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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A213747 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 23, 16, 7, 72, 57, 36, 15, 201, 170, 125, 76, 31, 522, 459, 366, 261, 156, 63, 1291, 1164, 975, 758, 533, 316, 127, 3084, 2829, 2448, 2007, 1542, 1077, 636, 255, 7181, 6670, 5905, 5016, 4071, 3110, 2165, 1276, 511, 16398, 15375, 13842
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Comments

Principal diagonal: A213748.
Antidiagonal sums: A213749.
Row 1, (1,3,7,15,31,...)**(1,3,7,15,31,...): A045618.
Row 2, (1,3,7,15,31,...)**(3,7,15,31,...).
Row 3, (1,3,7,15,31,...)**(7,15,31,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6.....23....72.....201
3....16....57....170....459
7....36....125...366....975
15...76....261...758....1007
31...156...533...1542...4071

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := -1 + 2^n; c[n_] := -1 + 2^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}]  (* A213747 *)
Table[t[n, n], {n, 1, 40}] (* A213748 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213749 *)

Formula

T(n,k) = 6*T(n,k-1)-13*T(n,k-2)+12*T(n,k-3)-4*T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = -1 + 2^n - (-2 - 2^n)*x and g(x) = (1 - 3*x + 2*x^2 )^2.

A213753 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = -1 + 2^(n-1+h), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 21, 16, 7, 58, 51, 36, 15, 141, 132, 111, 76, 31, 318, 307, 280, 231, 156, 63, 685, 672, 639, 576, 471, 316, 127, 1434, 1419, 1380, 1303, 1168, 951, 636, 255, 2949, 2932, 2887, 2796, 2631, 2352, 1911, 1276, 511, 5998, 5979, 5928, 5823
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Comments

Principal diagonal: A213754.
Antidiagonal sums: A213755.
Row 1, (1,3,5,7,9,...)**(1,3,7,15,...): A047520.
Row 2, (1,3,5,7,9,...)**(3,7,15,31,...).
Row 3, (1,3,5,7,9,...)**(7,15,31,63...).
Ror a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6.....21....58.....141
3....16....51....132....307
7....36....111...280....639
15...76....231...576....1303
31...156...471...1168...2631

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := 2 n - 1; c[n_] := -1 + 2^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}]  (* A213753 *)
Table[t[n, n], {n, 1, 40}] (* A213754 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213755 *)

Formula

T(n,k) = 5*T(n,k-1)-9*T(n,k-2)+7*T(n,k-3)-2*T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(-1 + 2^n + x + (-2 + 2^n)*x^2) and g(x) = (1 - 2*x)(1 - x )^3.

A213759 Principal diagonal of the convolution array A213783.

Original entry on oeis.org

1, 4, 11, 22, 39, 62, 93, 132, 181, 240, 311, 394, 491, 602, 729, 872, 1033, 1212, 1411, 1630, 1871, 2134, 2421, 2732, 3069, 3432, 3823, 4242, 4691, 5170, 5681, 6224, 6801, 7412, 8059, 8742, 9463, 10222, 11021, 11860, 12741, 13664, 14631
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2012

Keywords

Links

Crossrefs

Cf. A213783, A213500.
Partial sums of A047838. - Guenther Schrack, May 24 2018

Programs

  • Mathematica
    b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/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}]  (* A213783 *)
Table[t[n, n], {n, 1, 40}] (* A213759 *)
LinearRecurrence[{3,-2,-2,3,-1},{1,4,11,22,39},50] (* Harvey P. Dale, Jul 22 2014 *)

Formula

a(n) = (3 - 3*(-1)^n - 4*n + 18*n^2 + 4*n^3)/24.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
G.f.: x*(1 + x + x^2 - x^3)/((1 - x)^4 *(1 + x)).
a(n+1) = a(n) + A047838(n+2) for n > 0. - Guenther Schrack, May 24 2018
a(n) = A212964(n+2) - n for n > 0. - Guenther Schrack, May 30 2018

A213762 Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 3, 15, 11, 5, 37, 29, 17, 7, 83, 67, 43, 23, 9, 177, 145, 97, 57, 29, 11, 367, 303, 207, 127, 71, 35, 13, 749, 621, 429, 269, 157, 85, 41, 15, 1515, 1259, 875, 555, 331, 187, 99, 47, 17, 3049, 2537, 1769, 1129, 681, 393, 217, 113, 53, 19, 6119
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Comments

Principal diagonal: A213763.
Antidiagonal sums: A213764.
Row 1, (1,2,4,8,16,...)**(1,3,5,7,9,...): A050488.
Row 2, (1,2,4,8,16,...)**(3,5,7,9,11,...).
Row 3, (1,2,4,8,16,...)**(5,7,9,11,13,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....5....15...37....83....177
3....11...29...67....145...303
5....17...43...97....207...429
7....23...57...127...269...555
9....29...71...157...331...681
11...35...85...187...393...807

Links

Crossrefs

Cf. A213500, A213756.

Programs

  • Mathematica
    b[n_] := 2^(n - 1); c[n_] := 2 n - 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}]  (* A213762 *)
Table[t[n, n], {n, 1, 40}] (* A213763 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213764 *)

Formula

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

A213765 Rectangular array: (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 4, 1, 10, 5, 2, 21, 14, 9, 3, 40, 31, 24, 14, 5, 72, 61, 52, 38, 23, 8, 125, 112, 101, 83, 62, 37, 13, 212, 197, 184, 162, 135, 100, 60, 21, 354, 337, 322, 296, 263, 218, 162, 97, 34, 585, 566, 549, 519, 480, 425, 353, 262, 157, 55, 960, 939, 920, 886
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Comments

Principal diagonal: A213766.
Antidiagonal sums: A213767.
Row 1, (1,3,5,7,9,...)**(1,1,2,3,5,...): A001891.
Row 2, (1,3,5,7,9,...)**(1,2,3,5,8,...): A023652.
Row 3, (1,3,5,7,9,...)**(2,3,5,8,13,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....4....10....21....40....72
1....5....14....31....61....112
2....9....24....52....101...184
3....14...38....83....162...296
5....23...62....135...263...480
8....37...100...218...425...776
13...60...162...353...688...1256

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := 2 n - 1; c[n_] := Fibonacci[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}]  (* A213765 *)
Table[t[n, n], {n, 1, 40}] (* A213766 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213767 *)

Formula

T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(F(n) + F(n+1)*x - F(n-1)*x^2) and g(x) = (1 - x - x^2)(1 - x )^2.
T(n,k) = F(n+k+4) - 2*k*F(n+1) - F(n+4), F = A000045. - Ehren Metcalfe, Jul 10 2019

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.

A213774 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 3, 14, 11, 5, 31, 26, 17, 7, 61, 53, 38, 23, 9, 112, 99, 75, 50, 29, 11, 197, 176, 137, 97, 62, 35, 13, 337, 303, 240, 175, 119, 74, 41, 15, 566, 511, 409, 304, 213, 141, 86, 47, 17, 939, 850, 685, 515, 368, 251, 163, 98, 53, 19, 1545, 1401, 1134
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Comments

Principal diagonal: A213775.
Antidiagonal sums: A213776.
Row 1, (1,2,3,5,8,...)**(1,3,5,7,9,...): A023652.
Row 2, (1,2,3,5,8,...)**(3,5,7,9,11,...).
Row 3, (1,2,3,5,8,...)**(5,7,9,11,13,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....5....14...31....61....112
3....11...26...53....99....176
5....17...38...75....137...240
7....23...50...97....175...304
9....29...62...119...213...368
11...35...74...141...251...432

Links

Crossrefs

Cf. A023652, A213500, A213768, A213775, A213776.

Programs

  • Mathematica
    b[n_] := Fibonacci[n + 1]; c[n_] := 2 n - 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}]  (* A213774 *)
Table[t[n, n], {n, 1, 40}] (* A213775 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213776 *)

Formula

T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(2*n - 1 + 2*x - (2*n - 3)*x^2) and g(x) = (1 - x - x^2)*(1 - x )^2.
T(n,k) = 2*n*Fibonacci(k+3) + Lucas(k+3) - 4*(k+n+1). - Ehren Metcalfe, Jul 08 2019

A213781 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and ** = convolution.

Original entry on oeis.org

1, 4, 2, 9, 7, 3, 17, 14, 10, 4, 28, 25, 19, 13, 5, 43, 39, 33, 24, 16, 6, 62, 58, 50, 41, 29, 19, 7, 86, 81, 73, 61, 49, 34, 22, 8, 115, 110, 100, 88, 72, 57, 39, 25, 9, 150, 144, 134, 119, 103, 83, 65, 44, 28, 10, 191, 185, 173, 158, 138, 118, 94, 73, 49, 31
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2012

Keywords

Comments

Principal diagonal: A213782.
Antidiagonal sums: A005712.
row 1, (1,2,2,3,3,4,4,...)**(1,2,3,4,5,6,7,...): A005744.
row 2, (1,2,2,3,3,4,4,...)**(2,3,4,5,6,7,8,...).
row 3, (1,2,2,3,3,4,4,...)**(3,4,5,6,7,8,9,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...7....14...25...39...58....81
3...10...19...33...50...73....100
4...13...24...41...61...88....119
5...16...29...49...72...103...138
6...19...34...57...83...118...157
7...22...39...65...94...133...176

Links

Crossrefs

Cf. A213500, A213778.

Programs

  • Mathematica
    b[n_] := Floor[(n + 2)/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}]  (* A213781 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005712 *)

Formula

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

A213782 Principal diagonal of the convolution array A213781.

Original entry on oeis.org

1, 7, 19, 41, 72, 118, 176, 254, 347, 465, 601, 767, 954, 1176, 1422, 1708, 2021, 2379, 2767, 3205, 3676, 4202, 4764, 5386, 6047, 6773, 7541, 8379, 9262, 10220, 11226, 12312, 13449, 14671, 15947, 17313, 18736, 20254, 21832, 23510, 25251, 27097, 29009, 31031
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2012

Keywords

Links

Crossrefs

Cf. A213781, A213500.

Programs

  • Mathematica
    (See A213781.)
LinearRecurrence[{2,1,-4,1,2,-1},{1,7,19,41,72,118},50] (* Harvey P. Dale, Oct 17 2016 *)
  • PARI
    Vec(x*(1+5*x+4*x^2-2*x^4)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016

Formula

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: x*(1+5*x+4*x^2-2*x^4) / ((1-x)^4*(1+x)^2). - Corrected by Colin Barker, Jan 31 2016
From Colin Barker, Jan 31 2016: (Start)
a(n) = (16*n^3+66*n^2+6*(-1)^n*n-34*n-3*(-1)^n+3)/48.
a(n) = (8*n^3+33*n^2-14*n)/24 for n even.
a(n) = (8*n^3+33*n^2-20*n+3)/24 for n odd.
(End)

A213783 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.

Original entry on oeis.org

1, 3, 1, 6, 4, 2, 11, 8, 6, 2, 17, 14, 11, 7, 3, 26, 22, 19, 13, 9, 3, 36, 32, 28, 22, 16, 10, 4, 50, 45, 41, 33, 27, 18, 12, 4, 65, 60, 55, 47, 39, 30, 21, 13, 5, 85, 79, 74, 64, 56, 44, 35, 23, 15, 5, 106, 100, 94, 84, 74, 62, 50, 38, 26, 16, 6, 133, 126, 120, 108
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2012

Keywords

Comments

Principal diagonal: A213759.
Antidiagonal sums: A213760.
Row 1, (1,2,2,3,3,4,4,...)**(1,1,2,2,3,3,4,...): A005744.
Row 2, (1,2,2,3,3,4,4,5,...)**(1,2,2,3,3,4,4,5,...).
Row 3, (1,2,2,3,3,4,4,5,...)**(2,2,3,3,4,4,5,5,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1...3....6....11...17...26...36....50
1...4....8....14...22...32...45....60
2...6....11...19...28...41...55....74
2...7....13...22...33...47...64....84
3...9....16...27...39...56...74....98
3...10...18...30...44...62...83....108
4...12...21...35...50...71...93....122
4...13...23...38...55...77...102...132

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/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}]  (* A213783 *)
Table[t[n, n], {n, 1, 40}] (* A213759 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213760 *)

Formula

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