OFFSET
1,2
COMMENTS
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-1).
FORMULA
M = a matrix having the same eigenvalues as the roots of the characteristic polynomial of A095125 and A095126: (x^3 - 2x^2 - 3x + 1). Then M^n * [1 1 1] = [p q r] where q = a(n) and p, r, are offset members of the same sequence.
G.f.: x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3). - Colin Barker, Aug 31 2019
EXAMPLE
a(7) = 751 = 2*a(6) + 3*a(5) - a(4) = 2*259 + 3*88 - 31.
a(4) = 31 = center term in M^4 * [1 1 1] = [10 31 88].
MATHEMATICA
a[1] = 1; a[2] = 4; a[3] = 10; a[n_] := a[n] = 2a[n - 1] + 3a[n - 2] - a[n - 3]; Table[ a[n], {n, 25}] (* Robert G. Wilson v, Jun 01 2004 *)
nxt[{a_, b_, c_}]:={b, c, 2c+3b-a}; NestList[nxt, {1, 4, 10}, 30][[All, 1]] (* or *) LinearRecurrence[{2, 3, -1}, {1, 4, 10}, 30] (* Harvey P. Dale, Feb 08 2022 *)
PROG
(PARI) Vec(x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3) + O(x^30)) \\ Colin Barker, Aug 31 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, May 29 2004
EXTENSIONS
Edited, corrected and extended by Robert G. Wilson v, Jun 01 2004
STATUS
approved