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 1-3 of 3 results.

A378388 Decimal expansion of the surface area of a tetrakis hexahedron with unit shorter edge length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 27 2024

Keywords

Comments

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

Examples

			11.925695879998878380848926233233473255683297917928...

Links

Crossrefs

Cf. A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1), A378389 (dihedral angle).
Cf. A377341 (surface area of a truncated octahedron with unit edge).
Cf. A002163, A208899.

Programs

  • Mathematica
    First[RealDigits[16*Sqrt[5]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TetrakisHexahedron", "SurfaceArea"], 10, 100]]

Formula

Equals (16/3)*sqrt(5) = (16/3)*A002163 = 16*A208899.

A124397 Numerators of partial sums of a series for sqrt(5)/3.

Original entry on oeis.org

1, 3, 21, 17, 99, 2223, 12039, 56763, 59337, 286961, 7358781, 36088473, 183146521, 181066401, 36534213, 4535753121, 22798981683, 113528187171, 113891192583, 568042152363, 14228623114839, 71035463999307, 355598139789279, 14210752102407, 1777633916948199
Offset: 0

Views

Author

Wolfdieter Lang, Nov 10 2006

Keywords

Comments

Denominators are given by A124398.
The alternating sums over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/5^k, have the limit s = lim_{n-> infinity} r(n) = sqrt(5)/3. From the expansion of 1/sqrt(1+x) for x=4/5.

Examples

			a(3) = 17 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = a(3)/A124398(3).

Links

Crossrefs

Cf. A123747/A123748 partial sums for a series for sqrt(5).
Cf. A123749/A124396 partial sums for a series for 3/sqrt(5).
Cf. A124398 (denominators), A208899 (sqrt(5)/3).

Programs

  • GAP
    List([0..20], n-> NumeratorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019
  • Magma
    [Numerator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019
  • Maple
    seq(numer(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019
  • Mathematica
    Table[Numerator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k,0,n}]], {n,0,20}] (* G. C. Greubel, Dec 25 2019 *)
  • PARI
    a(n) = numerator(sum(k=0, n, ((-1)^k)*binomial(2*k,k)/5^k)); \\ Michel Marcus, Aug 11 2019
  • Sage
    [numerator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k * binomial(2*k,k)/5^k in lowest terms.
r(n) = Sum_{k=0..n} (-1)^k*((2*k-1)!!/(2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.

A124398 Denominators of partial sums of a series for sqrt(5)/3.

Original entry on oeis.org

1, 5, 25, 25, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 244140625, 244140625, 48828125, 6103515625, 30517578125, 152587890625, 152587890625, 762939453125, 19073486328125, 95367431640625, 476837158203125, 19073486328125, 2384185791015625
Offset: 0

Views

Author

Wolfdieter Lang, Nov 10 2006

Keywords

Comments

Denominators of alternating sums over central binomial coefficients scaled by powers of 5.
Numerators are given by A124397.
For the rationals r(n) see the W. Lang link under A124397.
r(n) is not 1/3 times the rational sequence A123747/A123748 which converges to sqrt(5).

Examples

			a(3) = 25 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = A124397(3)/a(3).

Links

Crossrefs

Cf. A124397 (numerators), A208899 (sqrt(5)/3).

Programs

  • GAP
    List([0..20], n-> DenominatorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019
  • Magma
    [Denominator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019
  • Maple
    seq(denom(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019
  • Mathematica
    Table[Denominator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k,0,n}]], {n,0,20}] (* G. C. Greubel, Dec 25 2019 *)
  • PARI
    a(n) = denominator(sum(k=0, n, ((-1)^k)*binomial(2*k,k)/5^k)); \\ Michel Marcus, Aug 11 2019
  • Sage
    [denominator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k * binomial(2*k,k)/5^k, in lowest terms.
r(n) = Sum_{k=0..n} (-1)^k*((2*k-1)!!/(2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.
Showing 1-3 of 3 results.

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