A(n+1, k).
%C A357743 This sequence is closely related to A002487 and A355855: we can build this sequence:
%C A357743 - by starting from an equilateral triangle with values 0, 1, 1:
%C A357743 0
%C A357743 / \
%C A357743 1---1
%C A357743 - and repeatedly applying the following substitution:
%C A357743 t
%C A357743 / \
%C A357743 t / \
%C A357743 / \ --> t+u---t+v
%C A357743 u---v / \ / \
%C A357743 / \ / \
%C A357743 u----u+v----v
%C A357743 The sequence reduced modulo an odd prime number presents rich nonperiodic patterns (see illustrations in Links section).
%H A357743 Rémy Sigrist, <a href="/A357743/a357743.png">Colored representation of the first 512 antidiagonals</a> (where the color is function of A(n, k) mod 3)
%H A357743 Rémy Sigrist, <a href="/A357743/a357743_1.png">Colored representation of the first 512 antidiagonals</a> (where the color is function of A(n, k) mod 5)
%H A357743 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F A357743 A(n, k) = A(k, n).
%F A357743 A(n, 0) = A002487(n).
%F A357743 A(n, 1) = A007306(n+1) for any n > 0.
%e A357743 Array A(n, k) begins:
%e A357743 n\k | 0 1 2 3 4 5 6 7 8 9 10
%e A357743 ----+---------------------------------------
%e A357743 0 | 0 1 1 2 1 3 2 3 1 4 3
%e A357743 1 | 1 2 3 3 4 5 5 4 5 7 8
%e A357743 2 | 1 3 2 5 3 6 3 7 4 9 5
%e A357743 3 | 2 3 5 6 5 5 8 9 7 8 11
%e A357743 4 | 1 4 3 5 2 7 5 8 3 9 6
%e A357743 5 | 3 5 6 5 7 10 11 9 8 11 11
%e A357743 6 | 2 5 3 8 5 11 6 11 5 10 5
%e A357743 7 | 3 4 7 9 8 9 11 10 7 7 12
%e A357743 8 | 1 5 4 7 3 8 5 7 2 9 7
%e A357743 9 | 4 7 9 8 9 11 10 7 9 14 17
%e A357743 10 | 3 8 5 11 6 11 5 12 7 17 10
%e A357743 .
%e A357743 The first antidiagonals are:
%e A357743 0
%e A357743 1 1
%e A357743 1 2 1
%e A357743 2 3 3 2
%e A357743 1 3 2 3 1
%e A357743 3 4 5 5 4 3
%e A357743 2 5 3 6 3 5 2
%e A357743 3 5 6 5 5 6 5 3
%e A357743 1 4 3 5 2 5 3 4 1
%e A357743 4 5 7 8 7 7 8 7 5 4
%o A357743 (PARI) A(n,k) = { if (n==0 && k==0, 0, n==1 && k==0, 1, n==0 && k==1, 1, n%2==0 && k%2==0, A(n/2,k/2), n%2==0, A(n/2,(k-1)/2) + A(n/2,(k+1)/2), k%2==0, A((n-1)/2,k/2) + A((n+1)/2,k/2), A((n+1)/2,(k-1)/2) + A((n-1)/2,(k+1)/2)); }
%Y A357743 See A358871 for a similar sequence.
%Y A357743 Cf. A002487, A007306, A355855.
%K A357743 nonn,tabl
%O A357743 0,5
%A A357743 _Rémy Sigrist_, Nov 29 2022