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 A358871 #10 Jan 18 2023 03:29:08
%S A358871 0,1,1,1,2,1,2,3,3,2,1,3,2,3,1,3,4,5,5,4,3,2,4,3,4,3,4,2,3,5,6,5,5,6,
%T A358871 5,3,1,4,3,5,2,5,3,4,1,4,5,7,8,7,7,8,7,5,4,3,5,4,6,5,6,5,6,4,5,3,5,7,
%U A358871 8,7,8,9,9,8,7,8,7,5,2,6,4,7,3,7,4,7,3,7,4,6,2
%N A358871 Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, A(1, 1) = 2, 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+1, k+(1+(-1)^(n+k))/2) + A(n, k+(1-(-1)^(n+k))/2).
%C A358871 This sequence is a variant of A357743: we can build this sequence:
%C A358871 - by starting from an isosceles right triangle with values 0, 1, 1:
%C A358871 0 <- right angle
%C A358871 / \
%C A358871 / \
%C A358871 1-----1
%C A358871 - and repeatedly applying the following substitution to each isosceles right triangle:
%C A358871 t t
%C A358871 / \ --> /|\
%C A358871 / \ / | \
%C A358871 u-----v u-u+v-v
%C A358871 ^
%C A358871 | right angles
%C A358871 The sequence presents rich patterns (see Links section).
%H A358871 Rémy Sigrist, <a href="/A358871/a358871.png">Colored representation of the first 512 antidiagonals</a> (where the color is function of A(n, k) mod 2)
%H A358871 Rémy Sigrist, <a href="/A358871/a358871_1.png">Colored representation of the first 512 antidiagonals</a> (where the color is function of A(n, k) mod 3)
%H A358871 Rémy Sigrist, <a href="/A358871/a358871_2.png">Colored representation of the first 512 antidiagonals</a> (where the color is function of A(n, k) mod 5)
%H A358871 Rémy Sigrist, <a href="https://arxiv.org/abs/2301.06039">Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of Fp</a>, arXiv:2301.06039 [math.CO], 2023.
%F A358871 A(n, k) = A(k, n).
%F A358871 A(n, 0) = A002487(n).
%F A358871 A(n, n) = 2*A002487(n).
%e A358871 Array A(n, k) begins:
%e A358871 n\k | 0 1 2 3 4 5 6 7 8 9 10
%e A358871 ----+---------------------------------------
%e A358871 0 | 0 1 1 2 1 3 2 3 1 4 3
%e A358871 1 | 1 2 3 3 4 4 5 4 5 5 7
%e A358871 2 | 1 3 2 5 3 6 3 7 4 8 4
%e A358871 3 | 2 3 5 4 5 5 8 6 7 7 10
%e A358871 4 | 1 4 3 5 2 7 5 8 3 9 6
%e A358871 5 | 3 4 6 5 7 6 9 7 8 8 11
%e A358871 6 | 2 5 3 8 5 9 4 9 5 10 5
%e A358871 7 | 3 4 7 6 8 7 9 6 7 7 12
%e A358871 8 | 1 5 4 7 3 8 5 7 2 9 7
%e A358871 9 | 4 5 8 7 9 8 10 7 9 8 13
%e A358871 10 | 3 7 4 10 6 11 5 12 7 13 6
%e A358871 .
%e A358871 The first antidiagonals are:
%e A358871 0
%e A358871 1 1
%e A358871 1 2 1
%e A358871 2 3 3 2
%e A358871 1 3 2 3 1
%e A358871 3 4 5 5 4 3
%e A358871 2 4 3 4 3 4 2
%e A358871 3 5 6 5 5 6 5 3
%e A358871 1 4 3 5 2 5 3 4 1
%e A358871 4 5 7 8 7 7 8 7 5 4
%o A358871 (PARI) A(n,k) = { my (nn = n\2, kk=k\2); if (n<=1 && k<=1, n+k, n%2==0 && k%2==0, A(n/2,k/2), n%2==0, A(n/2,k\2)+A(n/2,k\2+1), k%2==0, A(n\2,k\2)+A(n\2+1,k\2), A(n\2+1,k\2+(1+(-1)^(n\2+k\2))/2) + A(n\2, k\2+(1-(-1)^(n\2+k\2))/2)); }
%Y A358871 Cf. A002487, A357743.
%K A358871 nonn,tabl
%O A358871 0,5
%A A358871 _Rémy Sigrist_, Dec 04 2022