This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A198586 #51 Oct 24 2024 03:12:11
%S A198586 5,85,341,5461,21845,349525,1398101,22369621,89478485,1431655765,
%T A198586 5726623061,91625968981,366503875925,5864062014805,23456248059221,
%U A198586 375299968947541,1501199875790165,24019198012642645,96076792050570581,1537228672809129301
%N A198586 a(n) = (4^A001651(n+1) - 1)/3: numbers (4^k-1)/3 for k > 1, not multiples of 3.
%C A198586 Numbers coprime to 6 producing 2 odd numbers in the Collatz iteration.
%C A198586 Numbers appearing in A198585 (sorted and duplicates removed). These numbers occur in A002450, numbers of the form (4^k-1)/3, for k = 2, 4, 5, 7, 8, 10, ... (note that k a multiple of 3 does not appear).
%C A198586 A124477 \ {0,1} is a subset: for these n, 3n+1 = 2^(p-3) with p > 3 prime, whence also n !== 0 (mod 3). - _M. F. Hasler_, Oct 16 2018
%C A198586 These are exactly the odd non-multiples of 3 such 3n+1 = 2^m for some m, i.e., n = (2^m-1)/3. This is possible iff m = 2k, so we get n = (4^k-1)/3. Then n == 0 (mod 3) <=> 4^k == 1 (mod 9) <=> k == 0 (mod 3) <=> k not in A001651. This yields the FORMULA. (Multiples of 3 are excluded because the original definition implied that the terms are in the Collatz-orbit of another odd number, i.e., of the form n = (3x+1)/2^r, which is impossible for x a multiple of 3.) - _M. F. Hasler_, Oct 16 2018
%C A198586 From _Wolfdieter Lang_, Jan 14 2022: (Start)
%C A198586 a(n) mod 8 = 5. As subsequence of A002450 for n >= 1.
%C A198586 {a(n) mod 6} == repeat{5, 1}. See the first comment, and the periodicity modulo 6 of A002450 for n >= 1.
%C A198586 {a(n) mod 72} == repeat{5, 13, 53, 61, 29, 37}. Proof by induction: First with the bisection formulas, a(1+2*k) = (4^(2+3*k) - 1)/3 and a(2+2*k) = (4^(3*k+4) - 1)/3, for k >= 0, then trisection, using (4^9 - 1)/3 = 873819 = 9*9709. (End)
%H A198586 T. D. Noe, <a href="/A198586/b198586.txt">Table of n, a(n) for n = 1..100</a>
%H A198586 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,64,-64).
%F A198586 a(n) = (4^A001651(n+1) - 1)/3. - _M. F. Hasler_, Oct 16 2018
%F A198586 From _Colin Barker_, Jan 17 2020: (Start)
%F A198586 G.f.: x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)).
%F A198586 a(n) = a(n-1) + 64*a(n-2) - 64*a(n-3) for n>3.
%F A198586 a(n) = (-1 + (-8)^n + 3*8^n) / 3.
%F A198586 (End)
%t A198586 e = 19; ex = Complement[Range[2,3*e], 3*Range[e]]; (4^ex - 1)/3
%t A198586 (* Second program: *)
%t A198586 Rest@ Map[(4^# - 1)/3 &, LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 21]] (* _Michael De Vlieger_, Oct 17 2018 *)
%o A198586 (PARI) is(n)=gcd(n,6)==1&&(n=3*n+1)>>valuation(n,2)==1 \\ _M. F. Hasler_, Oct 16 2018
%o A198586 (PARI) A198586(n)=4^(3*n\2+1)\3 \\ _M. F. Hasler_, Oct 16 2018
%o A198586 (PARI) Vec(x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)) + O(x^20)) \\ _Colin Barker_, Jan 17 2020
%o A198586 (Magma) [4^(3*n div 2 + 1) div 3: n in [1..25]]; // _Vincenzo Librandi_, Oct 20 2018
%Y A198586 Cf. A001651, A002450, A124477, A198584.
%K A198586 nonn,easy
%O A198586 1,1
%A A198586 _T. D. Noe_, Oct 30 2011
%E A198586 Definition corrected by _M. F. Hasler_, Oct 16 2018