OFFSET
1,1
COMMENTS
Maximum number of Pythagorean triples in an equilateral triangle.
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).
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.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Craig Knecht, Equilateral triangle tiled with 3,4,5 Pythagorean triples.
Craig Knecht, Interlocked up/down Pythagorean pairs.
Craig Knecht, Linear and triangular triples.
Craig Knecht, Incarcerated numbers.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
FORMULA
From Colin Barker, Oct 26 2015: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4) for n>4.
G.f.: x*(5*x^2+2*x+5) / ((x-1)^2*(x^2+x+1)).
(End)
EXAMPLE
Triangle (T(n,k): Row sum
5; 5
3, 4; 7
4, 5, 3; 12
5, 3, 4, 5; 17
3, 4, 5, 3, 4; 19
4, 5, 3, 4, 5, 3; 24
PROG
(PARI) Vec(x*(5*x^2+2*x+5)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 26 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Craig Knecht, Oct 20 2015
STATUS
approved