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 1-6 of 6 results.

A113436 First row of A113435.

Original entry on oeis.org

1, 2, 7, 26, 98, 371, 1406, 5329, 20196, 76532, 289997, 1098826, 4163483, 15775426, 59772826, 226477879, 858118966, 3251390237, 12319431012, 46677994276, 176861668393, 670124115506, 2539082288671, 9620514646154, 36451871795186
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Links

Crossrefs

Cf. A113435.

Programs

  • Mathematica
    CoefficientList[Series[(1 - 5*x + 8*x^2 - 4*x^3)/(1 - 7*x + 15*x^2 - 11*x^3 + x^4), {x,0,50}], x] (* or *) LinearRecurrence[{7,-15,11,-1}, {1, 2,7,26}, 50] (* G. C. Greubel, Mar 10 2017 *)
  • PARI
    my(x='x+ O(x^50)); Vec((1 -5*x +8*x^2 -4*x^3)/(1 -7*x +15*x^2 -11*x^3 +x^4)) \\ G. C. Greubel, Mar 10 2017

Formula

a(n) = A113435(3*n).
a(n) = 7*a(n-1) - 15*a(n-2) + 11*a(n-3) - a(n-4).
G.f.: (1 -5*x +8*x^2 -4*x^3)/(1 -7*x +15*x^2 -11*x^3 +x^4).

A113437 Second row of A113435.

Original entry on oeis.org

1, 3, 11, 41, 154, 581, 2197, 8317, 31501, 119338, 452141, 1713111, 6490879, 24593701, 93184802, 353074657, 1337790401, 5068852073, 19205744921, 72770053106, 275723780329, 1044710007771, 3958378188707, 14998188734897
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Links

Crossrefs

Cf. A113435.

Programs

  • Mathematica
    CoefficientList[Series[(1 - 4*x + 5*x^2 - 2*x^3)/(1 - 7*x + 15*x^2 - 11*x^3 + x^4), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
LinearRecurrence[{7,-15,11,-1},{1,3,11,41},30] (* Harvey P. Dale, May 09 2018 *)
  • PARI
    x='x+O('x^50); Vec((1-4*x+5*x^2-2*x^3)/(1-7*x+15*x^2-11*x^3+x^4)) \\ G. C. Greubel, Mar 11 2017

Formula

a(n) = A113435(3*n+1).
a(n) = 7*a(n-1) - 15*a(n-2) + 11*a(n-3) - a(n-4).
G.f.: (1-4*x+5*x^2-2*x^3)/(1-7*x+15*x^2-11*x^3+x^4).

A113438 Third row of A113435.

Original entry on oeis.org

1, 4, 16, 62, 237, 901, 3418, 12956, 49096, 186029, 704861, 2670692, 10119152, 38341126, 145273353, 550436561, 2085588866, 7902239404, 29941371656, 113447051497, 429847830217, 1628681887876, 6171031956688, 23381874459566, 88593294803061, 335677616363629
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Links

Crossrefs

Cf. A113435.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x)^3/(x^4 - 11*x^3 + 15*x^2 - 7*x + 1), {x, 0, 30}], x] (* Wesley Ivan Hurt, May 28 2015 *)
  • PARI
    Vec(-(x-1)^3 / (x^4-11*x^3+15*x^2-7*x+1) + O(x^100)) \\ Colin Barker, May 28 2015

Formula

a(n) = 7*a(n-1)-15*a(n-2)+11*a(n-3)-a(n-4), n>4.
a(n) = A113435(3n+2).
G.f.: (1-x)^3 / (x^4-11*x^3+15*x^2-7*x+1).

Extensions

Error in recursion and strange things in formula fixed by Colin Barker, May 28 2015

A197450 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 2,1,1,0,0 for x=0,1,2,3,4.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 12, 12, 4, 7, 36, 81, 36, 7, 11, 107, 474, 474, 107, 11, 16, 321, 2678, 4837, 2678, 321, 16, 26, 957, 15410, 52117, 52117, 15410, 957, 26, 41, 2868, 88767, 585194, 1108699, 585194, 88767, 2868, 41, 62, 8581, 510482, 6455759, 23829080
Offset: 1

Views

Author

R. H. Hardin Oct 15 2011

Keywords

Comments

Every 0 is next to 0 2's, every 1 is next to 1 1's, every 2 is next to 2 1's, every 3 is next to 3 0's, every 4 is next to 4 0's
Table starts
..1.....2........3..........4.............7...............11
..2.....5.......12.........36...........107..............321
..3....12.......81........474..........2678............15410
..4....36......474.......4837.........52117...........585194
..7...107.....2678......52117.......1108699.........23829080
.11...321....15410.....585194......23829080........982926107
.16...957....88767....6455759.....504568165......40087361395
.26..2868...510482...71202438...10724642167....1640477445822
.41..8581..2936714..787298158..228248265080...67160665746869
.62.25694.16895217.8700460833.4853181015432.2747685400705119

Examples

			Some solutions containing all values 0 to 4 for n=6 k=4
..2..1..1..0....1..2..1..0....1..0..0..1....0..0..3..0....0..0..1..0
..1..0..0..0....1..2..1..0....1..3..0..1....0..3..0..1....0..0..1..0
..1..0..4..0....0..3..0..3....0..0..3..2....0..3..0..1....0..4..0..3
..0..3..0..0....0..0..4..0....0..4..0..1....1..0..4..0....0..0..3..0
..0..3..0..3....0..0..0..1....0..0..0..1....1..0..0..0....0..1..2..1
..0..0..0..0....0..0..0..1....0..0..0..0....2..1..1..0....0..1..2..1

Links

Crossrefs

Column 1 is A113435(n+1)

A113439 a(n) = a(n-1) + Sum_{k=1..floor(n/4)} a(n-4k), with a(0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 12, 17, 23, 34, 50, 72, 101, 146, 212, 306, 436, 627, 905, 1305, 1871, 2689, 3872, 5577, 8014, 11521, 16576, 23858, 34309, 49337, 70968, 102108, 146868, 211233, 303832, 437080, 628708, 904306, 1300737, 1871065, 2691401
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Comments

If presented in four rows a(4k), a(4k+1), a(4k+2) and a(4k+3), each term is the sum of the previous term in the sequence and the partial sum of its row; see Example section.

Examples

			From _Jon E. Schoenfield_, Mar 11 2017: (Start)
Table of values T(j,k) = a(4k+j) in 4 rows:
.
  j | k=0     1     2     3     4     5     6     7
----+--------------------------------------------------
  0 |   1     2     8    34   146   627  2689 11521 ...
  1 |   1     3    12    50   212   905  3872 16576 ...
  2 |   1     4    17    72   306  1305  5577 23858 ...
  3 |   1     5    23   101   436  1871  8014 34309 ...
.
    T(2,4) = T(1,4) + T(2,0) + T(2,1) + T(2,2) + T(2,3)
      306  =   212  +    1   +    4   +   17   +   72
(End)

Links

Crossrefs

Cf. A028495, A113435, A113444.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x^4)/(1 - x - 2*x^4 + x^5), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
LinearRecurrence[{1,0,0,2,-1},{1,1,1,1,2},50] (* Harvey P. Dale, Nov 10 2019 *)
  • PARI
    x='x+O('x^50); Vec((1-x^4)/(1-x-2*x^4+x^5)) \\ G. C. Greubel, Mar 11 2017

Formula

a(n) = a(n-1) + 2*a(n-4) - a(n-5).
a(n) = 9*a(n-4) - 28*a(n-8) + 38*a(n-12) - 20*a(n-16) +a(n-20).
G.f.: (1-x^4)/(1-x-2*x^4+x^5).

A113444 a(n) = a(n-1) + Sum_{0

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 13, 18, 24, 31, 43, 60, 83, 113, 151, 206, 283, 389, 532, 721, 982, 1342, 1837, 2512, 3422, 4665, 6367, 8699, 11886, 16218, 22126, 30195, 41226, 56299, 76849, 104883, 143147, 195404, 266776, 364175, 497092, 678503, 926164
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Comments

If presented in five rows a(5n) a(5n+1).. a(5n+4) each term is the sum of the previous term in the sequence and the partial sum of its row.

Links

Crossrefs

Cf. A113435, A113439, A028495.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x^5)/(1 - x - 2*x^5 + x^6), {x,0,50}], x] (* G. C. Greubel, Mar 11 2017 *)
  • PARI
    x='x+O('x^50); Vec((1-x^5)/(1-x-2*x^5+x^6)) \\ G. C. Greubel, Mar 11 2017

Formula

G.f.: (1-x^5)/(1-x-2*x^5+x^6).
a(n) = a(n-1) + 2*a(n-5) - a(n-6).
a(n) = 11*a(n-5) -45*a(n-10) +90*a(n-15) -90*a(n-20) +37*a(n-25)-a(n-30).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.