The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A278711 Triangle T read by rows: T(n, m), for n >= 2, and m=1, 2, ..., n-1, equals the positive integer solution x of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists. 2
 12, 0, 45, 240, 0, 112, 0, 525, 0, 225, 1260, 0, 0, 0, 396, 0, 2205, 0, 1617, 0, 637, 4032, 0, 3520, 0, 2496, 0, 960, 0, 6237, 0, 5265, 0, 0, 0, 1377, 9900, 0, 9100, 0, 0, 0, 5100, 0, 1900, 0, 14157, 0, 12705, 0, 10285, 0, 6897, 0, 2541, 20592, 0, 0, 0, 17136, 0, 13680, 0, 0, 0, 3312, 0, 27885, 0, 25857, 0, 22477, 0, 17745, 0, 11661, 0, 4225, 38220, 0, 36652, 0, 33516, 0, 0, 0, 22540, 0, 14700, 0, 5292, 0, 49725, 0, 47025, 0, 0, 0, 36225, 0, 0, 0, 0, 0, 6525 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS The corresponding triangle with the square root of the positive integer solutions y is A278712. A primitive Pythagorean triangle is characterized by two integers n > m >= 1, gcd(n, m) = 1 and n+m odd. See A249866, also for references. For the one-to-one correspondence between rational Pythagorean triangles with area A > 0 and rational points on the elliptic curve y^2 = x^3 - A^2*x with y not vanishing see Theorem 4.1 of the Keith Conrad link or Theorem 15.6, p. 212, of the Ash-Gross reference. LINKS Table of n, a(n) for n=2..106. Avner Ash and Robert Gross, Elliptic tales : curves, counting, and number theory, Princeton University Press, 2012 Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008 FORMULA T(n, m) = (n^2 - m^2)*n^2 if n > m >= 1, gcd(n, m) = 1 and n+m is odd, and T(n, m) = 0 otherwise. EXAMPLE The triangle T(n, m) begins: n\m 1 2 3 4 5 6 7 8 2: 12 3: 0 45 4: 240 0 112 5: 0 525 0 225 6: 1260 0 0 0 396 7: 0 2205 0 1617 0 637 8: 4032 0 3520 0 2496 0 960 9 0 6237 0 5265 0 0 0 1377 ........................................... n = 10: 9900 0 9100 0 0 0 5100 0 1900, n = 11: 0 14157 0 12705 0 10285 0 6897 0 2541, n = 12: 20592 0 0 0 17136 0 13680 0 0 0 3312, n = 13: 0 27885 0 25857 0 22477 0 17745 0 11661 0 4225, n = 14: 38220 0 36652 0 33516 0 0 0 22540 0 14700 0 5292, n = 15: 0 49725 0 47025 0 0 0 36225 0 0 0 0 0 6525. ... ------------------------------------------- The triangle of solutions [x,y] begins ([0,0] if there is no primitive Pythagorean): n\m 1 2 3 4 2: [12,36] 3: [0,0] [45,225] 4:[240,3600] [0,0] [112,784] 5: [0,0] [525,11025] [0,0] [225, 2025] ... n=6: [1260,44100] [0,0] [0,0] [0,0] [396,4356], n=7: [0,0] [2205,99225] [0,0] [1617,53361] [0.0] [637,8281], n=8: [4032,254016] [0,0] [3520,193600] [0,0] [2496,97344] [0,0] [960,14400], n=9: [0,0] [6237,480249] [0,0] [5265,342225] [0,0] [0,0] [0,0] [1377,23409], n=10: [9900,980100] [0,0] [9100,828100] [0,0] [0,0] [0,0] [5100,260100] [0,0] [1900, 36100]. ... ------------------------------------------- CROSSREFS Cf. A249866, A249869, A278712. Sequence in context: A332053 A225951 A333577 * A331911 A307841 A257949 Adjacent sequences: A278708 A278709 A278710 * A278712 A278713 A278714 KEYWORD nonn,tabl,easy AUTHOR Wolfdieter Lang, Nov 27 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 13 08:31 EDT 2024. Contains 373383 sequences. (Running on oeis4.)