A157240 a(n) = A128018(n) + 1.

2, -1, -7, -7, 17, 65, 65, -127, -511, -511, 1025, 4097, 4097, -8191, -32767, -32767, 65537, 262145, 262145, -524287, -2097151, -2097151, 4194305, 16777217, 16777217, -33554431, -134217727, -134217727, 268435457, 1073741825

Offset: 0

Author

Creighton Dement, Feb 25 2009

Keywords

• easy
• sign

Comments

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.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-6,4).

Crossrefs

Cf. A128018, A157241.

Programs

Mathematica

LinearRecurrence[{3,-6,4},{2,-1,-7},40] (* Harvey P. Dale, May 30 2021 *)

PARI

Vec((2 - 7*x + 8*x^2) / ((1 - x)*(1 - 2*x + 4*x^2)) + O(x^35)) \\ Colin Barker, May 22 2019

Formula

G.f.: (2-7*x+8*x^2) / ((1-x)*(4*x^2-2*x+1)).

From Colin Barker, May 22 2019: (Start)

a(n) = (2 + (1-i*sqrt(3))^(1+n) + (1+i*sqrt(3))^(1+n)) / 2 where i=sqrt(-1).

a(n) = 3*a(n-1) - 6*a(n-2) + 4*a(n-3) for n>2.

(End)

a(n) = A138230(n+1)+1. - R. J. Mathar, Sep 11 2019