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.

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A213771 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 2, 18, 11, 3, 40, 30, 16, 4, 75, 62, 42, 21, 5, 126, 110, 84, 54, 26, 6, 196, 177, 145, 106, 66, 31, 7, 288, 266, 228, 180, 128, 78, 36, 8, 405, 380, 336, 279, 215, 150, 90, 41, 9, 550, 522, 472, 406, 330, 250, 172, 102, 46
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2012

Keywords

Comments

Principal diagonal: A213772
Antidiagonal sums: A132117
Row 1, (1,4,7,10,...)**(1,2,3,4,...): A002411
Row 2, (1,4,7,10,...)**(2,3,4,5,...): A162260
Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 - 2*k)/2
Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 - 3*k)/2
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6....18...40....75....126
2....11...30...62....110...177
3....16...42...84....145...228
4....21...54...106...180...279
5....26...66...128...215...330

Links

Crossrefs

Cf. A212500

Programs

  • Mathematica
    b[n_]:=3n-2;c[n_]:=n;
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}] (* A213771 *)
Table[t[n,n],{n,1,40}] (* A213772 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A132117 *)

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*(n + (n+1)*x - (n+2)*x^2) and g(x) = (1 - x)^4.

A213838 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 8, 3, 29, 20, 5, 72, 59, 32, 7, 145, 128, 89, 44, 9, 256, 235, 184, 119, 56, 11, 413, 388, 325, 240, 149, 68, 13, 624, 595, 520, 415, 296, 179, 80, 15, 897, 864, 777, 652, 505, 352, 209, 92, 17, 1240, 1203, 1104, 959, 784
Offset: 1

Views

Author

Clark Kimberling, Jul 05 2012

Keywords

Comments

Principal diagonal: A213839.
Antidiagonal sums: A213840.
Row 1, (1,5,9,13,...)**(1,3,5,7,...): A100178.
Row 2, (1,5,9,13,...)**(3,5,7,9,...): (4*k^3 + 9*k^2 - 4*k)/3.
Row 3, (1,5,9,13,...)**(5,7,9,11,...): (4*k^3 + 21*k^2 - 10*k)/3.
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....8....29....72....145
3....20...59....128...235
5....32...89....184...325
7....44...119...240...415
9....56...149...296...505
11...68...179...352...595

Links

Crossrefs

Cf. A212500.

Programs

  • Mathematica
    b[n_]:=4n-3; 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}] (* A213838 *)
Table[t[n,n],{n,1,40}] (* A213839 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213840 *)

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*(2*n-1 + 4*n*x - (6*n-9)*x^2) and g(x) = (1-x)^4.

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.

A212593 a(n) is the difference between multiples of 9 with even and odd digit sum in base 8 in interval [0,8^n).

Original entry on oeis.org

1, 8, 15, 232, 449, 7400, 14351, 237832, 461313, 7648968, 14836623, 246015528, 477194433, 7912700328, 15348206223, 254499628104, 493651049985, 8185582834056, 15877514618127, 263276481572712, 510675448527297, 8467876653984360
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Keywords

Links

Crossrefs

Cf. A038754, A212500, A212592, A091042.

Programs

  • Mathematica
    LinearRecurrence[{0, 36, 0, -126, 0, 84, 0, -9}, {1, 8, 15, 232, 449, 7400, 14351, 237832}, 22] (* Bruno Berselli, May 22 2012 *)

Formula

For n>=9, a(n) = 36*a(n-2)-126*a(n-4)+84*a(n-6)-9*a(n-8).
G.f.: x*(1+8*x-21*x^2-56*x^3+35*x^4+56*x^5-7*x^6-8*x^7)/((1-3*x^2)*(1-33*x^2+27*x^4-3*x^6)). [Bruno Berselli, May 22 2012]

A212670 a(n) = 1/15*(128*n^5 + 320*n^4 + 80*n^3 - 200*n^2 + 92*n - 15).

Original entry on oeis.org

27, 615, 3843, 14351, 40363, 94711, 195859, 368927, 646715, 1070727, 1692195, 2573103, 3787211, 5421079, 7575091, 10364479, 13920347, 18390695, 23941443, 30757455, 39043563, 49025591, 60951379, 75091807, 91741819, 111221447, 133876835, 160081263, 190236171
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

Keywords

Comments

a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 128*n^7).

Links

Crossrefs

Cf. A038754, A084990, A091042, A212500, A212592, A212593, A212594, A212668, A212669.

Programs

  • Mathematica
    Table[(1/15) (8 n^2 - 4 n + 1) (16 n^3 + 48 n^2 + 32 n - 15), {n, 29}] (* Bruno Berselli, May 24 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{27,615,3843,14351,40363,94711},30] (* Harvey P. Dale, Apr 30 2018 *)
  • PARI
    Vec(x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6 + O(x^50)) \\ Colin Barker, Dec 01 2015

Formula

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^7(Pi*i/(2*n+1))*sin(2*Pi*i/(2*n+1)).
G.f.: x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6. [Bruno Berselli, May 24 2012]

A213773 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 8, 4, 30, 23, 7, 76, 66, 38, 10, 155, 142, 102, 53, 13, 276, 260, 208, 138, 68, 16, 448, 429, 365, 274, 174, 83, 19, 680, 658, 582, 470, 340, 210, 98, 22, 981, 956, 868, 735, 575, 406, 246, 113, 25, 1360, 1332, 1232, 1078
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2012

Keywords

Comments

Principal diagonal: A213782
Antidiagonal sums: A214092
Row 1, (1,4,7,10,…)**(1,4,7,10,…): A100175
Row 2, (1,4,7,10,…)**(4,7,10,13,…): (3*k^3 + 6*k^2 - k)/2
Row 3, (1,4,7,10,…)**(7,10,13,16,…): (3*k^3 + 15*k^2 - 4*k)/2
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....8....30....76....155...276
4....23...66....142...260...429
7....38...102...208...365...582
10...53...138...274...470...735
13...68...174...340...575...888

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_]:=3n-2;c[n_]:=3n-2;
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}] (* A213773 *)
Table[t[n,n],{n,1,40}] (* A214092 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213818 *)

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(3*n-2 + (3*n+1)*x - (6*n-10)*x^2) and g(x) = (1-x)^4.

A213821 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

2, 9, 4, 24, 16, 6, 50, 39, 23, 8, 90, 76, 54, 30, 10, 147, 130, 102, 69, 37, 12, 224, 204, 170, 128, 84, 44, 14, 324, 301, 261, 210, 154, 99, 51, 16, 450, 424, 378, 318, 250, 180, 114, 58, 18, 605, 576, 524, 455, 375, 290, 206
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2012

Keywords

Comments

Principal diagonal: A033431.
Antidiagonal sums: A176060.
Row 1, (2,5,8,11,…)**(1,2,3,4,…): A006002.
Row 2, (2,5,8,11,…)**(2,3,4,5,…): (k^3 + 5*k^2 + 2*k)/2.
Row 3, (1,2,3,4,…)**(8,11,14,17,…): (k^3 + 8*k^2 + 3*k)/2.
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
2….9….24…50….90
4….16…39…76…130
6….23…54…102…170
8….30…69…128…210
10…37…84…154…250
12…44…99…180…290

Crossrefs

Cf. A212500

Programs

  • Mathematica
    b[n_]:=3n-1;c[n_]:=n;
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}] (* A213821 *)
Table[t[n,n],{n,1,40}] (* A033431 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A176060 *)

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*(2*n - (n-2)*x - (n-1)*x^2) and g(x) = (1-x)^4.

A213822 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

4, 20, 10, 57, 41, 16, 124, 102, 62, 22, 230, 202, 147, 83, 28, 384, 350, 280, 192, 104, 34, 595, 555, 470, 358, 237, 125, 40, 872, 826, 726, 590, 436, 282, 146, 46, 1224, 1172, 1057, 897, 710, 514, 327, 167, 52, 1660, 1602
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2012

Keywords

Comments

Principal diagonal: A213823.
Antidiagonal sums: A213824.
Row 1, (2,5,8,11,...)**(2,5,8,11,...): (3*k^3 + 3*k^2 + 2*k)/2.
Row 2, (2,5,8,11,...)**(5,8,11,14,...): (3*k^3 + 12*k^2 + 5*k)/2.
Row 3, (2,5,8,11,...)**(8,11,14,17,...): (3*k^3 + 21*k^2 + 8*k)/2.
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
4....20....57....124...230
10...41....102...202...350
16...62....147...280...470
22...83....192...358...590
28...104...237...436...710

Links

Crossrefs

Cf. A212500.

Programs

  • Mathematica
    b[n_]:=3n-1;c[n_]:=3n-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}] (* A213822 *)
Table[t[n,n],{n,1,40}] (* A213823 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213824 *)

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-2) - (3*n-7)*x - (3*n-4)*x^2) and g(x) = (1-x)^4.

A213825 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

2, 13, 8, 42, 34, 14, 98, 87, 55, 20, 190, 176, 132, 76, 26, 327, 310, 254, 177, 97, 32, 518, 498, 430, 332, 222, 118, 38, 772, 749, 669, 550, 410, 267, 139, 44, 1098, 1072, 980, 840, 670, 488, 312, 160, 50, 1505, 1476, 1372
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2012

Keywords

Comments

Principal diagonal: A213826
Antidiagonal sums: A213827
Row 1, (2,5,8,13,...)**(1,4,7,10,13,...): (3*k^2 + k)/2
Row 2, (2,5,8,13,...)**(4,7,10,13,...): (3*k^3 + 9*k^2 - 2*k)/2
Row 3, (2,5,8,13,...)**(7,10,13,16,...): (3*k^3 + 18*k^2 - 5*k)/2
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
2....13....42....98....190
8....34....87....176...310
14...55....132...254...430
20...76....177...332...550
26...97....222...410...670
32...118...267...488...790

Links

Crossrefs

Cf. A212500

Programs

  • Mathematica
    b[n_]:=3n-1;c[n_]:=3n-2;
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}] (* A213825 *)
d=Table[t[n,n],{n,1,40}] (* A213826 *)
d/2 (* A024215 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
s1=Table[s[n],{n,1,50}] (* A213827 *)

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*((3*n-1) + (3*n+2)*x - (6*n-8)*x^2) and g(x) = (1-x)^4.
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