cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A051277 Coefficients in 7-adic expansion of sqrt(2).

Original entry on oeis.org

3, 1, 2, 6, 1, 2, 1, 2, 4, 6, 6, 2, 1, 1, 0, 2, 1, 1, 4, 6, 1, 3, 2, 6, 6, 3, 5, 5, 6, 3, 4, 5, 0, 1, 6, 3, 0, 4, 6, 2, 4, 4, 6, 4, 2, 4, 4, 2, 6, 1, 3, 4, 1, 3, 1, 4, 2, 6, 6, 0, 3, 5, 5, 1, 1, 2, 0, 6, 6, 1, 1, 2, 4, 4, 4, 2, 3, 6, 6, 3, 6, 1, 4, 4, 2, 2, 1, 3
Offset: 0

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Author

N. J. A. Sloane

Keywords

Examples

			3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + ...

References

  • Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 76.

Links

Crossrefs

Cf. A034945, A290558.

Programs

  • Maple
     t := proc(n) option remember; if n = 1 then 3 else irem(t(n-1)^7 - 7*t(n-1)^5 + 14*t(n-1)^3 - 7*t(n-1), 7^n) end if; end:
convert(t(100), base, 7); # Peter Bala, Nov 20 2022
  • PARI
    Vecrev(digits(lift(sqrt(2+O(7^99))),7)) \\ Joerg Arndt, Aug 05 2017

Formula

Equals the 7-adic limit as n -> oo of 2*T(7^n,3/2) = the 7-adic limit as n -> oo of ((3 + sqrt(5))/2)^(7^n) + ((3 - sqrt(5))/2)^(7^n), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Nov 20 2022

Extensions

Missing terms=0 inserted by Seiichi Manyama, Aug 04 2017

A290557 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(2). These are the numbers congruent to 3 mod 7 (except for the initial 0).

Original entry on oeis.org

0, 3, 10, 108, 2166, 4567, 38181, 155830, 1802916, 24862120, 266983762, 1961835256, 5916488742, 19757775943, 116646786350, 116646786350, 9611769806236, 42844700375837, 275475214363044, 6789129606004840, 75182500718243698, 154974767015855699
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2017

Keywords

Comments

x = ...216213,
x^2 = ...000002 = 2.

Examples

			a(1) = (    3)_7 = 3,
a(2) = (   13)_7 = 10,
a(3) = (  213)_7 = 108,
a(4) = ( 6213)_7 = 2166,
a(5) = (16213)_7 = 4567.

Links

Crossrefs

Cf. A034945, A051277, A290559.

Programs

Formula

a(0) = 0 and a(1) = 3, a(n) = a(n-1) + (a(n-1)^2 - 2) mod 7^n for n > 1.
a(n) == 2*T(7^n, 3/2) (mod 7^n) == ((3 + sqrt(5))/2)^(7^n) + ((3 - sqrt(5))/2)^(7^n) (mod 7^n), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Dec 03 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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