A065445 -id:A065445 - cp's OEIS frontend

A065045 Continued fraction expansion of the constant Product_{k>=0} (1 + 1/2^(2k))^(1/2) (A065445).

1, 1, 1, 1, 4, 1, 10, 1, 3, 3, 1, 2, 10, 1, 1, 1, 13, 1, 18, 1, 1, 2, 16, 1, 223, 1, 2, 3, 2, 6, 56, 1, 3, 5, 3, 14, 1, 8, 1, 1, 8, 135, 8, 1, 1, 2, 6, 1, 3, 2, 5, 6, 3, 1, 2, 1, 12, 1, 1, 72, 2, 2, 1, 4, 1, 10, 1, 2, 14, 1, 2, 1, 1, 2, 6, 2, 10, 9, 47, 1, 9, 3, 3, 1, 2, 5, 2, 1, 5, 1, 1, 3, 15, 1, 2, 2, 8

Offset: 0

Robert G. Wilson v, Nov 19 2001

			1.646760258121065648366051222... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, Oct 04 2009

Harry J. Smith, Table of n, a(n) for n = 0..1999

Cf. A065445 (decimal expansion).

ContinuedFraction[ N[ Product[ Sqrt[ (1 + 1/2^(2k) ) ], {k, 0, Infinity} ], 500 ], 100 ]

{ allocatemem(932245000); default(realprecision, 2300); x=contfrac(prodinf(k=0, sqrt(1 + 1/2^(2*k)))); for (n=1, 2000, write("b065045.txt", n-1, " ", x[n])) } \\ Harry J. Smith, Oct 04 2009

Terms corrected and terms added by Harry J. Smith, Oct 04 2009

Offset changed by Andrew Howroyd, Aug 05 2024

A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

Original entry on oeis.org

2, 7, 1, 1, 8, 1, 9, 3, 4, 7, 7, 2, 6, 9, 5, 8, 7, 6, 0, 6, 9, 1, 0, 8, 8, 4, 6, 9, 7, 0, 7, 9, 1, 8, 6, 0, 2, 4, 4, 3, 3, 9, 9, 0, 8, 5, 6, 7, 4, 8, 8, 5, 4, 9, 4, 6, 9, 3, 0, 8, 0, 6, 2, 9, 0, 0, 6, 0, 2, 6, 2, 3, 6, 1, 3, 0, 5, 9, 7, 7, 8, 0, 0, 9, 7, 8, 7, 7, 4, 0, 5, 2, 5, 2, 1, 4, 6, 0, 4, 6

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			2.71181934772695876069108846970791860244...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000302, A065445, A081845, A100221, A132323, A371747, A371749, A371751.

RealDigits[QPochhammer[-1, 1/4], 10, 100][[1]]

Equals A065445^2. - Hugo Pfoertner, Apr 05 2024

A273413 Decimal expansion of Product_{k>=0} (1 + 1/2^(2k))^(-1/2).

Original entry on oeis.org

6, 0, 7, 2, 5, 2, 9, 3, 5, 0, 0, 8, 8, 8, 1, 2, 5, 6, 1, 6, 9, 4, 4, 6, 7, 5, 2, 5, 0, 4, 9, 2, 8, 2, 6, 3, 1, 1, 2, 3, 9, 0, 8, 5, 2, 1, 5, 0, 0, 8, 9, 7, 7, 2, 4, 5, 6, 9, 7, 6, 0, 1, 3, 1, 1, 0, 1, 4, 7, 8, 8, 1, 2, 0, 8, 4, 2, 4, 9, 0, 6, 9, 0, 6, 2, 2, 7, 4, 2, 5, 9, 0, 8, 0, 3, 8, 4, 0, 5, 2, 7, 4

Offset: 0

Jeremy Tan, May 22 2016

This constant is multiplied into the CORDIC algorithm to obtain the correct sine or cosine. See p. 647 of the fxtbook (below).

			0.60725293500888125616944675250492826311239085215008977245...

Jeremy Tan, Table of n, a(n) for n = 0..249
Joerg Arndt, Matters Computational (The Fxtbook), section 33.2
Wikipedia, CORDIC

Cf. A065445.

pent(z, n)= 1+sum(k=1, n, (-1)^k*(z^(k*(3*k-1)/2) + z^(k*(3*k+1)/2)));
/* == prod(n>=1, 1-z^n) via pentagonal number theorem */
N=30; u=0.25; K1=1/sqrt( 2 * pent(u^2, N)/pent(u, N) )
/* using prod(n>=1, 1+z^2) = prod(n>=1, 1-(z^2)^2)/prod(n>=1, 1-z^n) */
\\ Joerg Arndt, May 23 2016

Equals 1/A065445.

A344906 Decimal expansion of Sum_{k>=0} arctan(1/2^k).

Original entry on oeis.org

1, 7, 4, 3, 2, 8, 6, 6, 2, 0, 4, 7, 2, 3, 4, 0, 0, 0, 3, 5, 0, 4, 3, 3, 7, 6, 5, 6, 1, 3, 6, 4, 1, 6, 2, 8, 5, 8, 1, 3, 8, 3, 1, 1, 8, 5, 4, 2, 8, 2, 0, 6, 5, 2, 3, 0, 0, 4, 5, 6, 9, 5, 7, 2, 0, 5, 6, 5, 5, 1, 7, 6, 5, 2, 2, 7, 4, 9, 2, 0, 5, 5, 8, 1, 6, 5, 8, 6, 8

Offset: 1

Daniel Hoyt, Jun 01 2021

This number can be interpreted geometrically as the angle in radians of a fan made of stacked right triangles, with the length to height ratio doubling each successive triangle as seen in the illustration.

Since this angle exceeds Pi/2, the set of rotation angles used in the CORDIC algorithm covers an angle range sufficient to compute sine and cosine for any angle between 0 and Pi/2. This means the algorithm can converge to any angle in that range through appropriate combinations of these basic rotations. - Daniel Hoyt, Oct 25 2024

			1.743286620472340003...

Daniel Hoyt, Illustration of this angle's arctan relationship.
Wikipedia, CORDIC.

Cf. A003881, A065445, A073000, A195727, A195782.

Digits:= 140:
evalf(sum(arccot(2^k), k=0..infinity));  # Alois P. Heinz, Jun 02 2021

PARI
```
suminf(k=0, atan(1/2^k))
```

sumalt(k=1, ((-1)^(k+1))*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)))

Equals Sum_{k>=1} (-1)^(k+1)*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)).

cp's OEIS Frontend

A065045 Continued fraction expansion of the constant Product_{k>=0} (1 + 1/2^(2k))^(1/2) (A065445).

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A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

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A273413 Decimal expansion of Product_{k>=0} (1 + 1/2^(2k))^(-1/2).

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A344906 Decimal expansion of Sum_{k>=0} arctan(1/2^k).

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