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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 Harvey P. Dale, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (9,-9,1). 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 Cf. A095003, A095004, A076765. Sequence in context: A058859 A291964 A333094 * A293527 A080833 A073516 Adjacent sequences: A094999 A095000 A095001 * A095003 A095004 A095005 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.)