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.

Showing 1-9 of 9 results.

A248289 Egyptian fraction representation of sqrt(65) (A010517) using a greedy function.

Original entry on oeis.org

8, 17, 292, 104588, 38180791782, 3220186027640389204438, 514699020130621511259820819971940751063386467, 352263737947121519527774929870101098823418099762680744113382295431246430941544915986778001
Offset: 0

Views

Author

Robert G. Wilson v, Oct 04 2014

Keywords

Crossrefs

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.
Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Programs

  • Mathematica
    Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 65]]

A041113 Denominators of continued fraction convergents to sqrt(65).

Original entry on oeis.org

1, 16, 257, 4128, 66305, 1065008, 17106433, 274767936, 4413393409, 70889062480, 1138638393089, 18289103351904, 293764292023553, 4718517775728752, 75790048703683585, 1217359297034666112, 19553538801258341377, 314073980117168128144, 5044737220675948391681, 81029869510932342395040
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Sqrt(65) = 16/2 + 16/257 + 16/(257*66305) + 16/(66305*17106433) + ... - Gary W. Adamson, Jun 13 2008
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 16's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n on alphabet {0,1,...,16} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, May 01 2023: (Start)
Also called the 16-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 16 kinds of squares available. (End)

Links

Crossrefs

Cf. A010517, A020822, A040056, A041112.
Row n=16 of A073133, A172236 and A352361 and column k=16 of A157103.

Programs

  • Magma
    [n le 2 select (16)^(n-1) else 16*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013
  • Mathematica
    Denominator[Convergents[Sqrt[65], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
Fibonacci[Range[30], 16] (* G. C. Greubel, Sep 29 2024 *)
  • SageMath
    A041113=BinaryRecurrenceSequence(16,1,1,16)
[A041113(n) for n in range(0,31)] # G. C. Greubel, Sep 29 2024

Formula

a(n) = Fibonacci(n+1, 16). - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 16*a(n-1) + a(n-2) for n > 1; a(0) = 1, a(1) = 16.
G.f.: 1/(1 - 16*x - x^2). (End)
a(n) = ((8+sqrt(65))^(n+1) - (8-sqrt(65))^(n+1))/(2*sqrt(65)). - Rolf Pleisch, May 14 2011
E.g.f.: exp(8*x)*(cosh(sqrt(65)*x) + 8*sinh(sqrt(65)*x)/sqrt(65)). - Stefano Spezia, Oct 28 2022

A040056 Continued fraction for sqrt(65).

Original entry on oeis.org

8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			8.06225774829854965236661... = 8 + 1/(16 + 1/(16 + 1/(16 + 1/(16 + ...)))).

Links

Crossrefs

Cf. A010517 (decimal expansion), A041112/A041113 (convergents), A248289 (Egyptian fraction).
Cf. A040000, A040002, A040012.

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[65],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16}] (* Harvey P. Dale, Nov 27 2020 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 49000); x=contfrac(sqrt(65)); for (n=0, 20000, write("b040056.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 07 2009

Formula

From Elmo R. Oliveira, Feb 10 2024: (Start)
a(n) = 16 = A010855(n) for n >= 1.
G.f.: 8*(1+x)/(1-x).
E.g.f.: 16*exp(x) - 8.
a(n) = 8*A040000(n) = 4*A040002(n) = 2*A040012(n). (End)

A041112 Numerators of continued fraction convergents to sqrt(65).

Original entry on oeis.org

8, 129, 2072, 33281, 534568, 8586369, 137916472, 2215249921, 35581915208, 571525893249, 9179996207192, 147451465208321, 2368403439540328, 38041906497853569, 611038907405197432, 9814664424981012481, 157645669707101397128, 2532145379738603366529, 40671971745524755261592
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010517, A041113.

Programs

  • Mathematica
    CoefficientList[Series[(8 + x)/(1 - 16 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 29 2013 *)
Numerator[Convergents[Sqrt[65],20]] (* or *) LinearRecurrence[{16,1},{8,129},20] (* Harvey P. Dale, Nov 12 2013 *)

Formula

From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 16*a(n-1) + a(n-2), with n > 1, a(0) = 8, a(1) = 129.
G.f.: (8 + x)/(1 - 16*x - x^2). (End)
E.g.f.: exp(8*x)*(8*cosh(sqrt(65)*x) + sqrt(65)*sinh(sqrt(65)*x)). - Stefano Spezia, Oct 28 2022

Extensions

More terms from Colin Barker, Nov 05 2013

A171417 Decimal expansion of (5+sqrt(65))/4.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Dec 08 2009

Keywords

Comments

Continued fraction expansion of (5+sqrt(65))/4 is A144437.

Examples

			(5+sqrt(65))/4 = 3.26556443707463741309....

Crossrefs

Cf. A010517 (decimal expansion of sqrt(65)), A144437 (repeat 3, 3, 1).

Programs

  • Mathematica
    RealDigits[(5+Sqrt[65])/4,10,120][[1]] (* Harvey P. Dale, Jun 17 2011 *)

A177707 Decimal expansion of (5+sqrt(65))/8.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 11 2010

Keywords

Comments

Continued fraction expansion of (5+sqrt(65))/8 is A177706.

Examples

			(5+sqrt(65))/8 = 1.63278221853731870654...

Crossrefs

Cf. A010517 (decimal expansion of sqrt(65)), A177706 (repeat 1, 1, 1, 1, 2).

A171419 Decimal expansion of (5+sqrt(65))/10.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Dec 08 2009

Keywords

Comments

Continued fraction expansion of (5+sqrt(65))/10 is A169609.

Examples

			(5+sqrt(65))/10 = 1.30622577482985496523....

Crossrefs

Cf. A010517 (decimal expansion of sqrt(65)), A169609 (repeat 1, 3, 3).

Programs

  • Mathematica
    RealDigits[(5+Sqrt[65])/10,10,120][[1]]  (* Harvey P. Dale, Apr 02 2011 *)

A176976 Decimal expansion of (4+sqrt(65))/7.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 30 2010

Keywords

Comments

Continued fraction expansion of (4+sqrt(65))/7 is A130782.

Examples

			(4+sqrt(65))/7 = 1.72317967832836423605...

Crossrefs

Cf. A010517 (decimal expansion of sqrt(65)), A130782 (repeat 1, 1, 2, 1, 1).

Programs

  • Mathematica
    RealDigits[(4 + Sqrt[65])/7, 10, 111][[1]] (* Robert G. Wilson v, Aug 19 2011 *)

A379534 Decimal expansion of (9*sqrt(65) - 55)/320.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, Dec 24 2024

Keywords

Comments

Lower bound to the 9th Heilbronn triangle constant.

Examples

			0.054875999170896708972810997102293563063154896095241...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.16, p. 527.

Links

Crossrefs

Cf. A010517, A248866, A343851, A379533.

Programs

  • Mathematica
    RealDigits[(9Sqrt[65]-55)/320,10,100][[1]]

Formula

Minimal polynomial: 320*x^2 + 110*x - 7. - Stefano Spezia, Aug 03 2025
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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