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 A095003 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3). 2
 1, 6, 45, 352, 2769, 21798, 171613, 1351104, 10637217, 83746630, 659335821, 5190939936, 40868183665, 321754529382, 2533168051389, 19943589881728, 157015551002433, 1236180818137734, 9732430994099437, 76623267134657760, 603253706083162641, 4749406381530643366 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n)/a(n-1) tends to 7.87298... = 4 + sqrt(15) = C (having the property that C + 1/C = 8). Eigenvalues of M are C, 1/C, 1; being roots of x^3 - 9x^2 + 9x - 1. LINKS Table of n, a(n) for n=1..22. Index entries for linear recurrences with constant coefficients, signature (9,-9,1). FORMULA a(n+3) = 9*a(n+2) - 9*a(n+1) + a(n); given a(1) = 1, a(2) = 6, a(3) = 45. Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [A095002(n) a(n) A095004(n)]. EXAMPLE a(4) = 352 since 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}})[[2, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *) LinearRecurrence[{9, -9, 1}, {1, 6, 45}, 30] (* Harvey P. Dale, Nov 12 2022 *) CROSSREFS Cf. A095002, A095004, A076765. Sequence in context: A024077 A097129 A048663 * A004988 A257633 A371779 Adjacent sequences: A095000 A095001 A095002 * A095004 A095005 A095006 KEYWORD nonn AUTHOR Gary W. Adamson, May 27 2004 EXTENSIONS Edited and extended by Robert G. Wilson v, May 29 2004 Definition corrected and edited by Georg Fischer, Jun 06 2021 STATUS approved

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Last modified June 20 23:53 EDT 2024. Contains 373535 sequences. (Running on oeis4.)