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%I #18 Jan 03 2019 05:35:32
%S 5,17,0,0,65,0,0,29,65,185,0,0,169,0,0,0,221,333,0,0,273,0,0,0,157,
%T 481,0,0,1189,0,0,641,1353,629,0,0,1517,0,0,425,1681,777,0,0,1845,0,0,
%U 0,205,925,0,0,2173,0,0,0,2337,1073,0,0,2501,0,0,0,2665,1221,0,0,2829,0,0,1405,2993,1369,0
%N Least hypotenuse, c, of a Primitive Pythagorean Triangle (PPT) such that the difference between it, c, and its greater leg, b, is n; or 0 if no such PPT exists.
%C n = 1, 2, 5, 8, 9, 10, 13, 17, 18, 21, 25, ..., satisfies the first criterion;
%C a(n) = 0 for n = 3, 4, 6, 7, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, ..., ;
%C a(n) = 0 for 5832 of the first 10000 terms;
%C a(8n) = 0 for 832 of the first 10000 terms;
%C a(8n) = 0 for n: 2, 3, 6, 7, 8, 10, 11, 12, 14, 15, 18, 19, 22, 23, 24, ..., ;
%C a(8n+1) > 0;
%C a(8n+2) > 0; a linear 2nd-order recurrence: a(n) = 2*a(n-1) - a(n-2) with a(1) = 185 & a(2) = 333;
%C a(8n+3) = 0;
%C a(8n+4) = 0;
%C a(8n+5) > 0;
%C a(8n+6) = 0;
%C a(8n+7) = 0;
%C Prime terms: 5, 17, 29, 157, 641, 3821, 4201, 17749, 21601, 31981, 38273, 44789, 61129, 66173, 72161, 100673, 108541, 114553, 121421, 142973, 165541, 173777, 182141, 204733, 213881, 225889, 235493, 281837, ..., .
%H Ron Knott and Robert G. Wilson v, <a href="/A277534/b277534.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%e a(1) is 5 since the PPT (3,4,5) satisfies the first stated criterion; a(2) is 17 since the PPT (8,15,17) satisfies the first stated criterion; a(3) = 0 since there exists no PPT that satisfies the stated criteria; etc.
%t f[n_] := FindInstance[ a^2 + b^2 == c^2 && Mod[c, 4] == 1 && 0 < a < b < c && c - b == n, {a, b, c}, Integers][[1, 3, 2, 1, 1, 3]] + 1 /. 1 + {}[[1, 3, 2, 1, 1, 3]] -> 0; f[1] = 5; Array[f, 75]
%Y Cf. A008846, A020882, A242219.
%K nonn,easy
%O 1,1
%A _Ron Knott_ and _Robert G. Wilson v_, Jun 05 2014
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