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.

A213844 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-5+4*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

3, 16, 7, 47, 32, 11, 104, 83, 48, 15, 195, 168, 119, 64, 19, 328, 295, 232, 155, 80, 23, 511, 472, 395, 296, 191, 96, 27, 752, 707, 616, 495, 360, 227, 112, 31, 1059, 1008, 903, 760, 595, 424, 263, 128, 35, 1440, 1383, 1264
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2012

Keywords

Comments

Principal diagonal: A213845.
Antidiagonal sums: A213846.
Row 1, (1,3,5,7...)**(3,7,11,15,...): A172482.
Row 2, (1,3,5,7,...)**(7,11,15,19,...): (4*k^3 + 15*k^2 + 2*k)/3.
Row 3, (1,3,5,7,...)**(11,15,19,23,...): (4*k^3 + 27*k^2 + 2*k)/3.
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
3....16...47....104...195...328
7....32...83....168...295...472
11...48...119...232...395...616
15...64...155...296...495...760

Links

Crossrefs

Cf. A212500.

Programs

  • Mathematica
    b[n_]:=2n-1;c[n_]:=4n-1;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213844 *)
Table[t[n,n],{n,1,40}] (* A213845 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213846 *)

Formula

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(4*n-1 + 4*x - (4*n-5)*x^2) and g(x) = (1-x)^4.

A213847 Rectangular array: (row n) = b**c, where b(h) = 4*h-1, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

3, 16, 9, 47, 36, 15, 104, 89, 56, 21, 195, 176, 131, 76, 27, 328, 305, 248, 173, 96, 33, 511, 484, 415, 320, 215, 116, 39, 752, 721, 640, 525, 392, 257, 136, 45, 1059, 1024, 931, 796, 635, 464, 299, 156, 51, 1440, 1401
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2012

Keywords

Comments

Principal diagonal: A213848.
Antidiagonal sums: A180324.
Row 1, (3,7,11,15,...)**(1,3,5,7,...): A172482.
Row 2, (3,7,11,15,...)**(3,5,7,9,...): (4*k^3 + 15*k^2 + 8*k)/3.
Row 3, (3,7,11,15,...)**(5,7,9,13,...): (4*k^3 + 27*k^2 + 14*k)/3.
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
3....16...47....104...195...328
9....36...89....176...305...484
15...56...131...248...415...640
21...76...173...320...525...796

Links

Crossrefs

Cf. A212500.

Programs

  • Mathematica
    b[n_]:=4n-1;c[n_]:=2n-1;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213847 *)
Table[t[n,n],{n,1,40}] (* A213848 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A180324 *)

Formula

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(6*n-3 + 4*(n-2)x - (2*n-3)*x^2) and g(x) = (1-x)^4.

A345897 a(n) = 2*n^4/3 - 4*n^3/3 + 11*n^2/6 - 13*n/6 + 1.

Original entry on oeis.org

1, 0, 4, 29, 107, 286, 630, 1219, 2149, 3532, 5496, 8185, 11759, 16394, 22282, 29631, 38665, 49624, 62764, 78357, 96691, 118070, 142814, 171259, 203757, 240676, 282400, 329329, 381879, 440482, 505586, 577655, 657169, 744624, 840532, 945421, 1059835, 1184334, 1319494
Offset: 0

Views

Author

Michel Marcus, Jun 29 2021

Keywords

Comments

For n >=1, a(n) is the number of divisions of a 2 X n board into 3 pieces. See Jacob Brown article.

Links

Crossrefs

Cf. A172482 (same but where the rightmost squares separate).

Programs

  • Mathematica
    CoefficientList[Series[(1 - 5 x + 14 x^2 - x^3 + 7 x^4)/(1 - x)^5, {x, 0, 38}], x] (* Michael De Vlieger, Apr 28 2023 *)
  • PARI
    a(n) = 2*n^4/3 - 4*n^3/3 + 11*n^2/6 - 13*n/6 + 1;

Formula

From Chai Wah Wu, Jun 29 2021: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (1 - 5*x + 14*x^2 - x^3 + 7*x^4)/(1-x)^5. (End)

A173809 a(2n+1) = 1+A131941(2n+1). a(2n) = A131941(2n).

Original entry on oeis.org

2, 3, 9, 16, 30, 47, 73, 104, 146, 195, 257, 328, 414, 511, 625, 752, 898, 1059, 1241, 1440, 1662, 1903, 2169, 2456, 2770, 3107, 3473, 3864, 4286, 4735, 5217, 5728, 6274, 6851, 7465, 8112, 8798, 9519, 10281, 11080
Offset: 1

Views

Author

Paul Curtz, Feb 25 2010

Keywords

Comments

An array T(n,k) of a(n) and its successive differences is T(1,k) = a(k), T(n,k) = T(n-1,k+1)-T(n-1,k) and starts:
2, 3, 9, 16, 30, 47, 73, 104, 146, 195, 257, 328,...
1, 6, 7, 14, 17, 26, 31, 42, 49, 62, 71, 86, 97, 114,... interleaved A056220 and A051890
5, 1, 7, 3, 9, 5, 11, 7, 13, 9, 15, 11, 17, 13, 19, 15.... A158552
-4, 6, -4, 6, -4, 6, -4, 6, -4, 6, -4, 6, -4, 6, -4,... A010711
10, -10, 10, -10, 10, -10, 10, -10, 10, -10, 10,..

Links

Programs

  • Mathematica
    LinearRecurrence[{3,-2,-2,3,-1},{2,3,9,16,30},60] (* Harvey P. Dale, Aug 10 2021 *)

Formula

G.f.: -x*(-2+3*x+x^3-4*x^2) / ( (1+x)*(x-1)^4 ). - R. J. Mathar, Jan 13 2011
a(2n) = A172482(n-1).
a(n)+a(n+1) = A116731(n+2). - R. J. Mathar, Jan 13 2011
Showing 1-4 of 4 results.

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