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A263536 Row sum of an equilateral triangle tiled with the 3,4,5 Pythagorean triple. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 6 12:00 EDT 2022. Contains 357264 sequences. (Running on oeis4.)