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A326825 a(n) is the number of iterations needed to reach 1 or 5 starting at n and using the map k -> (k/2 if k is even, otherwise k + (smallest triangular number > k)). Set a(n) = -1 if the trajectory never reaches 1 or 5.

%I A326825 #32 May 11 2025 11:34:22
%S A326825 0,1,6,2,0,7,7,3,5,1,12,8,10,8,8,4,6,6,4,2,15,13,58,9,56,11,52,9,36,9,
%T A326825 15,5,28,7,13,7,20,5,18,3,18,16,16,14,59,59,59,10,14,57,57,12,55,53,
%U A326825 51,10,22,37,55,10,24,16,22,6,35,29,14,8,27,14,12,8,10,21,23,6,27,19
%N A326825 a(n) is the number of iterations needed to reach 1 or 5 starting at n and using the map k -> (k/2 if k is even, otherwise k + (smallest triangular number > k)). Set a(n) = -1 if the trajectory never reaches 1 or 5.
%C A326825 It is conjectured that this algorithm will always terminate at 1 or 5.
%C A326825 _Jim Nastos_ verified the conjecture for n <= 354999.
%C A326825 The conjecture holds for all n <= 10^9. - _Jon E. Schoenfield_, Oct 20 2019
%H A326825 M. F. Hasler, <a href="/A326825/b326825.txt">Table of n, a(n) for n = 1..1000</a>, May 08 2025
%F A326825 For n = 11: 11+15 = 26; 26/2 = 13; 13+15 = 28; 28/2 = 14; 14/2 = 7; 7+10 = 17; 17+21 = 38; 38/2 = 19; 19+21 = 40; 40/2 = 20; 20/2 = 10; 10/2 = 5; taking 12 steps to reach 5, so a(11) = 12.
%o A326825 (PARI) M326825=Map([1, 0; 5, 0]); apply( {A326825(n)=if(mapisdefined(M326825, n, &n), n, mapput(M326825, n, 1+n=A326825(if(n%2, n+binomial((sqrtint(8*n+8)+3)\2, 2), n\2))); 1+n)}, [1..77]) \\ M. F. Hasler, May 08 2025
%Y A326825 Cf. A006577, A326823 (similar with squares instead of triangular numbers).
%K A326825 nonn
%O A326825 1,3
%A A326825 _Ali Sada_, Oct 20 2019
%E A326825 Edited by _Jon E. Schoenfield_ and _N. J. A. Sloane_, Oct 20 2019