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A077034 a(1)=3; a(2n), a(2n+1) are smallest integers > a(2n-1) such that a(2n-1)^2+a(2n)^2=a(2n+1)^2.

%I A077034 #18 Jan 14 2023 10:53:58
%S A077034 3,4,5,12,13,84,85,132,157,12324,12325,15960,20165,26280,33125,79500,
%T A077034 86125,95400,128525,152040,199085,477804,517621,871500,1013629,
%U A077034 513721874820,513721874821,4351526469072,4381745402885,10516188966924,11392538047501
%N A077034 a(1)=3; a(2n), a(2n+1) are smallest integers > a(2n-1) such that a(2n-1)^2+a(2n)^2=a(2n+1)^2.
%C A077034 Note that each time two more terms are added simultaneously. The sequence is infinite.
%C A077034 Smallest sequence of Pythagorean triples {a(k-1),a(k),a(k+1)},with k=2n,such that the hypotenuse of one triangle is the short leg of the next one. Such a sequence is called 2-prime Pythagorean because only the first two triangles (3,4,5),(5,12,13) both have prime hypotenuse and short leg. The next such sequence is given by A076604. Actually, the starting terms for all 2-prime and 3-prime Pythagorean triangles are given respectively by A048270 and A048295. The starting term for the smallest n-prime Pythagorean triangle is A105318. - _Lekraj Beedassy_, Sep 16 2005
%C A077034 a(2n) <= (a(2n-1)^2-1)/2; a(2n+1) <= (a(2n-1)^2+1)/2. [_Max Alekseyev_, May 11 2011]
%e A077034 a(1)=3 implies a(2)=4 and a(3)=5: 3^2+4^2=5^2.
%e A077034 a(3)=5 implies a(4)=12 and a(5)=13: 5^2+12^2=13^2.
%Y A077034 Cf. A048270, A048295, A076604, A105318.
%K A077034 nonn
%O A077034 1,1
%A A077034 _Zak Seidov_, Oct 21 2002
%E A077034 More terms from _Max Alekseyev_, May 11 2011