A263536 Row sum of an equilateral triangle tiled with the 3,4,5 Pythagorean triple.

5, 7, 12, 17, 19, 24, 29, 31, 36, 41, 43, 48, 53, 55, 60, 65, 67, 72, 77, 79, 84, 89, 91, 96, 101, 103, 108, 113, 115, 120, 125, 127, 132, 137, 139, 144, 149, 151, 156, 161, 163, 168, 173, 175, 180, 185, 187, 192, 197, 199, 204, 209, 211, 216, 221, 223, 228

Offset: 1

Author

Craig Knecht, Oct 20 2015

Keywords

• nonn
• easy

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.

Examples

			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
		

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).

Crossrefs

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

Programs

PARI

Vec(x*(5*x^2+2*x+5)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 26 2015

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)