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 1-5 of 5 results.

A120785 Denominators of partial sums of Catalan numbers scaled by powers of 1/16.

Original entry on oeis.org

1, 16, 128, 4096, 32768, 524288, 4194304, 268435456, 2147483648, 34359738368, 274877906944, 8796093022208, 70368744177664, 1125899906842624, 9007199254740992, 1152921504606846976
Offset: 0

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

Numerators are given under A120784.
The listed numbers coincide with the denominators of sum(C(k)/(-16)^k,k=0..n). For the numerators see A120794. In general these denominators may be different. See e.g. A120783 versus A120793 and A120787 versus A120796.
Also one fourth of denominators of partial sums for sqrt(2)+sqrt(3). See A121503 for the numerators.

Formula

a(n)=denominator(r(n)), with the rationals r(n):=sum(C(k)/16^k,k=0..n) with C(k):=A000108(k) (Catalan numbers).

A138281 a(n) = floor((sqrt(2) + sqrt(3))^n).

Original entry on oeis.org

1, 3, 9, 31, 97, 308, 969, 3051, 9601, 30210, 95049, 299052, 940897, 2960313, 9313929, 29304086, 92198401, 290080547, 912670089, 2871501385, 9034502497, 28424933309, 89432354889, 281377831710, 885289046401, 2785353383794, 8763458109129, 27572156006234
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 12 2008

Keywords

Links

Crossrefs

Cf. A089078, A110117, A121503, A135611, A135798.

Programs

  • Magma
    [Floor((Sqrt(2) + Sqrt(3))^n): n in [0..50]]; // G. C. Greubel, Jan 27 2018
  • Mathematica
    Table[Floor[(Sqrt[2] + Sqrt[3])^n], {n, 0, 50}] (* G. C. Greubel, Jan 27 2018 *)
  • PARI
    for(n=0,50, print1(floor((sqrt(2) + sqrt(3))^n), ", ")) \\ G. C. Greubel, Jan 27 2018

Formula

a(2*n) = floor(A001079(n) + A001078(n)*sqrt(6));
(sqrt(2) + sqrt(3))^(2*n) = A001079(n) + A001078(n)*sqrt(6);
a(2*n+1) = floor(A054320(n)*sqrt(2) + A138288(n)*sqrt(3));
(sqrt(2)+sqrt(3))^(2*n+1) = A054320(n)*sqrt(2) + A138288(n)*sqrt(3).

Extensions

Terms a(16) and a(18) corrected, terms a(19) onward added by G. C. Greubel, Jan 27 2018

A121500 Minimal polygon values for a certain polygon problem leading to an approximation of Pi.

Original entry on oeis.org

3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42
Offset: 3

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

For a regular n-gon inscribed in a unit circle (area Pi), the arithmetic mean of the areas of this n-gon with a regular circumscribed m-gon is nearest to Pi for m=a(n).
This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with 0.15% relative error. See the Popper reference under A121503.

Examples

			n=8, a(8)=6: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error 0.001487 (rounded). All other circumscribed m-gons with inscribed octagon lead to a larger relative error.
n=21, a(21)=15: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error 0.0000147 (rounded).

Crossrefs

Cf. A121501 (positions n where relative errors decrease).

Formula

a(n) = min(abs(E(n,m)),m >= 3), n>=3 (checked for m=3..3+500), with E(n,m):= ((Fin(n)+Fout(m))/2-Pi)/Pi), where Fin(n):=(n/2)*sin(2*Pi/n) and Fout(m):= m*tan(Pi/m). Fin(n) is the area of the regular n-gon inscribed in the unit circle. Fout(n) is the area of a regular n-gon circumscribing the unit circle.

A121504 Numerators of partial sums of a series used for the series of sqrt(2) + sqrt(3) involving Catalan numbers.

Original entry on oeis.org

3, 53, 433, 13941, 111759, 1789509, 14320329, 916611309, 7333257267, 117334608047, 938685468127, 30038055403185, 240304869286059, 3844880951681069, 30759058568289097, 3937160132112061181
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

The corresponding denominators are A120785(n).
The limit of this series is 4*(4-(sqrt(2)+sqrt(3))) = 3.414942518 (maple10, 10 digits).
See A121503 for the geometric interpretation of sqrt(2)+sqrt(3) and a Popper reference.

Examples

			Rationals r(n): [3, 53/16, 433/128, 13941/4096, 111759/32768,
1789509/524288, 14320329/4194304, 916611309/268435456,...].

Links

Crossrefs

A121503/(4*A120785) are the partial sums of a series for sqrt(2)+sqrt(3).

Programs

  • Magma
    [Numerator( (&+[Binomial(2*k,k)*(1 + 2^(k+1))/(16^k*(k+1)): k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, Sep 27 2018
  • Mathematica
    Table[Numerator[Sum[CatalanNumber[k]*(1 + 2^(k + 1))/16^k, {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Sep 27 2018 *)
  • PARI
    for(n=0, 30, print1(numerator(sum(k=0,n, binomial(2*k,k)*(1 + 2^(k+1))/(16^k*(k+1)))), ", ")) \\ G. C. Greubel, Sep 27 2018

Formula

a(n) = numerator(r(n)) with r(n):= sum(C(k)*(1+2^(k+1))/16^k,k=0..n), n>=0, with C(k)=A000108(k) (Catalan numbers).

A121506 Minimal polygon values appearing in a certain polygon problem leading to an approximation of Pi.

Original entry on oeis.org

3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66
Offset: 3

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Analog of A121500 with n and m roles interchanged.
For a regular m-gon circumscribed around a unit circle (area Pi) the arithmetic mean of the areas of this m-gon with a regular inscribed n-gon is nearest to Pi for n=a(m).
This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with a 0.15% relative error. See the Popper reference under A121503.

Examples

			m=15, a(15)=21=n: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error E(21,15)= 0.0000147(rounded).
m=7, a(7)=9=n: F(9,7) leads to error E(9,7)= 0.003122 (rounded).
This is larger than E(8,6), therefore the m value 7 does not appear in A121502.
m=6, a(6)=8=n: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error E(8,6)= 0.001487 (rounded). All other inscribed n-gons with circumscribed hexagon lead to a larger relative error.

Crossrefs

Cf. A121502 (values for m for which relative errors E(n, m) decrease).

Formula

a(m)=min(abs(F(n,m)),n=3..infinity), m>=3 (checked for n=3..3+500), with F(nm):= ((Fin(n)+Fout(m))/2-Pi)/Pi), where Fin(n):=(n/2)*sin(2*Pi/n) and Fout(m):= m*tan(Pi/m). Fin(n) is the area of the regular n-gon inscribed in the unit circle. Fout(n) is the area of an regular n-gon circumscribing the unit circle. E(n,m) = (F(n,m)-pi)/pi is the relative error.
Showing 1-5 of 5 results.

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