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 1-4 of 4 results.

A189970 Decimal expansion of (1 + x + sqrt(14+10*x))/4, where x=sqrt(5).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 05 2011

Keywords

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (1 + x + sqrt(14+10*x))/4, where x=sqrt(5). This rectangle can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,r,1,r,1,r,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,3,6,3,...] at A189971. For details, see A188635.
Decimal expansion of sqrt(r + r*sqrt(r + r*sqrt(r + ...))), where r = (1 + sqrt(5))/2 = golden ratio. - Ilya Gutkovskiy, Aug 24 2015
A quartic integer. - Charles R Greathouse IV, Aug 29 2015

Examples

			2.31651242917313233045161321161782337624579...

Links

Crossrefs

Cf. A188635, A001622, A189971, A190157.

Programs

  • Magma
    (1 + Sqrt(5) + Sqrt(14 + 10*Sqrt(5)) )/4; // G. C. Greubel, Jan 12 2018
  • Mathematica
    r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, {r, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A189971 *)
RealDigits[N[%%, 120]]     (* A189970 *)
N[%%%, 40]
RealDigits[(1+Sqrt[5]+Sqrt[14+10Sqrt[5]])/4,10,120][[1]] (* Harvey P. Dale, Sep 24 2015 *)
  • PARI
    default(realprecision,1000);x=sqrt(5);(1+x+sqrt(14+10*x))/4 \\ Anders Hellström, Aug 24 2015
  • PARI
    polrootsreal(x^4-x^3-2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Aug 29 2015

A189971 Continued fraction of (1 + x + sqrt(14 + 10*x))/4, where x=sqrt(5).

Original entry on oeis.org

2, 3, 6, 3, 1, 2, 15, 2, 3, 6, 1, 7, 1, 4, 2, 3, 1, 4, 2, 1, 1, 1, 2, 1, 20, 17, 3, 1, 2, 3, 1, 1, 3, 1, 4, 9, 73, 1, 37, 192, 3, 1, 1, 1, 1, 5, 1, 21, 1, 6, 7, 1, 3, 3, 1, 8, 2, 2, 1, 1, 8, 1, 2, 1, 1, 8, 1, 2, 1, 20, 2, 16, 3, 19, 2, 1, 3, 7, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 9, 32, 1, 1, 10, 5, 1, 7, 5, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling, May 05 2011

Keywords

Comments

Equivalent to the periodic continued fraction [r,1,r,1,...] where r=(1+sqrt(5))/2, the golden ratio. For geometric interpretations of both continued fractions, see A189970 and A188635.

Links

Crossrefs

Cf. A001622, A189970, A188635, A190157, A190158.

Programs

  • Magma
    ContinuedFraction( (1 + Sqrt(5) + Sqrt(14 + 10*Sqrt(5)) )/4 ); // G. C. Greubel, Jan 12 2018
  • Mathematica
    (See A189970.)
ContinuedFraction[(1+Sqrt[5]+Sqrt[14+10Sqrt[5]])/4,120] (* Harvey P. Dale, Jul 31 2013 *)
  • PARI
    contfrac((1+sqrt(5)+sqrt(14+10*sqrt(5)))/4) \\ G. C. Greubel, Jan 12 2018

A272362 Expansion of (1 + x - x^2 - x^3 - x^4)/((1 - x)*(1 - x - 2*x^2 - 2*x^3 - x^4)).

Original entry on oeis.org

1, 3, 6, 14, 32, 74, 171, 396, 917, 2124, 4920, 11397, 26401, 61158, 141673, 328187, 760249, 1761126, 4079670, 9450606, 21892446, 50714123, 117479896, 272143639, 630424122, 1460385314, 3383000731, 7836763241, 18153959452, 42053872709, 97418318825, 225670746387, 522769088906, 1211001092038
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 27 2016

Keywords

Comments

Partial sums of A272642. - Wolfdieter Lang, May 06 2016

Links

Crossrefs

Cf. A000045, A000201, A001622, A189970, A190157, A272642.

Programs

  • Magma
    I:=[1,3,6,14,32]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5): n in [1..30]]; // Vincenzo Librandi, May 08 2016
  • Mathematica
    LinearRecurrence[{2, 1, 0, -1, -1}, {1, 3, 6, 14, 32}, 34]
RecurrenceTable[{a[n] == Floor[GoldenRatio a[n - 1] + GoldenRatio a[n - 2]], a[0] == 1, a[1] == 3}, a, {n, 33}]
CoefficientList[Series[(1 + x - x^2 - x^3 - x^4)/((1 - x) (1 - x - 2 x^2 - 2 x^3 - x^4)), {x, 0, 50}], x] (* Vincenzo Librandi, May 08 2016 *)
  • PARI
    Vec((1+x-x^2-x^3-x^4)/(1-2*x-x^2+x^4+x^5) + O(x^99)) \\ Altug Alkan, Apr 27 2016

Formula

G.f.: (1 + x - x^2 - x^3 - x^4)/((1 - x)*(1 - x - 2*x^2 - 2*x^3 - x^4)).
a(n) = 2*a(n-1) + a(n-2) - a(n-4) - a(n-5).
a(n) = floor(phi*a(n-1) + phi*a(n-2)), a(0)=1, a(1)=3, where phi is the golden ratio (A001622).
Limit_{n->infinity} a(n)/a(n-1) = 2/(sqrt(2*sqrt(5)-1) - 1) = sqrt(phi + phi*sqrt(phi + phi*sqrt(phi + ...))) = A189970.
Limit_{n->infinity} a(n-1)/a(n) = (sqrt(2*sqrt(5)-1) - 1)/2 = 1 + A190157.

A298738 Decimal expansion of (1/2)(1 + sqrt(7 + 2*sqrt(5))).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 13 2018

Keywords

Examples

			constant = 2.1935270853310539386... = positive zero of x^2 - x - r^2, where r = golden ratio = (1+ sqrt(5))/2; see A001622.

Links

Crossrefs

Cf. A001622, A190157, A298522, A298523, A275828.

Programs

  • Mathematica
    r = (1/2)*(1 + Sqrt[7 + 2*Sqrt[5]])
RealDigits[N[r, 100], 10][[1]];  (* A298738 *)
RealDigits[(1+Sqrt[7+2*Sqrt[5]])/2,10,120][[1]] (* Harvey P. Dale, Jun 13 2025 *)
Showing 1-4 of 4 results.

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