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 A208854 Array of odd catheti of primitive Pythagorean triangles when read by SW-NE diagonals. 4
 3, 5, 15, 21, 7, 35, 45, 9, 0, 63, 77, 33, 11, 55, 99, 117, 65, 13, 39, 91, 143, 165, 105, 0, 15, 0, 0, 195, 221, 153, 85, 17, 51, 119, 187, 255, 285, 209, 133, 57, 19, 95, 171, 247, 323, 357, 273, 0, 105, 21, 0, 0, 231, 0, 399 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS See the comments, reference and links in A208853. The present array is a(n,m) = abs((2*n-1)^2 - (2*m)^2) if gcd((2*n-1,2*m)=1 and 0 otherwise. Put u=2*n-1 and v=2*m. The array read by SW-NE diagonals is T(n,m):=a(n-m+1,m), n>=m>=1. All primitive Pythagorean triples are given by (a(n,m),b(n,m):=A208855(n,m), c(n,m):=A208853(n,m)), n>=1, m>=1. If the entry is (0,0,0) there is no primitive Pythagorean triple for these n and m values. See the example section of A208853 for the array of triples. Every odd number a=2*k+1, k>=1, appears at least in one primitive triple, namely in (2*k+1, 4*T(k),4*T(k)+1), with the triangular numbers T(k) := A000217(k). This a-value is a=u^2-v^2 with (u,v)=(k+1,k). It may appear in other primitive triples. E.g. a=33=2*16+1 appears in (u,v)=(17,16) ((n,m)= (9,8)) as (33,544,545), and also in (33,56,65) with (n,m)=(4,2) (maybe others). LINKS Table of n, a(n) for n=1..55. FORMULA T(n,m)=a(n-m+1,m), n>=m>=1, with a(n,m):=abs((2*n-1)^2 - (2*m)^2) if gcd((2*n-1,2*m)=1 and 0 otherwise. EXAMPLE Array a(n,m): .....m| 1 2 3 4 5 6 7 8 9 10 .....v| 2 4 6 8 10 12 14 16 18 20 n, u 1, 1 3 15 35 63 99 143 195 255 323 399 2, 3 5 7 0 55 91 0 187 247 0 391 3, 5 21 9 11 39 0 119 171 231 299 0 4, 7 45 33 13 15 51 95 0 207 275 351 5, 9 77 65 0 17 19 0 115 175 0 319 6, 11 117 105 85 57 21 23 75 135 203 279 7, 13 165 153 133 105 69 25 27 87 155 231 8, 15 221 209 0 161 0 0 29 31 0 0 9, 17 285 273 253 225 189 145 93 33 35 111 10,19 357 345 325 297 261 217 165 105 37 39 ... Triangle T(n,m): .....m| 1 2 3 4 5 6 7 8 9 10 .....v| 2 4 6 8 10 12 14 16 18 20 n, u 1, 1 3 2, 3 5 15 3, 5 21 7 35 4, 7 45 9 0 63 5, 9 77 33 11 55 99 6 11 117 65 13 39 91 143 7, 13 165 105 0 15 0 0 195 8, 15 221 153 85 17 51 119 187 255 9, 17 285 209 133 57 19 95 171 247 323 10,19 357 273 0 105 21 0 0 231 0 399 ... For the array of triples see the example section of A208853. CROSSREFS Cf. A208853, A208855. Sequence in context: A290297 A344991 A063185 * A165260 A201874 A059528 Adjacent sequences: A208851 A208852 A208853 * A208855 A208856 A208857 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Mar 05 2012 STATUS approved

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Last modified September 22 05:50 EDT 2023. Contains 365519 sequences. (Running on oeis4.)