A263536 - OEIS

%I #42 Jan 21 2021 04:21:30

%S 5,7,12,17,19,24,29,31,36,41,43,48,53,55,60,65,67,72,77,79,84,89,91,

%T 96,101,103,108,113,115,120,125,127,132,137,139,144,149,151,156,161,

%U 163,168,173,175,180,185,187,192,197,199,204,209,211,216,221,223,228

%N Row sum of an equilateral triangle tiled with the 3,4,5 Pythagorean triple.

%C Maximum number of Pythagorean triples in an equilateral triangle.

%C 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 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 Colin Barker, <a href="/A263536/b263536.txt">Table of n, a(n) for n = 1..1000</a>

%H Craig Knecht, <a href="/A263536/a263536.jpg">Equilateral triangle tiled with 3,4,5 Pythagorean triples</a>.

%H Craig Knecht, <a href="/A263536/a263536_1.jpg">Interlocked up/down Pythagorean pairs</a>.

%H Craig Knecht, <a href="/A263536/a263536_2.jpg">Linear and triangular triples</a>.

%H Craig Knecht, <a href="/A263536/a263536_4.jpg">Incarcerated numbers</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F From _Colin Barker_, Oct 26 2015: (Start)

%F a(n) = a(n-1)+a(n-3)-a(n-4) for n>4.

%F G.f.: x*(5*x^2+2*x+5) / ((x-1)^2*(x^2+x+1)).

%F (End)

%e Triangle (T(n,k):           Row sum

%e 5;                             5

%e 3, 4;                          7

%e 4, 5, 3;                      12

%e 5, 3, 4, 5;                   17

%e 3, 4, 5, 3, 4;                19

%e 4, 5, 3, 4, 5, 3;             24

%o (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 Cf. A136289 (every triple contains 1,2,3), A008854 (every triple contains 1,2,2), A259052 (sum of Pascal triples).

%K nonn,easy

%O 1,1

%A _Craig Knecht_, Oct 20 2015