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.

A373017 Decimal expansion of Sum_{k>=0} (sin(k*Pi/5))^2/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 16 2024

Keywords

Comments

Minimal polynomial is 124x^2 - 150x + 45. - Charles R Greathouse IV, Jul 16 2024

Examples

			0.5507402908669405718610683789823078330135334267835...

Links

Crossrefs

Cf. A373021.

Programs

  • Mathematica
    s = Sum[(Sin[k  Pi/5])^2/2^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d]]
  • PARI
    suminf(k=0,(sin(k*Pi/5))^2>>k) \\ Charles R Greathouse IV, Jul 16 2024
  • PARI
    (75-sqrt(45))/124 \\ Charles R Greathouse IV, Jul 16 2024

Formula

Equals (3/124) (25 - sqrt(5)).

A373022 Decimal expansion of Sum_{k>=0} sin(2*k*Pi/5)/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 20 2024

Keywords

Examples

			0.505352652816194920203386273737665...

Crossrefs

Cf. A373021 (includes guide to related sequences).

Programs

  • Mathematica
    {b, m, h} = {2, 5, 2}; s = Sum[Sin[ h k Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals sqrt(2*(5 + sqrt(5)))/(12 - 2*sqrt(5)).
Equals (-1)*Sum_{k>=0} sin(8*k*Pi/5)/2^k.
Minimal polynomial: 5 - 265*x^2 + 961*x^4.

A373023 Decimal expansion of Sum_{k>=0} sin(3*k*Pi/5)/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 20 2024

Keywords

Examples

			0.30501800805463899236580110742909107145...

Crossrefs

Cf. A373021 (includes guide to related sequences).

Programs

  • Mathematica
    {b, m, h} = {2, 5, 3}; s = Sum[Sin[ h k Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals sqrt(2*(5 + sqrt(5)))/(8 + 2*sqrt(5)).
Equals (-1)*Sum_{k>=0} sin(7*k*Pi/5)/2^k.
Minimal polynomial: 5 - 65*x^2 + 121*x^4.

A373024 Decimal expansion of Sum_{k>=0} sin(4*k*Pi/5)/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 20 2024

Keywords

Examples

			0.14273443441658095225471374294933839312078514...

Crossrefs

Cf. A373021 (includes guide to related sequences).

Programs

  • Mathematica
    {b, m, h} = {2, 5, 4}; s = Sum[Sin[ h k Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals sqrt(10 - 2*sqrt(5))/(12 + 2*sqrt(5)).
Equals (-1)*Sum_{k>=0} sin(6*k*Pi/5)/2^k.
Minimal polynomial: 5 - 265*x^2 + 961*x^4.

A373025 Decimal expansion of Sum_{k>=0} cos(k*Pi/5)/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 20 2024

Keywords

Examples

			1.3503729060226985949648873184633558...

Crossrefs

Cf. A373021 (includes guide to related sequences).

Programs

  • Mathematica
    {b, m, h} = {2, 5, 1}; s = Sum[Cos[ h  k  Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals (23 + 3*sqrt(5))/22.
Equals Sum_{k>=0} cos(9 k*Pi/5)/2^k.
Minimal polynomial: 11 - 23 x + 11 x^2.

A373018 Decimal expansion of Sum_{k>=0} |sin(k*Pi/3)|/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 16 2024

Keywords

Examples

			0.7423074889580902686546198606453738715469165373...

Crossrefs

Cf. A373021.

Programs

  • Mathematica
    s = Sum[Abs[Sin[k Pi/3]]/2^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d]]

Formula

Equals (3/7)*sqrt(3).

A374813 Decimal expansion of Sum_{k>=0} cos(2k*Pi/5)/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 20 2024

Keywords

Examples

			0.8985194182661188562778632420353843339729...

Crossrefs

Cf. A373021 (includes guide to related sequences).

Programs

  • Mathematica
    {b, m, h} = {2, 5, 2}; s = Sum[Cos[ h  k  Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals (49 + 3*sqrt(5))/62.
Equals Sum_{k>=0} cos(8 k*Pi/5)/2^k.
Minimal polynomial: 19 - 49 x + 31 x^2.
Equals 2 - 2*A373017. - R. J. Mathar, Aug 27 2024

A374814 Decimal expansion of Sum_{k>=0} cos(3k*Pi/5)/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 20 2024

Keywords

Examples

			0.74053618488639231412602177244573...

Crossrefs

Cf. A373021 (includes guide to related sequences).

Programs

  • Mathematica
    {b, m, h} = {2, 5, 3}; s = Sum[Cos[ h  k  Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals (23 - 3*sqrt(5))/22.
Equals Sum_{k>=0} cos(7 k*Pi/5)/2^k.
Minimal polynomial: 11 - 23 x + 11 x^2.

A374815 Decimal expansion of Sum_{k>=0} cos(4k*Pi/5)/2^k.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 20 2024

Keywords

Examples

			0.6821257430242037243672980482871963111...

Crossrefs

Cf. A373021 (includes guide to related sequences).

Programs

  • Mathematica
    {b, m, h} = {2, 5, 4}; s = Sum[Cos[ h  k  Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals (49 - 3*sqrt(5))/62.
Equals Sum_{k>=0} cos(6 k*Pi/5)/2^k.
Minimal polynomial: 19 - 49 x + 31 x^2.
Showing 1-9 of 9 results.

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