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.

A075682 First differences of A075681.

Original entry on oeis.org

0, 2, 19, 39, 61, 86, 114, 145, 179, 216, 256, 299, 345, 394, 446, 501, 559, 620, 684, 751, 821, 894, 970, 1049, 1131, 1216, 1304, 1395, 1489, 1586, 1686, 1789, 1895, 2004, 2116, 2231, 2349, 2470, 2594, 2721, 2851, 2984, 3120, 3259, 3401, 3546
Offset: 1

Views

Author

Jon Perry, Oct 12 2002

Keywords

Links

Crossrefs

Cf. A075681, A075683.

Programs

  • Magma
    [0,2,19] cat [n*(3*n+17)/2 -19: n in [4..50]]; // G. C. Greubel, Jan 01 2023
  • Mathematica
    Join[{0,2,19},LinearRecurrence[{3,-3,1},{39,61,86},50]] (* Harvey P. Dale, Aug 26 2014 *)
  • SageMath
    [0,2,19]+[n*(3*n+17)/2 -19 for n in range(4,51)] # G. C. Greubel, Jan 01 2024

Formula

a(n) = (1/2)*(3*n^2 + 17*n - 38), for n > 3. - Ralf Stephan
From G. C. Greubel, Jan 01 2024: (Start)
G.f.: x^2*(2 + 13*x - 12*x^2 + x^3 - x^4)/(1-x)^3.
E.g.f.: (1/2)*(-38 + 20*x + 3*x^2)*exp(x) + 19 + 9*x - x^2 - x^3/3!. (End)

Extensions

More terms from Ralf Stephan

A075683 2nd differences of A075681.

Original entry on oeis.org

2, 17, 20, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178, 181, 184, 187, 190, 193, 196
Offset: 1

Views

Author

Jon Perry, Oct 12 2002

Keywords

Links

Crossrefs

Cf. A075681, A075682.

Programs

  • Magma
    [2,17,20] cat [3*n+10: n in [4..70]]; // G. C. Greubel, Jan 02 2024
  • Mathematica
    Join[{2,17,20}, 10+3*Range[4,70]] (* G. C. Greubel, Jan 02 2024 *)
  • SageMath
    [2,17,20]+[3*n+10 for n in range(4,71)] # G. C. Greubel, Jan 02 2024

Formula

From G. C. Greubel, Jan 02 2024: (Start)
a(n) = 2*a(n-1) - a(n-2), for n >= 6.
a(n) = A075681(n+2) - 2*A075681(n+1) + A075681(n).
a(n) = A075682(n+1) - A075682(n).
a(n) = 3*n + 10 - 11*[n=1] + [n=2] + [n=3].
G.f.: x*(2 + 13*x - 12*x^2 - x^3 + x^4)/(1 - x)^2.
E.g.f.: (10 + 3*x)*exp(x) - 10 - 11*x + x^2/2! + x^3/3!. (End)

Extensions

Terms a(11) onward added by G. C. Greubel, Jan 02 2024

A003878 a(n) = n^4 + (9/2)*n^3 + n^2 - (9/2)*n + 1.

Original entry on oeis.org

1, 3, 48, 199, 543, 1191, 2278, 3963, 6429, 9883, 14556, 20703, 28603, 38559, 50898, 65971, 84153, 105843, 131464, 161463, 196311, 236503, 282558, 335019, 394453, 461451, 536628, 620623, 714099, 817743, 932266, 1058403, 1196913, 1348579, 1514208, 1694631
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Old name was: "Number of stacks of n pikelets, distance 4 flips from a well-ordered stack".

Links

Crossrefs

Cf. A075681.

Programs

  • Magma
    [(2*n^4+9*n^3+2*n^2-9*n+2)/2: n in [0..40]]; // G. C. Greubel, Jan 03 2024
  • Mathematica
    Table[n^4+(9/2)(n^3-n)+n^2+1,{n,0,30}] (* Harvey P. Dale, Dec 01 2020 *)
  • SageMath
    [(2*n^4+9*n^3+2*n^2-9*n+2)/2 for n in range(41)] # G. C. Greubel, Jan 03 2024

Formula

G.f.: (1 - 2*x + 43*x^2 - 21*x^3 + 3*x^4)/(1-x)^5. [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]
E.g.f.: (1/2)*(2 + 4*x + 43*x^2 + 21*x^3 + 2*x^4)*exp(x). - G. C. Greubel, Jan 03 2024

Extensions

Offset corrected by G. C. Greubel, Jan 03 2024
Showing 1-3 of 3 results.

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