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 A356096 #11 Jan 18 2023 03:28:47 %S A356096 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,-1,5,-1,1,1,5,5,5,5, %T A356096 1,1,-1,5,3,5,-1,1,1,1,5,5,5,5,1,1,1,3,1,-1,5,-1,1,3,1,1,1,1,1,1,1,1, %U A356096 1,1,1,1,1,1,1,3,1,1,1,1,1,1,-1,5,-1,1 %N A356096 A family of triangles T(m), m >= 0, read by triangles and then by rows; triangle T(0) is [1; 1, 1]; for m >= 0, triangle T(m+1) is obtained by replacing each subtriangle [t; u, v] in T(m) by [t; 2*t-u, 2*t-v; 2*u-t, t+u+v, 2*v-t; u, 2*u-v, 2*v-u, v]. %C A356096 We apply the following substitutions to transform T(m) into T(m+1): %C A356096 t %C A356096 / \ %C A356096 / \ %C A356096 t 2*t-u 2*t-v %C A356096 / \ ___\ / \ / \ %C A356096 / \ / / \ / \ %C A356096 u-----v 2*u-t t+u+v 2*v-t %C A356096 / \ / \ / \ %C A356096 / \ / \ / \ %C A356096 u---2*u-v--2*v-u--v %C A356096 and: %C A356096 u---2*u-v--2*v-u--v %C A356096 \ / \ / \ / %C A356096 \ / \ / \ / %C A356096 u-----v 2*u-t t+u+v 2*v-t %C A356096 \ / ___\ \ / \ / %C A356096 \ / / \ / \ / %C A356096 t 2*t-u 2*t-v %C A356096 \ / %C A356096 \ / %C A356096 t %C A356096 T(m) has 3^m+1 rows. %C A356096 All terms are odd. %C A356096 As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section). %H A356096 Rémy Sigrist, <a href="/A356096/a356096.png">Colored representation of T6</a> (the color is function of T(6)(n, k)) %H A356096 Rémy Sigrist, <a href="/A356096/a356096_1.png">Representation of the multiples of 3 in T(7)</a> %H A356096 Rémy Sigrist, <a href="/A356096/a356096_2.png">Representation of the negative terms in T(7)</a> %H A356096 Rémy Sigrist, <a href="/A356096/a356096.gp.txt">PARI program</a> %H A356096 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. %e A356096 Triangle T(0) is: %e A356096 1 %e A356096 1 1 %e A356096 Triangle T(1) is: %e A356096 1 %e A356096 1 1 %e A356096 1 3 1 %e A356096 1 1 1 1 %e A356096 Triangle T(2) is: %e A356096 1 %e A356096 1 1 %e A356096 1 3 1 %e A356096 1 1 1 1 %e A356096 1 -1 5 -1 1 %e A356096 1 5 5 5 5 1 %e A356096 1 -1 5 3 5 -1 1 %e A356096 1 1 5 5 5 5 1 1 %e A356096 1 3 1 -1 5 -1 1 3 1 %e A356096 1 1 1 1 1 1 1 1 1 1 %o A356096 (PARI) See Links section. %Y A356096 See A355855, A356002, A356097 and A356098 for similar sequences. %K A356096 sign,tabf %O A356096 0,8 %A A356096 _Rémy Sigrist_, Jul 26 2022