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.

A228044 Decimal expansion of sum of reciprocals, row 2 of the natural number array, A185787.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 06 2013

Keywords

Comments

Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)*(n+k-1)/2, and let r = (2*Pi/sqrt(7))*tanh(Pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047.
Let c(k) be the sum of reciprocals of the numbers in column k of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*Pi^2; d(3) = 1/2, and d(1) is given by A228048.
It appears that Mathematica gives closed-form exact expressions for s(n), c(n) for 1<=n<=20 and further. The same holds for diagonal sums dr(n,n+k), k>=0; and for diagonal sums and dc(n+k,n), k>=0. In any case, general terms for all four sequences can be represented using the digamma function. The representations imply that c(n) is rational if and only if n is a term of A000124, and that dr(n) is rational if and only if n has the form k^2 + 2 for k >= 1.

Examples

			1.12229460660350434354343218597925...

Crossrefs

Cf. A185787, A000027, A228040, A226985, A228045.

Programs

  • Mathematica
    $MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2;
u = N[Sum[1/t[2, k], {k, 1, Infinity}], 130]
RealDigits[u, 10]
  • PARI
    sumnumrat(2/(n*(n+1)+4),1) \\ Charles R Greathouse IV, Feb 08 2023

Formula

Equals 1/3 + 1/5 + 1/8 + ...
Equals (1/30)*(-15 + 8*r*tanh(r)), where r = (Pi/2)*sqrt(15).

A228041 Decimal expansion of sum of reciprocals, row 3 of Wythoff array, W = A035513.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 05 2013

Keywords

Comments

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.
Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

Examples

			1/6 + 1/10 + 1/16 + ...  = 0.4299428331215887765860056514594635898444...

Crossrefs

Cf. A035513, A079586, A228040, A228042, A228043.

Programs

  • Mathematica
    f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 3; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Formula

Equals A079586/2 - 5/4. - Amiram Eldar, May 22 2021

A228042 Decimal expansion of sum of reciprocals, row 4 of Wythoff array, W = A035513.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 05 2013

Keywords

Comments

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.
Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

Examples

			1/9 + 1/15 + 1/24 + ... = 0.28662855541439251772400376763964239322963...

Crossrefs

Cf. A035513, A079586, A228040, A228041, A228043.

Programs

  • Mathematica
    f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 4; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Formula

Equals A079586/3 - 5/6. - Amiram Eldar, May 22 2021

A228043 Decimal expansion of sum of reciprocals, row 5 of Wythoff array, W = A035513.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 05 2013

Keywords

Comments

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.
Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

Examples

			1/12 + 1/20 + 1/32 + ... = 0.21497141656079438829300282572973179492222...

Crossrefs

Cf. A035513, A079586, A228040, A228041, A228042.

Programs

  • Mathematica
    f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 5; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Formula

Equals A079586/4 - 5/8. - Amiram Eldar, May 22 2021
Showing 1-4 of 4 results.

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