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 A263536 #45 Mar 05 2025 01:11:56 %S A263536 5,7,12,17,19,24,29,31,36,41,43,48,53,55,60,65,67,72,77,79,84,89,91, %T A263536 96,101,103,108,113,115,120,125,127,132,137,139,144,149,151,156,161, %U A263536 163,168,173,175,180,185,187,192,197,199,204,209,211,216,221,223,228 %N A263536 Row sum of an equilateral triangle tiled with the 3,4,5 Pythagorean triple. %C A263536 Maximum number of Pythagorean triples in an equilateral triangle. %C A263536 Two rules are used to construct this equilateral triangle: #1. Start with the number 5 at the top. #2. Require every "triple" to contain the Pythagorean triple 3, 4, 5 (see link below). %C A263536 Up and down Pythagorean triples consist of two terms below and one above when k is odd (an up triple), and two terms above and one below when k is even (a down triple). Three adjacent terms in a straight line within the triangle form a linear triple. %H A263536 Colin Barker, <a href="/A263536/b263536.txt">Table of n, a(n) for n = 1..1000</a> %H A263536 Craig Knecht, <a href="/A263536/a263536.jpg">Equilateral triangle tiled with 3,4,5 Pythagorean triples</a>. %H A263536 Craig Knecht, <a href="/A263536/a263536_1.jpg">Interlocked up/down Pythagorean pairs</a>. %H A263536 Craig Knecht, <a href="/A263536/a263536_2.jpg">Linear and triangular triples</a>. %H A263536 Craig Knecht, <a href="/A263536/a263536_4.jpg">Incarcerated numbers</a>. %H A263536 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1). %F A263536 From _Colin Barker_, Oct 26 2015: (Start) %F A263536 a(n) = a(n-1)+a(n-3)-a(n-4) for n>4. %F A263536 G.f.: x*(5*x^2+2*x+5) / ((x-1)^2*(x^2+x+1)). %F A263536 (End) %e A263536 Triangle T(n,k): Row sum %e A263536 5; 5 %e A263536 3, 4; 7 %e A263536 4, 5, 3; 12 %e A263536 5, 3, 4, 5; 17 %e A263536 3, 4, 5, 3, 4; 19 %e A263536 4, 5, 3, 4, 5, 3; 24 %o A263536 (PARI) Vec(x*(5*x^2+2*x+5)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Oct 26 2015 %Y A263536 Cf. A136289 (every triple contains 1,2,3), A008854 (every triple contains 1,2,2), A259052 (sum of Pascal triples). %K A263536 nonn,easy %O A263536 1,1 %A A263536 _Craig Knecht_, Oct 20 2015