A006242 Extracting a square root.

10, 970, 912670090, 760223786832147978143718730, 439363892017598816969702791108195858981800447259539613873486126455827777484460810

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Jeffrey Shallit, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..7
• E. B. Escott, Rapid method for extracting a square root, Amer. Math. Monthly, 44 (1937), 644-646.

Crossrefs

Cf. A006243.

Programs

Magma

[n eq 1 select 10 else Self(n-1)^3-3*Self(n-1): n in [1..5]]; // Vincenzo Librandi, Feb 09 2017

Mathematica

RecurrenceTable[{a[1]==10, a[n]==a[n-1]^3 - 3 a[n-1]}, a, {n, 8}] (* Vincenzo Librandi, Feb 09 2017 *)

Formula

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

Extensions

New offset and a(5) from Sean A. Irvine, Feb 08 2017