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.

A204202 Triangle based on (0,2/3,1) averaging array.

Original entry on oeis.org

2, 2, 5, 2, 7, 11, 2, 9, 18, 23, 2, 11, 27, 41, 47, 2, 13, 38, 68, 88, 95, 2, 15, 51, 106, 156, 183, 191, 2, 17, 66, 157, 262, 339, 374, 383, 2, 19, 83, 223, 419, 601, 713, 757, 767, 2, 21, 102, 306, 642, 1020, 1314, 1470, 1524, 1535, 2, 23, 123, 408, 948
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2012

Keywords

Comments

See A204201 for a discussion of averaging arrays and related triangles

Examples

			First six rows:
2
2...5
2...7....11
2...9....18...23
2...11...27...41...47
2...13...38...68...88..95

Crossrefs

Cf. A204201.

Programs

  • Mathematica
    a = 0; r = 2/3; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]   (* averaging array *)
u = Table[(1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204202 triangle *)
Flatten[u]    (* A204202 sequence *)

Formula

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A055010(n) = A083329(n) = A153893(n-1).
Sum_{k=1..n} T(n,k) = A066373(n+1).
T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=2, T(2,1)=2, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. (End)

A204203 Triangle based on (0,1/4,1) averaging array.

Original entry on oeis.org

1, 1, 5, 1, 6, 13, 1, 7, 19, 29, 1, 8, 26, 48, 61, 1, 9, 34, 74, 109, 125, 1, 10, 43, 108, 183, 234, 253, 1, 11, 53, 151, 291, 417, 487, 509, 1, 12, 64, 204, 442, 708, 904, 996, 1021, 1, 13, 76, 268, 646, 1150, 1612, 1900, 2017, 2045, 1, 14, 89, 344, 914
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2012

Keywords

Comments

See A204201 for a discussion and guide to other averaging arrays.

Examples

			First six rows:
1
1...5
1...6...13
1...7...19...29
1...8...26...48...61
1...9...34...74...109...125

Crossrefs

Cf. A204201.

Programs

  • Mathematica
    a = 0; r = 1/4; b = 1;  t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204203 triangle *)
Flatten[u]    (* A204203 sequence *)

Formula

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A036563(n+1).
Sum_{k=1..n} T(n,k) = A014480(n-1).
T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. (End)

A204204 Triangle based on (0,3/4,1) averaging array.

Original entry on oeis.org

3, 3, 7, 3, 10, 15, 3, 13, 25, 31, 3, 16, 38, 56, 63, 3, 19, 54, 94, 119, 127, 3, 22, 73, 148, 213, 246, 255, 3, 25, 95, 221, 361, 459, 501, 511, 3, 28, 120, 316, 582, 820, 960, 1012, 1023, 3, 31, 148, 436, 898, 1402, 1780, 1972, 2035, 2047, 3, 34, 179
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2012

Keywords

Comments

See A204201 for a discussion and guide to other averaging arrays.

Examples

			First six rows:
3
3...7
3...10...15
3...13...25...31
3...16...38...56...63
3...19...54...94...119..127

Crossrefs

Cf. A204201.

Programs

  • Mathematica
    a = 0; r = 3/4; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]   (* averaging array *)
u = Table[3 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204204 triangle *)
Flatten[u]    (* A204204 sequence *)

Formula

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A000225(n+1).
Sum_{k=1..n} T(n,k) = A128135(n+1).
T(n,k)=T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=3, T(2,1)=3, T(2,2)=7, T(n,k)=0 if k<1 or if k>n. (End)

A204205 Triangle based on (0,1/5,1) averaging array.

Original entry on oeis.org

1, 1, 6, 1, 7, 16, 1, 8, 23, 36, 1, 9, 31, 59, 76, 1, 10, 40, 90, 135, 156, 1, 11, 50, 130, 225, 291, 316, 1, 12, 61, 180, 355, 516, 607, 636, 1, 13, 73, 241, 535, 871, 1123, 1243, 1276, 1, 14, 86, 314, 776, 1406, 1994, 2366, 2519, 2556, 1, 15, 100, 400
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2012

Keywords

Comments

See A204201 for a discussion and guide to other averaging arrays.

Examples

			First six rows:
1
1...6
1...7...16
1...8...23...36
1...9...31...59...76
1...10..40...90...135...156

Crossrefs

Cf. A204201.

Programs

  • Mathematica
    a = 0; r = 1/5; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]   (* A204205 triangle *)
Flatten[u]     (* A204205 sequence *)

Formula

T(n,n) = A048487(n-1). - Philippe Deléham, Dec 24 2013
T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=6, T(n,k)=0 if k<1 or if k>n. - Philippe Deléham, Dec 24 2013

A204206 Triangle based on (1,3/2,2) averaging array.

Original entry on oeis.org

3, 5, 7, 9, 12, 15, 17, 21, 27, 31, 33, 38, 48, 58, 63, 65, 71, 86, 106, 121, 127, 129, 136, 157, 192, 227, 248, 255, 257, 265, 293, 349, 419, 475, 503, 511, 513, 522, 558, 642, 768, 894, 978, 1014, 1023, 1025, 1035, 1080, 1200, 1410, 1662, 1872
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2012

Keywords

Comments

See A204201 for a discussion and guide to other averaging arrays.

Examples

			First six rows:
3
5...7
9...12...15
17..21...27...31
33..38...48...58...63
65..71...86...106..121..127

Crossrefs

Cf. A204201.

Programs

  • Mathematica
    a = 1; r = 3/2; b = 2;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[3 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]   (* A204206 triangle *)
Flatten[u]     (* A204206 sequence *)

Formula

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A000225(n+1).
Sum_{k=1..n} T(n,k) = A167667(n).
T(n,k)=T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=3, T(2,1)=5, T(2,2)=7, T(n,k)=0 if k<1 or if k>n. (End)

A204207 Triangle based on (1,2,3) averaging array.

Original entry on oeis.org

2, 3, 5, 5, 8, 11, 9, 13, 19, 23, 17, 22, 32, 42, 47, 33, 39, 54, 74, 89, 95, 65, 72, 93, 128, 163, 184, 191, 129, 137, 165, 221, 291, 347, 375, 383, 257, 266, 302, 386, 512, 638, 722, 758, 767, 513, 523, 568, 688, 898, 1150, 1360, 1480, 1525, 1535, 1025
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2012

Keywords

Comments

See A204201 for a discussion and guide to other averaging arrays.

Examples

			First six rows:
2
3....5
5....8....11
9....13...19...23
17...22...32...42...47
33...39...54...74...89...95

Crossrefs

Cf. A204201.

Programs

  • Mathematica
    a = 1; r = 2; b = 3;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[2 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u] (* A204207 triangle *)
Flatten[u]   (* A204207 sequence *)

Formula

T(n,n) = A083329(n). - Philippe Deléham, Dec 24 2013
T(n,1) = A000051(n-1). - Philippe Deléham, Dec 24 2013
Sum_{k=1..n} T(n,k)=A036289(n). - Philippe Deléham, Dec 24 2013
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k-1) - 2*T(n-2,k-2), T(1,1)=2, T(2,1)=3, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. - Philippe Deléham, Dec 24 2013

Extensions

Example corrected by Philippe Deléham, Dec 22 2013
Showing 1-6 of 6 results.

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