A362545 -id:A362545 - cp's OEIS frontend

A377726 Lengths of the long leg of the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

84, 3280, 113764, 3878112, 131820084, 4478459440, 152138450884, 5168244315840, 175568258308884, 5964153062868112, 202605638937276964, 6882627588628286880, 233806732478308836084, 7942546277279354556400, 269812766698548756220804, 9165691521493946935370112

Offset: 1

Miguel-Ángel Pérez García-Ortega, Nov 05 2024

			Triangles begins:
  n=1:     13,        84,        85;
  n=2:     81,      3280,      3281;
  n=3:    477,    113764,    113765;
  ...
This sequence gives the middle column.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Cf. A002315, A377016, A377017, A377725, A362545 (short legs).

ra[n_]:=ra[n]=Module[{ra},ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{2ra-1,2ra^2-2ra,2ra^2-2ra+1}];exradio={};Do[exradio=Join[exradio,FullSimplify[ra[n]]],{n,0,10}];exradio

a(n) = 2 * A002315(n) * (A002315(n) - 1).

A362540 Number of chordless cycles of length >= 4 in the n-flower graph.

Original entry on oeis.org

3, 23, 63, 127, 273, 583, 1287, 2975, 6993, 16535, 39525, 95071, 229029, 552199, 1332375, 3215807, 7762611, 18739607, 45240309, 109217983, 263673699, 636563527, 1536798717, 3710157407, 8957109801, 21624374039, 52205854257, 126036078751, 304278008331, 734592089095

Offset: 2

Eric W. Weisstein, Apr 24 2023

The n-flower graph can be defined for n >= 3 without multiple edges. It is a snark for odd n >= 5. The sequence has been extended to n=2 using the recurrence. - Andrew Howroyd, Apr 26 2023

Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Flower Graph
Index entries for linear recurrences with constant coefficients, signature (4,-5,4,-2,-4,7,-8,6,4,-6,0,1).

Cf. A362545.

LinearRecurrence[{4, -5, 4, -2, -4, 7, -8, 6, 4, -6, 0, 1}, {3, 23, 63, 127, 273, 583, 1287, 2975, 6993, 16535, 39525, 95071}, 20]
CoefficientList[Series[(-3 - 11 x + 14 x^2 + 22 x^3 + 6 x^4 + 68 x^5 - 9 x^6 - 19 x^7 + 25 x^8 - 13 x^9 - 9 x^10 + x^11)/((-1 + x)^3 (1 - x - x^2 - 3 x^3 - 5 x^4 - 3 x^5 - 4 x^6 + 3 x^8 + x^9)), {x, 0, 20}], x]
Table[(1 + (-1)^n)/2 + 2 (-I)^n ChebyshevT[n, I] + 3 (n - 2) n + RootSum[-1 + # + #^3 &, #^n &] + RootSum[1 + # + #^3 &, #^n &], {n, 2, 20}]

From Andrew Howroyd, Apr 26 2023: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - 2*a(n-4) - 4*a(n-5) + 7*a(n-6) - 8*a(n-7) + 6*a(n-8) + 4*a(n-9) - 6*a(n-10) + a(n-12).
G.f.: x^2*(3 + 11*x - 14*x^2 - 22*x^3 - 6*x^4 - 68*x^5 + 9*x^6 + 19*x^7 - 25*x^8 + 13*x^9 + 9*x^10 - x^11)/((1 - x)^3*(1 + x)*(1 - 2*x - x^2)*(1 + x^2 - x^3)*(1 + x^2 + x^3)). (End)
2*a(n) = -2*(-1)^n*A112455(n) +1+6*n^2-12*n+(-1)^n-2*A112455(n)+4*A001333(n). - R. J. Mathar, Feb 18 2024

a(2)-a(4) and a(17) and beyond from Andrew Howroyd, Apr 26 2023

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A377726 Lengths of the long leg of the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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