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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A193588 A Fibonacci triangle: T(n,k) = Fib(k+2) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 5, 1, 2, 3, 5, 8, 1, 2, 3, 5, 8, 13, 1, 2, 3, 5, 8, 13, 21, 1, 2, 3, 5, 8, 13, 21, 34, 1, 2, 3, 5, 8, 13, 21, 34, 55, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

n-th row sum: A001911, Fib(n+3)-2;
n-th alternating row sum: A000045, F(n).
The augmentation (as defined at A193091) of A193588 is A193589.

Examples

			First 5 rows of A193588:
  1;
  1, 2;
  1, 2, 3;
  1, 2, 3, 5;
  1, 2, 3, 5, 8;

Crossrefs

Cf. A193588.

Programs

  • Mathematica
    (See A193589, the augmentation of A193588.)
Table[Fibonacci[k+2],{n,0,20},{k,0,n}]//Flatten (* Harvey P. Dale, Nov 29 2017 *)
Module[{nn=15,fibs},fibs=Fibonacci[Range[2,nn]];Table[Take[fibs,n],{n,nn-1}]]// Flatten (* Harvey P. Dale, Mar 02 2023 *)

Formula

a(n) = A115346(n) + 1. - Filip Zaludek, Nov 19 2016

A193593 Augmentation of the triangle A193592. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 31, 40, 23, 1, 15, 75, 166, 187, 105, 1, 21, 155, 530, 958, 993, 549, 1, 28, 287, 1415, 3786, 5988, 5865, 3207, 1, 36, 490, 3311, 12441, 28056, 40380, 37947, 20577, 1, 45, 786, 7000, 35469, 109451, 217720, 292092
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193592, (column 1)=A014616, (column 2)=A090809, (right edge)=A113227.

Examples

			First 5 rows:
1
1...1
1...3...2
1...6...10...6
1...10..31...40...23
Rows reversed as in Callan's n-edge increasing ordered trees with outdegree k:
 1
0      1
0      1      1
0      2      3      1
0      6     10      6      1
0     23     40     31     10      1
0    105    187    166     75     15      1
0    549    993    958    530    155     21     1
0   3207   5865   5988   3786   1415    287    28    1
0  20577  37947  40380  28056  12441   3311   490   36   1
0 143239 265901 292092 217720 109451  35469  7000  786  45 1

Links

Crossrefs

Cf. A193091, A193592, A113227 (row sums and diagonal), A090809 (3rd col).

Programs

  • Mathematica
    p[n_, 0] := 1; p[n_, k_] := n + 1 - k /; k > 0;
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A193592 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 12}]]  (* A193593 *)
Flatten[Table[v[n], {n, 0, 10}]]

A193595 Augmentation of the Fibonacci triangle A058071. See Comments.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 8, 9, 13, 30, 42, 58, 56, 85, 240, 360, 480, 576, 533, 821, 3120, 4800, 6600, 7488, 8698, 7666, 12015, 65520, 102960, 141120, 165240, 178158, 200200, 171501, 271601, 2227680, 3538080, 4876560, 5670720, 6310590, 6513474
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193595, (column 1)=A003266, (column 2)=A191994.

Examples

			First 5 rows of A193589:
1
1....1
2....2....3
6....8....9....13
30...42...58...56...85

Crossrefs

Cf. A193091, A058071, A003266, A191994.

Programs

  • Mathematica
    p[n_, k_] := Fibonacci[k + 1]*Fibonacci[n + 1 - k]
Table[p[n, k], {n, 0, 5}, {k, 0,
  n}]  (* A058071, a Fibonacci triangle *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 10}]]  (* A193595 *)
Flatten[Table[v[n], {n, 0, 9}]]

A193597 Augmentation of the triangle A193596. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 6, 5, 1, 6, 17, 21, 16, 1, 9, 34, 78, 86, 61, 1, 12, 69, 201, 397, 401, 269, 1, 16, 116, 522, 1282, 2250, 2113, 1350, 1, 20, 194, 1074, 4099, 8900, 14187, 12509, 7650, 1, 25, 292, 2172, 10078, 34044, 67316, 99102, 82713, 48634, 1
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First 5 rows of A193596:
1
1...1
1...1...1
1...2...2...1
1...2...3...2...1

Crossrefs

Cf. A193596, A193091.

Programs

  • Mathematica
    p[n_, k_] := Ceiling[Binomial[n, k]/2]
Table[p[n, k], {n, 0, 10}, {k, 0, n}]
Flatten[%]  (* A193596 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 10}]] (* A193597 *)
Flatten[Table[v[n], {n, 0, 10}]]

A193606 Augmentation of the triangle A193605. See Comments.

Original entry on oeis.org

1, 1, 3, 1, 7, 17, 1, 12, 57, 127, 1, 18, 134, 531, 1125, 1, 25, 265, 1556, 5513, 11279, 1, 33, 470, 3793, 19152, 62675, 124837, 1, 42, 772, 8175, 55297, 250524, 771121, 1502679, 1, 52, 1197, 16087, 140269, 834879, 3478204, 10185019, 19480445
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First five rows of A193606:
1
1...3
1...7....17
1...12...57....127
1...18...134...531...1125

Crossrefs

Cf. A193605, A193091.

Programs

  • Mathematica
    u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}]
p[n_, k_] := Sum[u[n, h], {h, 0, k}]
Table[p[n, k], {n, 0, 12}, {k, 0, n}] (* A193695 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193606 *)
Flatten[Table[v[n], {n, 0, 8}]]

A193607 Augmentation of the triangle A011973. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 20, 7, 1, 16, 80, 131, 37, 1, 25, 220, 806, 1085, 265, 1, 36, 490, 3130, 9360, 10952, 2402, 1, 49, 952, 9325, 48224, 124498, 130852, 26371, 1, 64, 1680, 23317, 183569, 813886, 1876101, 1809430, 340272
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
(col 2): A000290 (the squares)
(col 3)= 2*A002419

Examples

			First five rows of A193607:
1
1...1
1...4....2
1...9....20...7
1...16...80...131...37

Crossrefs

Cf. A011973, A193091, A002419.

Programs

  • Mathematica
    p[n_, k_] := Binomial[2 n - k, k];
Table[p[n, k], {n, 0, 7}, {k, 0, n}]  (* A011973 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 9}]] (* A193607 *)
Flatten[Table[v[n], {n, 0, 8}]]

A193559 Augmentation of the triangular array |A123191|. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 7, 17, 7, 1, 10, 41, 82, 32, 1, 13, 74, 238, 434, 166, 1, 16, 116, 502, 1412, 2446, 926, 1, 19, 167, 901, 3317, 8587, 14405, 5419, 1, 22, 227, 1462, 6581, 21802, 53381, 87610, 32816, 1, 25, 296, 2212, 11717, 46681, 143666, 338038
Offset: 0

Views

Author

Clark Kimberling, Jul 30 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First five rows of |A123191|:
1
1...1
1...3...1
1...3...3...1
1...3...3...3...1
First 5 rows of A193559:
1
1...1
1...4...2
1...7...17...7
1...10..41...82...32

Crossrefs

Cf. A193091.

Programs

  • Mathematica
    p[n_, k_] := If[Or[k == 0, k == n], 1, 3]
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* Abs. value of A123191 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193559 *)
Flatten[Table[v[n], {n, 0, 10}]]

A193590 Augmentation of the Euler triangle A008292. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 5, 2, 1, 16, 33, 8, 1, 42, 275, 342, 58, 1, 99, 1669, 6441, 5600, 718, 1, 219, 8503, 82149, 217694, 143126, 14528, 1, 466, 39076, 843268, 5466197, 10792622, 5628738, 466220, 1, 968, 168786, 7621160, 107506633, 509354984, 788338180
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193590, (column 1)=A002662, with general term 2^n-1-n(n+1)/2.

Examples

			First 5 rows of A193589:
1
1....1
1....5....2
1....16...33....8
1....42...275...342....58

Crossrefs

Cf. A008292, A193091.

Programs

  • Mathematica
    p[n_, k_] :=
Sum[((-1)^j)*((k + 1 - j)^(n + 1))*Binomial[n + 2, j], {j, 0, k + 1}]
(* A008292, Euler triangle *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193590  *)
Flatten[Table[v[n], {n, 0, 8}]]

A193591 Augmentation of the Euler partition triangle A026820. See Comments.

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 7, 19, 31, 1, 10, 45, 103, 161, 1, 14, 82, 297, 617, 937, 1, 18, 146, 652, 2057, 4005, 5953, 1, 23, 228, 1395, 5251, 15004, 27836, 40668, 1, 28, 355, 2555, 13023, 43470, 115110, 205516, 295922, 1, 34, 509, 4689, 27327, 122006, 371942
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First 5 rows:
  1
  1...2
  1...4...7
  1...7...19...31
  1...10..45...103...161

Crossrefs

Cf. A014616 (column 1), A026820, A193091.

Programs

  • Mathematica
    p[n_, k_] := Length@IntegerPartitions[n + 1,
   k + 1] (* A026820, Euler partition triangle *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 12}]]  (* A193591 *)
Flatten[Table[v[n], {n, 0, 9}]]

A193601 Augmentation of the triangle A062344. See Comments.

Original entry on oeis.org

1, 1, 2, 1, 6, 10, 1, 12, 49, 76, 1, 20, 149, 508, 756, 1, 30, 354, 2082, 6353, 9192, 1, 42, 720, 6484, 32852, 92750, 131406, 1, 56, 1315, 16820, 127365, 580606, 1545757, 2153912, 1, 72, 2219, 38256, 404559, 2706150, 11385058, 28931758, 39768798
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First 5 rows of A193601:
1
1...2
1...6....10
1...12...49...76
1...20...149..508...756

Crossrefs

Cf. A193091, A062344.

Programs

  • Mathematica
    p[n_, k_] := Binomial[2 n, k] (* A062344 *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193601 *)
Flatten[Table[v[n], {n, 0, 8}]]
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