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

1,1

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

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

Sequence in context: A314309 A314310 A117140 * A314311 A031144 A314312

Adjacent sequences: A263533 A263534 A263535 * A263537 A263538 A263539

Keyword

nonn,easy

Author

Craig Knecht, Oct 20 2015

Status

approved