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Revision History for A014176

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A014176

Decimal expansion of the silver mean, 1+sqrt(2).
(history; published version)

#122 by N. J. A. Sloane at Mon Mar 25 21:39:51 EDT 2024

STATUS

proposed

approved

#121 by Peter Bala at Mon Mar 25 10:31:58 EDT 2024

STATUS

editing

proposed

#120 by Peter Bala at Mon Mar 25 10:31:52 EDT 2024

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.

FORMULA

An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25 , by setting n = 1/2 and x = 8*k + 6 for k >= 0.

For example, taking k = 0 and k = 1 yields

For example, taking k = 0 and k = 1 yieldssqrtsqrt(2) + 1 = 15/(6 + (1*3)/(12 + (5*7)/(12 + (9*11)/(12 + (13*15)/(12 + ... + (4*n + 1)*(4*n + 3)/(12 + ... )))))) and

#119 by Peter Bala at Mon Mar 25 09:53:03 EDT 2024

FORMULA

From Peter Bala, Mar 24 2024: (Start)

An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25 by setting n = 1/2 and x = 8*k + 6 for k >= 0.

For example, taking k = 0 and k = 1 yieldssqrt(2) + 1 = 15/(6 + (1*3)/(12 + (5*7)/(12 + (9*11)/(12 + (13*15)/(12 + ... + (4*n + 1)*(4*n + 3)/(12 + ... )))))) and

sqrt(2) + 1 = (715/21) * 1/(14 + (1*3)/(28 + (5*7)/(28 + (9*11)/(28 + (13*15)/(28 + ... + (4*n + 1)*(4*n + 3)/(28 + ... )))))). (End)

STATUS

approved

editing

#118 by R. J. Mathar at Fri Jan 12 07:10:35 EST 2024

STATUS

editing

approved

#117 by R. J. Mathar at Fri Jan 12 07:10:31 EST 2024

CROSSREFS

Apart from initial digit the same as A002193.

Cf. A002193, A000032, A006497, A080039, A179260, A121601.

STATUS

approved

editing

#116 by Wolfdieter Lang at Fri Nov 10 13:01:39 EST 2023

STATUS

editing

approved

#115 by Wolfdieter Lang at Fri Nov 10 13:01:35 EST 2023

CROSSREFS

Cf. A000129, A049310.

STATUS

approved

editing

#114 by Wolfdieter Lang at Fri Nov 10 13:00:32 EST 2023

STATUS

editing

approved

#113 by Wolfdieter Lang at Fri Nov 10 12:59:11 EST 2023

FORMULA

From Wolfdieter Lang, Nov 10 2023:(Start)

Equals lim_{n->oo} A000129(n+1)/A000129(n) (see A000129, Pell). - _Wolfdieter Lang_, Nov 02 2023

Equals lim_{n->oo} S(n+1, 2*sqrt(2))/S(n, 2*sqrt(2)), with the Chebyshev S(n,x) polynomial (see A049310). (End)

STATUS

approved

editing