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.

Previous Showing 31-36 of 36 results.

A117651 A002415 and A052472 interlaced.

Original entry on oeis.org

1, 0, 2, 1, 0, 6, 10, 20, 35, 50, 84, 105, 168, 196, 300, 336, 495, 540, 770, 825, 1144, 1210, 1638, 1716, 2275, 2366, 3080, 3185, 4080, 4200, 5304, 5440, 6783, 6936, 8550, 8721, 10640, 10830, 13090, 13300, 15939, 16170, 19228, 19481, 23000, 23276, 27300
Offset: 0

Views

Author

Roger L. Bagula, Apr 11 2006

Keywords

Links

Crossrefs

Cf. A002415, A052472, A018226.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x -2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10-4*x^11 )/( (1+x)^4*(1-x)^5) )); // G. C. Greubel, May 19 2019
  • Mathematica
    f[n_]:= n*(n+1)*(n+2)*(n-3)/12; g[n_]:= n^2*(n^2 -1)/12; Table[{Abs[f[n]], g[n]}, {n, 1, 25}]//Flatten
LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1}, {1,0,2,1,0,6,10,20,35,50,84, 105}, 50] (* Harvey P. Dale, Mar 05 2016 *)
  • PARI
    my(x='x+O('x^50)); Vec((1-x-2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10-4*x^11 )/((1+x)^4*(1-x)^5)) \\ G. C. Greubel, May 19 2019
  • Sage
    ((1-x-2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10 -4*x^11 )/((1+x)^4*(1-x)^5)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, May 19 2019

Formula

G.f.: (1 -x -2*x^2 +3*x^3 -3*x^4 +4*x^5 +16*x^6 -16*x^7 -14*x^8 +14*x^9 +4*x^10 -4*x^11 )/((1+x)^4*(1-x)^5). - Colin Barker, Mar 15 2013
a(n) = abs((2*n^4 +12*n^3 -2*n^2 -132*n -195 +(4*n^3 -6*n^2 -124*n -189)*(-1)^n))/384. - Luce ETIENNE, Jun 01 2015
a(n) = abs((-3*(65 +63*(-1)^n) -4*(33 +31*(-1)^n)*n -2*(1+3*(-1)^n)*n^2 +4*(3 +(-1)^n)*n^3 +2*n^4)/384). - Colin Barker, Jun 02 2015
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 11. - Charles R Greathouse IV, Jun 02 2015

A117652 a(n) = floor(n*(n+2)*(n+4)*(n-6)/192).

Original entry on oeis.org

0, -1, -1, -2, -2, -2, 0, 3, 10, 20, 35, 55, 84, 120, 168, 227, 300, 388, 495, 621, 770, 943, 1144, 1374, 1638, 1937, 2275, 2654, 3080, 3553, 4080, 4662, 5304, 6009, 6783, 7628, 8550, 9552, 10640, 11817, 13090, 14462, 15939, 17525, 19228, 21050, 23000, 25081
Offset: 0

Views

Author

Roger L. Bagula, Apr 11 2006

Keywords

Comments

Quasipolynomial with period 16. - Charles R Greathouse IV, Sep 06 2011

Links

Crossrefs

Cf. A052472, A018226.

Programs

  • Magma
    [Floor( n*(n+2)*(n+4)*(n-6)/192): n in [0..50]]; // Vincenzo Librandi, Sep 06 2011
  • Mathematica
    Table[Floor[n*(n+1)*(n+2)*(n-3)/12], {n, 0, 25, 1/2}]
LinearRecurrence[{4,-5,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,5,-4,1},{0,-1,-1,-2,-2,-2,0,3,10,20,35,55,84,120,168,227,300,388},50] (* Harvey P. Dale, Nov 02 2024 *)
  • PARI
    a(n)=n*(n+2)*(n+4)*(n-6)\192 \\ Charles R Greathouse IV, Sep 06 2011
  • Sage
    [floor(n*(n+2)*(n+4)*(n-6)/192) for n in (0..50)] # G. C. Greubel, May 20 2019

Formula

a(n) = floor( n*(n+2)*(n+4)*(n-6)/192).
a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-4) - 4*a(n-6) + 4*a(n-8) - 4*a(n-10) + 4*a(n-12) - 4*a(n-14) + 5*a(n-16) - 4*a(n-17) + a(n-18).

Extensions

More precise description, converted to a more regular signed sequence - the Assoc. Eds. of the OEIS, Jun 27 2010

A277131 Magic numbers of anti-Mackay icosahedra.

Original entry on oeis.org

45, 127, 279, 521, 873, 1355, 1987, 2789, 3781, 4983, 6415, 8097, 10049, 12291, 14843, 17725, 20957, 24559, 28551, 32953, 37785, 43067, 48819, 55061, 61813, 69095, 76927, 85329, 94321, 103923, 114155, 125037, 136589, 148831, 161783, 175465, 189897, 205099
Offset: 2

Views

Author

Felix Fröhlich, Oct 01 2016

Keywords

Links

Crossrefs

Cf. A005902, A008592, A018226, A018227.

Programs

  • Maple
    A277131:=n->11-(19*n)/3+5*n^2+(10*n^3)/3: seq(A277131(n), n=2..50); # Wesley Ivan Hurt, Oct 07 2016
  • Mathematica
    DeleteCases[CoefficientList[Series[x^2*(45 - 53 x + 41 x^2 - 13 x^3)/(1 - x)^4, {x, 0, 39}], x], 0] (* Michael De Vlieger, Oct 02 2016 *)
  • PARI
    a(n) = (2*n+1) * (5*n^2+5*n+3) / 3 - 10*(n-1)
  • PARI
    Vec(x^2*(45-53*x+41*x^2-13*x^3)/(1-x)^4 + O(x^50)) \\ Colin Barker, Oct 01 2016

Formula

a(n) = A005902(n) - A008592(n-1).
a(n) = 10/3*n^3 + 25*n^2 + 161/3*n + 45 with offset 0.
From Colin Barker, Oct 01 2016: (Start)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>5.
a(n) = 11-(19*n)/3+5*n^2+(10*n^3)/3.
G.f.: x^2*(45-53*x+41*x^2-13*x^3) / (1-x)^4.
(End)

A117580 A cubic quadratic sequence arranged so that the modulo-3 equals one cubic sequence is just ahead of the quadratic sequence (called here the Maestro sequence).

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 125, 169, 225, 343, 361, 441, 729, 729, 841, 1331, 1369, 1521, 2197, 2025
Offset: 0

Views

Author

Roger L. Bagula, Apr 08 2006

Keywords

Comments

Arranged so that they are near the Magic numbers (nuclear shell filling numbers): called Maestro as they have to be conducted like an orcestra to get them to behave this way.

Crossrefs

Cf. A018226.

Programs

  • Mathematica
    g[n_] := (n - Floor[n/3])^3 /; Mod[n, 3] - 1 == 0 g[n_] := (2*n - 1)^2 /; (n < 4) g[n_] := (2*n - 1)^2 /; (n > 13) && (n < 17) g[n_] := (2*n - 3)^2 /; (n > 4) && (n < 13) g[n_] := (2*n + 3)^2 /; (n >= 17) && (n < 19) g[n_] := (2*n + 5)^2 /; (n >= 18) a=Table[g[n], {n, 1, 20}]

Formula

g[n_] := (n - Floor[n/3])^3 /; Mod[n, 3] - 1 == 0 g[n_] := (2*n - 1)^2 /; (n < 4) g[n_] := (2*n - 1)^2 /; (n > 13) && (n < 17) g[n_] := (2*n - 3)^2 /; (n > 4) && (n < 13) g[n_] := (2*n + 3)^2 /; (n >= 17) && (n < 19) g[n_] := (2*n + 5)^2 /; (n >= 18) a(n) = g[n]

A219239 Double magic numbers (in physics).

Original entry on oeis.org

4, 10, 16, 22, 28, 30, 36, 40, 48, 52, 56, 58, 70, 78, 84, 90, 100, 102, 110, 128, 132, 134, 146, 154, 164, 176, 208, 252
Offset: 1

Views

Author

Wolfdieter Lang, Dec 12 2012

Keywords

Comments

For the magic numbers see A018226.
An atomic nucleus is called double magic if Z (number of protons in an atomic nucleus, atomic number) and N (number of neutrons) are both magic numbers. The nucleon or mass number (forget the Z electrons) is A = Z + N.
Each number a(n) is obtained in only one way as a sum of two (possibly equal) magic numbers. Only 28 is magic and double magic.

Examples

			Tin-132 is a double magic radionuclide (unstable isotope) with nucleon number A = 132 = a(21), Z = 50 and N = 82. Similarly for tin-100 with Z = N = 50. The stable primordial nuclide barium-132 is not double magic, because it has Z = 56 and N = 76.

Links

Crossrefs

Cf. A018226, A033547.

Formula

a(n) is the sum of two numbers from [2, 8, 20, 28, 50, 82, 126] (the magic numbers A018226).

A227416 Magic numbers from Smale’s 7th problem.

Original entry on oeis.org

6, 12, 24, 32, 48, 60, 67, 72, 80, 104, 108, 122, 132, 137
Offset: 1

Views

Author

Jonathan Vos Post, Jul 10 2013

Keywords

Comments

See pp. 16-17 of Nerattini et al. The sequence is defined as the numbers n of points distributed on the two-sphere in such a way that their average logarithmic pair-energy is minimal, and locally convex as a function of n. So far the available data for this sequence are empirical and should eventually be vindicated, or replaced if necessary, by rigorous data. The name "magic numbers" alludes to a similarity with the "magic numbers" in nuclear physics, see A018226.

Links

Crossrefs

Cf. A018226.

Extensions

Comment section and reference section corrected and revised by Michael Kiessling, Aug 20 2013
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