Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #10 May 30 2021 15:27:10
%S 2,-1,-7,-7,17,65,65,-127,-511,-511,1025,4097,4097,-8191,-32767,
%T -32767,65537,262145,262145,-524287,-2097151,-2097151,4194305,
%U 16777217,16777217,-33554431,-134217727,-134217727,268435457,1073741825
%N a(n) = A128018(n) + 1.
%C Generating floretion is Y = .5('i + 'j + 'k + i' + j' + k') + ee. ("tes"). Note: A current conjecture is that if X is a floretion for which 4*tes(X^n) is an integer for all n, then X+sigma(X) also has this property. "sigma" is the uniquely defined projection operator which "flips the arrows" of a floretion (i.e. sigma('i) = i', sigma('j) = j', etc.). Taking X = .5('i + 'j + 'k + ee), then tesseq(X) = [ -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, ...] is an integer sequence, thus by the conjecture 4*tes(Y^n) = 4*tes((X+sigma)^n) should also be an integer sequence for all n.
%H Colin Barker, <a href="/A157240/b157240.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,4).
%F G.f.: (2-7*x+8*x^2) / ((1-x)*(4*x^2-2*x+1)).
%F From _Colin Barker_, May 22 2019: (Start)
%F a(n) = (2 + (1-i*sqrt(3))^(1+n) + (1+i*sqrt(3))^(1+n)) / 2 where i=sqrt(-1).
%F a(n) = 3*a(n-1) - 6*a(n-2) + 4*a(n-3) for n>2.
%F (End)
%F a(n) = A138230(n+1)+1. - _R. J. Mathar_, Sep 11 2019
%t LinearRecurrence[{3,-6,4},{2,-1,-7},40] (* _Harvey P. Dale_, May 30 2021 *)
%o (PARI) Vec((2 - 7*x + 8*x^2) / ((1 - x)*(1 - 2*x + 4*x^2)) + O(x^35)) \\ _Colin Barker_, May 22 2019
%Y Cf. A128018, A157241.
%K easy,sign
%O 0,1
%A _Creighton Dement_, Feb 25 2009