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 A356002 #16 Jan 18 2023 03:28:54 %S A356002 1,1,1,1,3,3,3,3,3,1,3,3,1,1,5,5,7,7,7,3,9,9,3,9,9,9,9,9,9,9,9,9,9,9, %T A356002 3,9,9,3,9,9,3,7,9,9,9,9,9,9,7,5,7,9,9,9,9,9,7,5,1,5,7,3,9,9,3,7,5,1, %U A356002 1,7,7,11,11,11,5,15,15,5,17,17,17,17,17 %N A356002 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; t+2*u, t+u+v, t+2*v; u, 2*u+v, u+2*v, v]. %C A356002 We apply the following substitutions to transform T(m) into T(m+1): %C A356002 t %C A356002 / \ %C A356002 / \ %C A356002 t 2*t+u 2*t+v %C A356002 / \ ___\ / \ / \ %C A356002 / \ / / \ / \ %C A356002 u-----v t+2*u t+u+v t+2*v %C A356002 / \ / \ / \ %C A356002 / \ / \ / \ %C A356002 u---2*u+v--u+2*v--v %C A356002 and: %C A356002 u---2*u+v--u+2*v--v %C A356002 \ / \ / \ / %C A356002 \ / \ / \ / %C A356002 u-----v t+2*u t+u+v t+2*v %C A356002 \ / ___\ \ / \ / %C A356002 \ / / \ / \ / %C A356002 t 2*t+u 2*t+v %C A356002 \ / %C A356002 \ / %C A356002 t %C A356002 T(m) has 3^m+1 rows, and largest term 3^m. %C A356002 All terms are odd. %C A356002 As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section). %H A356002 Rémy Sigrist, <a href="/A356002/a356002.png">Colored representation of T(6)</a> (the color is function of T(6)(n,k)) %H A356002 Rémy Sigrist, <a href="/A356002/a356002_1.png">Colored representation of T(6)</a> (the color is function of the 3-adic valuation of T(6)(n,k)) %H A356002 Rémy Sigrist, <a href="/A356002/a356002_2.png">Representation of the terms congruent to 3 mod 4 in T(6)</a> %H A356002 Rémy Sigrist, <a href="/A356002/a356002.gp.txt">PARI program</a> %H A356002 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 A356002 Triangle T(0) is: %e A356002 1 %e A356002 1 1 %e A356002 Triangle T(1) is: %e A356002 1 %e A356002 3 3 %e A356002 3 3 3 %e A356002 1 3 3 1 %e A356002 Triangle T(2) is: %e A356002 1 %e A356002 5 5 %e A356002 7 7 7 %e A356002 3 9 9 3 %e A356002 9 9 9 9 9 %e A356002 9 9 9 9 9 9 %e A356002 3 9 9 3 9 9 3 %e A356002 7 9 9 9 9 9 9 7 %e A356002 5 7 9 9 9 9 9 7 5 %e A356002 1 5 7 3 9 9 3 7 5 1 %o A356002 (PARI) See Links section. %Y A356002 See A355855 for a similar sequence. %Y A356002 Cf. A177407. %K A356002 nonn,tabf %O A356002 0,5 %A A356002 _Rémy Sigrist_, Jul 22 2022