Search: a006243 -id:a006243
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A112845
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Recurrence a(n) = a(n-1)^3 - 3*a(n-1) with a(0) = 6.
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+10
7
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OFFSET
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0,1
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COMMENTS
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Identical to A006243 apart from the initial term. For some general remarks on this recurrence see A001999. - Peter Bala, Nov 13 2012
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LINKS
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FORMULA
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a(n) = -2*cos(3^n*arccos(-3)).
a(n) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n).
Product {n = 0..inf} (1 + 2/(a(n) - 1)) = sqrt(2).
(End)
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MATHEMATICA
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RecurrenceTable[{a[n] == a[n - 1]^3 - 3*a[n - 1], a[0] == 6}, a, {n,
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A006242
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Extracting a square root.
(Formerly M4758)
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+10
4
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OFFSET
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1,1
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REFERENCES
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Jeffrey Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(1) = 10, a(n) = a(n-1)^3 - 3*a(n-1) [From Escott]. - Sean A. Irvine, Feb 08 2017
a(n) = (5 + 2*sqrt(6))^(3^(n-1)) + (5 - 2*sqrt(6))^(3^(n-1)). - Bruno Berselli, Feb 10 2017
a(n) = 2*T(3^(n-1),5), where T(n,x) deotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Mar 29 2022
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MATHEMATICA
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RecurrenceTable[{a[1]==10, a[n]==a[n-1]^3 - 3 a[n-1]}, a, {n, 8}] (* Vincenzo Librandi, Feb 09 2017 *)
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PROG
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(Magma) [n eq 1 select 10 else Self(n-1)^3-3*Self(n-1): n in [1..5]]; // Vincenzo Librandi, Feb 09 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A282180
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a(n+1) = a(n)*(a(n)^2 - 3) with a(0) = 8.
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+10
1
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = (4 + sqrt(15))^(3^n) + (4 - sqrt(15))^(3^n). - Bruno Berselli, Feb 10 2017
a(n) = -2*cos(3^n * arccos(-4)). - Daniel Suteu, Feb 10 2017
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MATHEMATICA
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RecurrenceTable[{a[0] == 8, a[n] == a[n-1]^3 - 3 a[n-1]}, a, {n, 8}]
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PROG
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(Magma) [n eq 1 select 8 else Self(n-1)^3 - 3*Self(n-1): n in [1..6]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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