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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.)