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1, 10, 81, 640, 5041, 39690, 312481, 2460160, 19368801, 152490250, 1200553201, 9451935360, 74414929681, 585867502090, 4612525087041, 36314333194240, 285902140466881, 2250902790540810, 17721320183859601, 139519658680336000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A sequence derived from A076765, with a(n)/a(n-1) tending to 4 + sqrt(15).
a(n)/a(n-1) tends to C = 4 + sqrt(15) = 7.87298334...(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.
This is the r=10 member of the r-family of sequences S_r(n), n>=1, defined in A092184 where more information can be found.
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LINKS
| Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n+3) = 9*a(n+2) - 9*a(n+1) + 1; given a(1) = 1, a(2) = 10, a(3) = 81. Let M = the 3 X 3 symmetric matrix [1 1 1 / 1 2 3 / 1 3 6]; then M^n * [1 0 0] = [A095002(a) A095003(3) A095004(a)].
a(n)= (T(n, 4)-1)/3 with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n). a(0):=0. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004
G.f.: x*(1+x)/((1-x)*(1-8*x+x^2)) = x*(1+x)/(1-9*x+9*x^2-x^3).
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EXAMPLE
| a(4) = 640 = 568 + 72 = A076765(3) + A076765(2).
a(4) = 640 = 9*81 - 9*10 + 1.
a(4) = 640, rightmost term in M^4 * [1 0 0]: [145 352 640] = [A095002(4) A095003(4) A095004(4)].
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MATHEMATICA
| a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 20}]; (from Robert G. Wilson v May 29 2004)
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CROSSREFS
| Cf. A095002, A005003, A076765.
Sequence in context: A136870 A018202 A154236 * A037541 A037485 A193327
Adjacent sequences: A095001 A095002 A095003 * A095005 A095006 A095007
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), May 27 2004
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EXTENSIONS
| Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), May 29 2004
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