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A117474 The values of 'a' in a^2 + b^2 = c^2 where b - a = 7 and gcd(a,b,c)=1.

%I A117474 #20 May 07 2025 19:12:59
%S A117474 5,8,48,65,297,396,1748,2325,10205,13568,59496,79097,346785,461028,
%T A117474 2021228,2687085,11780597,15661496,68662368,91281905,400193625,
%U A117474 532029948,2332499396,3100897797,13594802765,18073356848
%N A117474 The values of 'a' in a^2 + b^2 = c^2 where b - a = 7 and gcd(a,b,c)=1.
%C A117474 The values of 'c' are in A060569.
%H A117474 Harvey P. Dale, <a href="/A117474/b117474.txt">Table of n, a(n) for n = 1..1000</a>
%H A117474 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-1,1).
%F A117474 a(n) = 6*a(n-2) - a(n-4) + 14; a(1)=5, a(2)=8, a(3)=48, a(4)=65.
%F A117474 G.f.: x*(3*x^4 + x^3 - 10*x^2 - 3*x-5) / ((x-1)*(x^2-2*x-1)*(x^2+2*x-1)). [_Colin Barker_, Dec 17 2012]
%e A117474 a(5) = 6*48 - 5 + 14 = 297, 297^2 + 304^2 = 425^2, 304 - 297 = 7, and gcd(297, 304, 425) = 1.
%p A117474 g:=proc(n) option remember; if n=1 then RETURN(5) fi; if n=2 then RETURN(8) fi; if n=3 then RETURN(48) fi; if n=4 then RETURN(65) fi; 6*g(n-2)-g(n-4)+14; end; # _N. J. A. Sloane_, Oct 06 2007
%t A117474 LinearRecurrence[{1,6,-6,-1,1},{5,8,48,65,297},40] (* _Harvey P. Dale_, May 07 2025 *)
%Y A117474 Cf. A060569.
%K A117474 nonn,easy
%O A117474 1,1
%A A117474 Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 19 2006