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 1-10 of 32 results. Next

A027129 a(n) = T(n,n) + T(n,n+1) + ... + T(n,2n), T given by A027113.

Original entry on oeis.org

1, 3, 9, 27, 79, 231, 672, 1951, 5662, 16439, 47776, 139025, 405110, 1182107, 3454035, 10105373, 29600389, 86800887, 254797772, 748646741, 2201587214, 6479524339, 19084115993, 56246912653, 165882843095, 489509315203
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027138 a(n) = Sum_{k=0..2n} (k+1) * A027113(n, 2n-k).

Original entry on oeis.org

1, 7, 26, 85, 264, 803, 2422, 7281, 21860, 65599, 196818, 590477, 1771456, 5314395, 15943214, 47829673, 143489052, 430467191, 1291401610, 3874204869, 11622614648, 34867843987, 104603532006, 313810596065, 941431788244, 2824295364783, 8472886094402
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • PARI
    Vec(-x*(x^3-2*x^2+2*x+1)/((x-1)^2*(3*x-1)) + O(x^50)) \\ Colin Barker, Jul 11 2015

Formula

For n>1, a(n) = 10*3^(n-2) - n - 1.
For n>4, a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - Colin Barker, Jul 11 2015
G.f.: -x*(x^3-2*x^2+2*x+1) / ((x-1)^2*(3*x-1)). - Colin Barker, Jul 11 2015

A027114 a(n) = A027113(n, n+2).

Original entry on oeis.org

3, 9, 20, 40, 77, 145, 270, 500, 923, 1701, 3132, 5764, 10605, 19509, 35886, 66008, 121411, 223313, 410740, 755472, 1389533, 2555753, 4700766, 8646060, 15902587, 29249421, 53798076, 98950092, 181997597, 334745773, 615693470
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Equals A089068(n+2) - 3.

Programs

  • Mathematica
    CoefficientList[Series[(3+3x+2x^2)/((1-x)(1-x-x^2-x^3)),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,0,-1},{3,9,20,40},40] (* Harvey P. Dale, Sep 26 2017 *)

Formula

G.f.: [3+3x+2x^2]/[(1-x)(1-x-x^2-x^3)].

A027115 a(n) = T(n,0) + T(n,1) + ... + T(n,2n), T given by A027113.

Original entry on oeis.org

1, 4, 11, 31, 87, 246, 700, 2003, 5758, 16616, 48102, 139625, 406214, 1184138, 3457771, 10112245, 29613029, 86824136, 254840534, 748725393, 2201731878, 6479790418, 19084605389, 56247812793, 165884498711, 489512360356
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027116 a(n) = A027113(n, n+3).

Original entry on oeis.org

7, 23, 56, 120, 241, 467, 886, 1660, 3087, 5715, 10552, 19452, 35825, 65943, 121342, 223240, 410663, 755391, 1389448, 2555664, 4700673, 8645963, 15902486, 29249316, 53797967, 98949979, 181997480, 334745652, 615693345
Offset: 3

Views

Author

Clark Kimberling

Keywords

Formula

Conjectures from Colin Barker, Feb 20 2016: (Start)
a(n) = 3*a(n-1)-2*a(n-2)-a(n-4)+a(n-5) for n>7.
G.f.: x^3*(7+2*x+x^2-2*x^3) / ((1-x)^2*(1-x-x^2-x^3)).
(End)

A027117 a(n) = A027113(n, n+4).

Original entry on oeis.org

16, 59, 155, 352, 738, 1475, 2861, 5444, 10232, 19079, 35395, 65452, 120786, 222615, 409965, 754616, 1388592, 2554723, 4699643, 8644840, 15901266, 29247995, 53796541, 98948444, 181995832, 334743887, 615691459, 1132434708
Offset: 4

Views

Author

Clark Kimberling

Keywords

Formula

Conjectures from Colin Barker, Feb 20 2016: (Start)
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3)-a(n-4)+2*a(n-5)-a(n-6) for n>9.
G.f.: x^4*(16-5*x-x^2-5*x^3+3*x^4) / ((1-x)^3*(1-x-x^2-x^3)).
(End)

A027118 a(n) = A027113(n, 2n).

Original entry on oeis.org

1, 2, 3, 7, 16, 39, 98, 252, 659, 1747, 4682, 12662, 34505, 94644, 261069, 723707, 2014952, 5631875, 15796210, 44444302, 125405217, 354765866, 1006003375, 2858939669, 8141135778, 23225997936, 66376634429, 190000938785
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027119 a(n) = A027113(n, 2n-1).

Original entry on oeis.org

1, 4, 9, 23, 59, 154, 407, 1088, 2935, 7980, 21843, 60139, 166425, 462638, 1291245, 3616923, 10164335, 28648092, 80960915, 229360649, 651237509, 1852936294, 5282196109, 15084862158, 43150636493, 123624304356, 354689105733
Offset: 1

Views

Author

Clark Kimberling

Keywords

A027120 a(n) = A027113(n, 2n-2).

Original entry on oeis.org

2, 7, 20, 56, 155, 429, 1188, 3298, 9181, 25634, 71781, 201569, 567538, 1601971, 4532460, 12851882, 36516613, 103955432, 296471643, 846932919, 2423256440, 6943726380, 19924638557, 57247669927, 164688166948, 474323305547
Offset: 2

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A027113.

A027121 a(n) = A027113(n, 2n-3).

Original entry on oeis.org

1, 4, 13, 40, 120, 352, 1022, 2948, 8473, 24304, 69649, 199544, 571795, 1639244, 4702499, 13500396, 38790727, 111555296, 321100627, 925086012, 2667533646, 7698716068, 22238169212, 64289860528, 186010834243, 538611262492
Offset: 2

Views

Author

Clark Kimberling

Keywords

Showing 1-10 of 32 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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