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A106709
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First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-2],[1,5]] and v is the column vector [0,1].
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2
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0, -2, -10, -46, -210, -958, -4370, -19934, -90930, -414782, -1892050, -8630686, -39369330, -179585278, -819187730, -3736768094, -17045465010, -77753788862, -354678014290, -1617882493726, -7380056440050, -33664517212798, -153562473183890, -700483331493854
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OFFSET
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0,2
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COMMENTS
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Real Pisot roots (the eigenvalues of M): 0.438447, 4.56155.
Let t(n,k) denote the k-th element of row n of Stern's triangle (see A337277). Then b(n) = Sum_k T(n,k)*T(n,k+1) gives the present sequence (without the signs). - N. J. A. Sloane, Nov 19 2020
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LINKS
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Table of n, a(n) for n=0..23.
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111.
Index entries for linear recurrences with constant coefficients, signature (5,-2).
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FORMULA
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a(n)=first entry of v[n], where v[n]=Mv[n-1], M is the 2 X 2 matrix [[0, -2], [1, 5]] and v[0] is the column vector [0,1].
G.f.: -2*x/(2*x^2-5*x+1). - Alois P. Heinz, Nov 26 2020
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MAPLE
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with(linalg): M:=matrix(2, 2, [0, -2, 1, 5]): v[0]:=matrix(2, 1, [0, 1]): for n from 1 to 22 do v[n]:=multiply(M, v[n-1]) od: seq(v[n][1, 1], n=0..22);
# second Maple program:
a:= n-> (<<0|-2>, <1|5>>^n)[1, 2]:
seq(a(n), n=0..25); # Alois P. Heinz, Nov 19 2020
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MATHEMATICA
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M = {{0, -2}, {1, 5}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}]
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CROSSREFS
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Equals -2*A107839(n-1), n>0.
Cf. A337277.
Sequence in context: A080643 A032389 A290923 * A204091 A221196 A137193
Adjacent sequences: A106706 A106707 A106708 * A106710 A106711 A106712
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KEYWORD
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sign
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AUTHOR
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Roger L. Bagula, May 30 2005
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EXTENSIONS
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Edited by N. J. A. Sloane, Apr 30 2006
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STATUS
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approved
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