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%I #15 Mar 30 2012 17:34:12
%S 2,2,2,3,4,4,5,11,15,7,22,42,11,56,176,13,79,299,17,137,697,19,172,
%T 988,23,254,1794,29,407,3683,31,466,4526,37,667,7807,41,821,10701,43,
%U 904,12384,47,1082,16262,53,1379,23479,59,1712,32568,61,1831,36051,67,2212
%N Triple Peano sequence: a list of triples (x,y,z) starting at (1,1,1); then x'=x+1, y'=y+x, z'=z+y, for x only ranging over the primes
%C a(3k+1) are the primes (A000040), by definition.
%C a(3k+2) are A072205. Second terms are (n^2+n+2)/2 by induction (for n prime).
%C a(3k) are A072206. Third terms are (n^3+5*n+6)/6 by induction (for n prime).
%e x'=x+1=1+1=2, y'=y+x=1+1=2, z'=z+y=1+1=2.
%t seq[n_Integer?Positive] := Module[{fn01 = 1, fn10 = 1, fnout = 1}, Do[{fn10, fn01, fnout} = {fn10 + 1, fn01 + fn10, fn01 + fnout}, {n - 1}]; {fn10, fn01, fnout}]; Flatten[ Table[ seq[ Prime[n]], {n, 1, 100}]]
%o (PARI) a(n)=subst([x,x*(x-1)/2+1,(x^3-3*x^2+8*x)/6],x, prime(1+(n-1)\3))[1+(n-1)%3]
%Y Cf. A072205 & A072206.
%K nonn
%O 1,1
%A _Roger L. Bagula_, Jun 17 2002
%E Edited by _Robert G. Wilson v_, Jul 03 2002
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