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A095002 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 3, a(3) = 19. 3
1, 3, 19, 145, 1137, 8947, 70435, 554529, 4365793, 34371811, 270608691, 2130497713, 16773373009, 132056486355, 1039678517827, 8185371656257, 64443294732225, 507360986201539, 3994444594880083, 31448195772839121, 247591121587832881, 1949280776929823923 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A companion to A095003, A005004; a(n)/a(n-1) tending to 4 + sqrt(15).
a(n)/a(n-1) tends to C = 4 + sqrt(15); C having the property that C + 1/C = 8. Eigenvalues of M (1, C, 1/C) are roots to x^3 - 9x^2 + 9x - 1.
LINKS
FORMULA
Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [a(n) A095003(n) A095004(n)].
From R. J. Mathar, Aug 22 2008: (Start)
O.g.f.: x*(1-6x+x^2)/((1-x)*(1-8x+x^2)).
a(n) = (2 + A001090(n+1) - 7*A001090(n))/3. (End)
EXAMPLE
a(4) = 145 = 9*19 - 9*3 + 1.
a(4) = 145, leftmost term in M^4 * [1 0 0] = [145 352 640].
MAPLE
a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1$2]:
seq(a(n), n=1..23); # Alois P. Heinz, Jun 06 2021
MATHEMATICA
a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0},
{0}})[[1, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)
nxt[{a_, b_, c_}]:={b, c, 9c-9b+a}; NestList[nxt, {1, 3, 19}, 30][[All, 1]] (* Harvey P. Dale, Sep 02 2022 *)
PROG
(PARI) Vec(x*(1-6*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^20)) \\ Michel Marcus, Mar 21 2015
CROSSREFS
Sequence in context: A058859 A291964 A333094 * A293527 A080833 A073516
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, May 27 2004
EXTENSIONS
Edited and extended by Robert G. Wilson v, May 29 2004
Edited by Georg Fischer, Jun 06 2021
STATUS
approved

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Last modified June 20 18:51 EDT 2024. Contains 373532 sequences. (Running on oeis4.)