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 1-3 of 3 results.

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

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}]]

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}]]
Showing 1-3 of 3 results.

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