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.

A248263 Egyptian fraction representation of sqrt(37) (A010491) using a greedy function.

Original entry on oeis.org

6, 13, 172, 39216, 11016972197, 134283233503741443791, 18872603108304707287590736836379382332539, 773806129529571836706640292961775806691343199188996534429569375589794450652266246
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[ 37]]

A015524 a(n) = 3*a(n-1) + 7*a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 3, 16, 69, 319, 1440, 6553, 29739, 135088, 613437, 2785927, 12651840, 57457009, 260933907, 1185000784, 5381539701, 24439624591, 110989651680, 504046327177, 2289066543291, 10395523920112, 47210037563373, 214398780130903, 973666603336320, 4421791270925281, 20081040036130083
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Linear 2nd order recurrence.

Links

Programs

  • Magma
    [n le 2 select n-1 else 3*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    a[n_]:=(MatrixPower[{{1,3},{1,-4}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3,7},{0,1},30] (* Harvey P. Dale, Jul 04 2011 *)
  • PARI
    x='x+O('x^30); concat([0], Vec(x/(1 - 3*x - 7*x^2))) \\ G. C. Greubel, Jan 01 2018
  • Sage
    [lucas_number1(n,3,-7) for n in range(0, 23)] #  Zerinvary Lajos, Apr 22 2009

Formula

From R. J. Mathar, Apr 21 2008: (Start)
O.g.f.: x/(1 - 3*x - 7*x^2).
a(n) = 14^n*(1/A^n -(-1)^n/B^n)/sqrt(37), where A = sqrt(37) - 3 = A010491 - 3 and B = sqrt(37) + 3 = A010491 + 3. (End)
a(n) = (7*(111+23*sqrt(37))*(1/2*(3+sqrt(37)))^n + (2553 + 431*sqrt(37)) * (1/2 (3-sqrt(37)))^n)/(518*(45+8*sqrt(37))). - Harvey P. Dale, Jul 04 2011

A041061 Denominators of continued fraction convergents to sqrt(37).

Original entry on oeis.org

1, 12, 145, 1752, 21169, 255780, 3090529, 37342128, 451196065, 5451694908, 65871534961, 795910114440, 9616792908241, 116197425013332, 1403985893068225, 16964028141832032, 204972323595052609, 2476631911282463340, 29924555258984612689, 361571295019097815608
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Sqrt(37) = 6.08276253... = 12/2 + 12/145 + 12/(145*21169) + 12/(21169*3090529) + ... - Gary W. Adamson, Jun 13 2008
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 12'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,...,12} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Apr 02 2023: (Start)
Also called the 12-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 12 kinds of squares available. (End)
Take any recurrence (t) of the form (12,1). Then a(n) = (t(i-n)*(-1)^n + t(i+n+2))/(t(i) + t(i+2)) always applies for integer i >= n >= 1. - Klaus Purath, Aug 02 2025

Links

Crossrefs

Cf. A010491, A041060.
Cf. A243399.
Row n=12 of A073133, A172236 and A352361 and column k=12 of A157103.

Programs

  • Mathematica
    Denominator[Convergents[Sqrt[37],30]] (* or *) LinearRecurrence[{12,1},{1,12},30] (* Harvey P. Dale, May 26 2014 *)
  • Sage
    [lucas_number1(n,12,-1) for n in range(1, 18)] # Zerinvary Lajos, Apr 28 2009

Formula

a(n) = F(n, 12), the n-th Fibonacci polynomial evaluated at x=12. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 12*a(n-1) + a(n-2), n>1; a(0)=1, a(1)=12.
G.f.: 1/(1 - 12*x - x^2). (End)
a(n) = ((6+sqrt(37))^(n+1) - (6-sqrt(37))^(n+1))/(2*sqrt(37)). - Rolf Pleisch, May 14 2011
a(2*n) = a(n-1)^2 + a(n)^2 = A097730(n), a(2*n+1) = 12*A097728(n). - Klaus Purath, Aug 02 2025
E.g.f.: exp(6*x)*(cosh(sqrt(37)*x) + 6*sinh(sqrt(37)*x)/sqrt(37)). - Stefano Spezia, Aug 09 2025

A040030 Continued fraction for sqrt(37).

Original entry on oeis.org

6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			6.08276253029821968899968... = 6 + 1/(12 + 1/(12 + 1/(12 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 2009

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Links

Crossrefs

Cf. A010491 (decimal expansion), A041060/A041061 (convergents), A248263 (Egyptian fraction).
Cf. A040000, A040002, A040006, A010851

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[37],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{6},120,{12}] (* Harvey P. Dale, Jan 02 2017 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 44000); x=contfrac(sqrt(37)); for (n=0, 20000, write("b040030.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

Formula

From Elmo R. Oliveira, Feb 06 2024: (Start)
a(n) = 12 for n >= 1.
G.f.: 6*(1+x)/(1-x).
E.g.f.: 12*exp(x) - 6.
a(n) = 6*A040000(n) = 3*A040002(n) = 2*A040006(n). (End)

A041060 Numerators of continued fraction convergents to sqrt(37).

Original entry on oeis.org

6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010491, A041061.
Cf. A089926. - R. J. Mathar, Sep 09 2008

Programs

  • Mathematica
    Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[37],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
CoefficientList[Series[(6 + x)/(1 - 12  x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

Formula

From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 12*a(n-1) + a(n-2), n > 1; a(0)=6, a(1)=73.
G.f.: (6+x)/(1-12*x-x^2). (End)

A176445 Decimal expansion of sqrt(1295).

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 19 2010

Keywords

Comments

Continued fraction expansion of sqrt(1295) is 35 followed by (repeat 1, 70).
sqrt(1295) = sqrt(5)*sqrt(7)*sqrt(37).

Examples

			sqrt(1295) = 35.98610843089316319412...

Crossrefs

Cf. A002163, A010465, A010491, A176444.

Programs

  • Mathematica
    RealDigits[Sqrt[1295],10,120][[1]] (* Harvey P. Dale, Apr 19 2019 *)

A176977 Decimal expansion of (3+sqrt(37))/7.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 30 2010

Keywords

Comments

Continued fraction expansion of (3+sqrt(37))/7 is A130784.

Examples

			1.29753750432831709842...

Links

Crossrefs

Cf. A010491 (decimal expansion of sqrt(37)), A130784 (repeat 1, 3, 2).

Programs

  • Mathematica
    RealDigits[(3+Sqrt[37])/7,10,103][[1]] (* Stefano Spezia, May 26 2025 *)

A177839 Decimal expansion of sqrt(2442).

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 14 2010

Keywords

Comments

Continued fraction expansion of sqrt(2442) is 49 followed by (repeat 2, 2, 2, 98).
sqrt(2442) = sqrt(2)*sqrt(3)*sqrt(11)*sqrt(37).

Examples

			sqrt(2442) = 49.41659640242334617762...

Links

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A010468 (decimal expansion of sqrt(11)), A010491 (decimal expansion of sqrt(37)), A177838 (decimal expansion of (44+sqrt(2442))/88).

Programs

  • Mathematica
    RealDigits[Sqrt[2442],10,120][[1]] (* Harvey P. Dale, Apr 01 2012 *)

A177036 Decimal expansion of (4+sqrt(37))/7.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 01 2010

Keywords

Comments

Continued fraction expansion of (4+sqrt(37))/7 is A010882.

Examples

			(4+sqrt(37))/7 = 1.44039464718545995557...

Crossrefs

Cf. A010491 (decimal expansion of sqrt(37)), A010882 (repeat 1, 2, 3).
Showing 1-9 of 9 results.

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

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