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 1-4 of 4 results.

A202453 Fibonacci self-fusion matrix, by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 6, 5, 5, 8, 8, 9, 9, 8, 8, 13, 13, 15, 15, 15, 13, 13, 21, 21, 24, 24, 24, 24, 21, 21, 34, 34, 39, 39, 40, 39, 39, 34, 34, 55, 55, 63, 63, 64, 64, 63, 63, 55, 55, 89, 89, 102, 102, 104, 104, 104, 102, 102, 89, 89, 144, 144, 165, 165
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2011

Keywords

Comments

The Fibonacci self-fusion matrix, F, is the fusion P**Q, where P and Q are the lower and upper triangular Fibonacci matrices. See A193722 for the definition of fusion of triangular arrays.
Every term F(n,k) of F is a product of two Fibonacci numbers; indeed,
F(n,k)=F(n)*F(k+1) if k is even;
F(n,k)=F(n+1)*F(k) if k is odd.
antidiagonal sums: (1,2,6,12,...), A054454
diagonal (1,2,6,15,...), A001654
diagonal (1,3,9,24,...), A064831
diagonal (2,5,15,39,..), A059840
diagonal (3,8,24,63,..), A080097
diagonal (5,13,39,102,...), A080143
diagonal (8,21,63,165,...), A080144
principal submatrix sums, A202462
All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.

Examples

			Northwest corner:
1...1....2....3....5....8....13
1...2....3....5....8...13....21
2...3....6....9...15...24....39
3...5....9...15...24...39....63
5...8...15...24...40...64...104

Links

Crossrefs

Cf. A000045, A202451, A202452, A202503 (Fibonacci fission array).

Programs

  • Mathematica
    n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
TableForm[P]  (* A202452, lower tri. Fibonacci array *)
TableForm[F]  (* A202453, Fibonacci fusion array *)
TableForm[FactorInteger[F]]

Formula

Matrix product P*Q, where P, Q are the lower and upper triangular Fibonacci matrices, A202451 and A202452.

A202451 Upper triangular Fibonacci matrix, by SW antidiagonals.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 19 2011

Keywords

Examples

			Northwest corner:
1...1...2...3...5...8...13...21...34
0...1...1...2...3...5....8...13...21
0...0...1...1...2...3....5....8...13
0...0...0...1...1...2....3....5....8

Links

Crossrefs

Cf. A000045, A188516, A202452, A202453, A202462.

Programs

  • Mathematica
    n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper triangular Fibonacci matrix *)
TableForm[P]  (* A202452, lower triangular Fibonacci matrix *)
TableForm[F]  (* A202453, Fibonacci self-fusion matrix *)
TableForm[FactorInteger[F]]

Formula

Row n consists of n-1 zeros followed by the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...).

A202452 Lower triangular Fibonacci matrix, by SW antidiagonals.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 19 2011

Keywords

Examples

			Northwest corner:
1...0...0...0...0...0...0...0...0
1...1...0...0...0...0...0...0...0
2...1...1...0...0...0...0...0...0
3...2...1...1...0...0...0...0...0
5...3...2...1...1...0...0...0...0

Links

Crossrefs

Cf. A202451, A202453, A202462, A188516, A000045.

Programs

  • Mathematica
    n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper triangular Fibonacci array *)
TableForm[P]  (* A202452, lower triangular Fibonacci array *)
TableForm[F]  (* A202453, Fibonacci self-fusion matrix *)
TableForm[FactorInteger[F]]

Formula

Column n consists of n-1 zeros followed by the Fibonacci sequence (1,1,2,3,5,8,...).

A202876 Symmetric matrix based on A000071, by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 10, 10, 7, 12, 18, 21, 18, 12, 20, 31, 38, 38, 31, 20, 33, 52, 66, 70, 66, 52, 33, 54, 86, 111, 122, 122, 111, 86, 54, 88, 141, 184, 206, 214, 206, 184, 141, 88, 143, 230, 302, 342, 362, 362, 342, 302, 230, 143, 232, 374, 493, 562, 602
Offset: 1

Views

Author

Clark Kimberling, Dec 26 2011

Keywords

Comments

Let s=A000071 (Fibonacci numbers -1), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202876 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202877 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

Examples

			Northwest corner:
1....2....4....7....12....20
2....5....10...18...31....52
4....10...21...38...66....111
7....18...38...70...122...206
12...31...66...122..214...362

Crossrefs

Cf. A202877, A202876.

Programs

  • Mathematica
    s[k_] := -1 + Fibonacci[k + 2];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A001924 *)
Table[m[1, j], {j, 1, 12}]     (* A000071 *)
Table[m[j, j], {j, 1, 12}]     (* A202462 *)
Showing 1-4 of 4 results.

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