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 1-4 of 4 results.

A200668 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with each no smaller than the sum of its three previous neighbors modulo (k+1).

Original entry on oeis.org

2, 3, 3, 4, 6, 5, 5, 10, 12, 8, 6, 15, 26, 24, 12, 7, 21, 45, 69, 46, 17, 8, 28, 75, 135, 175, 89, 25, 9, 36, 112, 267, 406, 432, 176, 36, 10, 45, 164, 448, 938, 1217, 1076, 350, 51, 11, 55, 225, 750, 1813, 3283, 3650, 2671, 697, 72, 12, 66, 305, 1125, 3414, 7322, 11516, 10959
Offset: 1

Views

Author

R. H. Hardin Nov 20 2011

Keywords

Comments

Table starts
..2....3.....4.....5......6.......7.......8........9.......10........11
..3....6....10....15.....21......28......36.......45.......55........66
..5...12....26....45.....75.....112.....164......225......305.......396
..8...24....69...135....267.....448.....750.....1125.....1690......2376
.12...46...175...406....938....1813....3414.....5682.....9412.....14443
.17...89...432..1217...3283....7322...15504....28743....52389.....87890
.25..176..1076..3650..11516...29536...70412...145431...291683....534853
.36..350..2671.10959..40399..119066..319532...735519..1623152...3252623
.51..697..6627.32941.141745..479993.1449895..3719534..9031554..19778869
.72.1391.16421.99044.497298.1935168.6578528.18809812.50252326.120270942

Examples

			Some solutions for n=7 k=6
..4....0....4....6....2....3....4....2....5....1....0....1....1....5....3....1
..5....0....5....6....6....6....6....5....6....2....4....2....4....5....5....4
..5....1....5....6....6....5....6....0....6....3....6....4....6....3....2....6
..6....1....4....4....2....6....6....0....4....6....4....1....5....6....4....6
..3....2....5....4....2....3....5....6....5....5....5....2....4....1....4....4
..2....4....5....6....6....4....3....6....5....5....4....0....2....4....5....4
..4....2....5....5....5....6....0....5....3....4....6....5....5....6....6....5

Links

Crossrefs

Row 2 is A000217(n+1)
Row 3 is A200252
Row 4 is A200253

A200469 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with each no smaller than the sum of its four previous neighbors modulo (k+1).

Original entry on oeis.org

2, 3, 3, 4, 6, 5, 5, 10, 12, 8, 6, 15, 26, 24, 13, 7, 21, 45, 69, 48, 20, 8, 28, 75, 135, 181, 98, 29, 9, 36, 112, 267, 405, 455, 199, 43, 10, 45, 164, 448, 951, 1213, 1120, 400, 63, 11, 55, 225, 750, 1792, 3328, 3627, 2794, 800, 91, 12, 66, 305, 1125, 3434, 7140, 11576, 10846
Offset: 1

Views

Author

R. H. Hardin Nov 18 2011

Keywords

Comments

Table starts
..2....3.....4.....5......6.......7.......8........9.......10........11
..3....6....10....15.....21......28......36.......45.......55........66
..5...12....26....45.....75.....112.....164......225......305.......396
..8...24....69...135....267.....448.....750.....1125.....1690......2376
.13...48...181...405....951....1792....3434.....5625.....9365.....14256
.20...98...455..1213...3328....7140...15446....28023....51356.....85228
.29..199..1120..3627..11576...28434...69136...139566...281169....509465
.43..400..2794.10846..40309..113193..309748...694953..1539522...3044998
.63..800..6955.32429.140298..450812.1386973..3460458..8429124..18199720
.91.1597.17254.96970.487872.1795581.6207948.17231016.46144179.108782707

Examples

			Some solutions for n=7 k=6
..1....0....0....3....3....0....3....0....3....3....1....1....2....4....2....4
..6....1....5....4....4....1....3....2....4....4....6....6....5....6....2....6
..2....6....5....5....2....2....6....5....2....2....3....1....2....4....4....5
..2....3....3....5....6....4....5....3....6....6....4....3....2....2....4....2
..5....4....6....3....5....0....5....4....1....1....4....6....6....2....5....5
..5....3....5....3....4....6....6....5....6....6....3....5....4....0....6....5
..1....2....6....3....4....5....3....5....3....6....1....4....3....1....6....5

Links

Crossrefs

Row 2 is A000217(n+1)
Row 3 is A200252
Row 4 is A200253
Row 5 is A200254

A199771 Row sums of the triangle in A199332.

Original entry on oeis.org

1, 5, 12, 26, 45, 75, 112, 164, 225, 305, 396, 510, 637, 791, 960, 1160, 1377, 1629, 1900, 2210, 2541, 2915, 3312, 3756, 4225, 4745, 5292, 5894, 6525, 7215, 7936, 8720, 9537, 10421, 11340, 12330, 13357, 14459, 15600, 16820, 18081, 19425, 20812, 22286, 23805
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 23 2011

Keywords

Comments

a(n) = Sum_{k=1..n} A199332(n,k);
a(2*n-1) = A015237(n); a(2*n) = A048395(n);
a(n+1) = A200252(n).

Links

Programs

  • Haskell
    a199771  = sum . a199332_row
  • Mathematica
    LinearRecurrence[{2,1,-4,1,2,-1},{1,5,12,26,45,75},50] (* Harvey P. Dale, Apr 27 2019 *)
  • PARI
    a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,2,1,-4,1,2]^(n-1)*[1;5;12;26;45;75])[1,1] \\ Charles R Greathouse IV, Jun 18 2017

Formula

G.f.: x*( 1+3*x+x^2+x^3 ) / ((1+x)^2*(x-1)^4). - R. J. Mathar, Nov 24 2011
a(n) = n*(3+2*n^2+4*n+(-1)^n)/8. - R. J. Mathar, Jun 23 2023

A212561 Number of (w,x,y,z) with all terms in {1,...,n} and w + x = 2y + 2z.

Original entry on oeis.org

0, 0, 1, 5, 12, 26, 45, 75, 112, 164, 225, 305, 396, 510, 637, 791, 960, 1160, 1377, 1629, 1900, 2210, 2541, 2915, 3312, 3756, 4225, 4745, 5292, 5894, 6525, 7215, 7936, 8720, 9537, 10421, 11340, 12330, 13357, 14459, 15600, 16820, 18081, 19425
Offset: 0

Views

Author

Clark Kimberling, May 21 2012

Keywords

Comments

Probably related to A199771 and A200252.
For a guide to related sequences, see A211795.
Except for the first term, a(n) is the number of undirected rook moves on an n X n chessboard, considered up to rotations but not reflections. - Hilko Koning, Aug 10 2025

Links

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x == 2 y + 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212561 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,0,1,5,12,26},50] (* Harvey P. Dale, Dec 04 2016 *)
a[n_Integer?NonNegative] := ((n - 1) (2 n^2 + 1 - (-1)^n))/8
Table[a[n], {n, 0, 100}] (* Hilko Koning, Aug 10 2025 *)
  • PARI
    concat([0,0], Vec(x^2*(x^3+x^2+3*x+1)/((x-1)^4*(x+1)^2) + O(x^100))) \\ Colin Barker, Feb 17 2015

Formula

a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).
a(n) = (2*n^3-2*n^2+n-1-(n-1)*(-1)^n)/8 = (n-1)*(2*n^2+1-(-1)^n)/8. - Luce ETIENNE, Jul 26 2014
G.f.: x^2*(x^3+x^2+3*x+1) / ((x-1)^4*(x+1)^2). - Colin Barker, Feb 17 2015
Showing 1-4 of 4 results.

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