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 1-5 of 5 results.

A049874 a(n)=b(n)-b(n-1), where b=A049862 (differences of products of Fibonacci numbers.)

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 2, 1, 5, 3, 2, 8, 5, 1, 2, 13, 8, 2, 3, 21, 13, 2, 1, 5, 34, 21, 3, 2, 8, 55, 34, 5, 1, 2, 13, 89, 55, 8, 2, 3, 21, 144, 89, 13, 2, 1, 5, 34, 233, 144, 21, 3, 2, 8, 55, 377, 233, 34, 5, 1, 2, 13, 89, 610, 377, 55, 8, 2, 3, 21, 144, 987, 610, 89, 13, 2, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049862.

Programs

  • PARI
    lista(nn) = {my(out = List([0])); for (i=0, nn, for (j=i+1, nn, listput(out, fibonacci(i)*fibonacci(j)););); my(v = Vec(vecsort(select(x->(x < fibonacci(nn+1)), out), , 8))); vector(#v-1, k, v[k+1] - v[k]);} \\ Michel Marcus, May 27 2019

Extensions

More terms from Michel Marcus, May 27 2019

A160009 Numbers that are the product of distinct Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 13, 15, 16, 21, 24, 26, 30, 34, 39, 40, 42, 48, 55, 63, 65, 68, 78, 80, 89, 102, 104, 105, 110, 120, 126, 130, 144, 165, 168, 170, 178, 195, 204, 208, 210, 233, 240, 267, 272, 273, 275, 288, 312, 315, 330, 336, 340, 377, 390, 432, 440, 442, 445
Offset: 1

Views

Author

T. D. Noe, Apr 29 2009

Keywords

Comments

Starts the same as A049862, the product of two distinct Fibonacci numbers. This sequence has an infinite number of consecutive terms that are consecutive numbers (such as 15 and 16) because fib(k)*fib(k+3) and fib(k+1)*fib(k+2) differ by one for all k >= 0.
It follows from Carmichael's theorem that if u and v are finite sets of Fibonacci numbers such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. The same holds for many other 2nd order linear recurrence sequences with constant coefficients. In the following guide to related "distinct product sequences", W = Wythoff array, A035513:
base sequence distinct-product sequence
A000045 (Fibonacci) A160009
A000032 (Lucas, without 2) A274280
A000032 (Lucas, with 2) A274281
A000285 (1,4,5,...) A274282
A022095 (1,5,6,...) A274283
A006355 (2,4,6,...) A274284
A013655 (2,5,7,...) A274285
A022086 (3,6,9,...) A274191
row 2 of W: (4,7,11,...) A274286
row 3 of W: (6,10,16,...) A274287
row 4 of W: (9,15,24,...) A274288
- Clark Kimberling, Jun 17 2016

Links

Crossrefs

A059844, A065108

Programs

  • Mathematica
    s={1}; nn=30; f=Fibonacci[2+Range[nn]]; Do[s=Union[s,Select[s*f[[i]],#<=f[[nn]]&]], {i,nn}]; s=Prepend[s,0]

A274374 Products of 2 distinct Fibonacci numbers and products of two distinct Lucas numbers (without 2), arranged in increasing order.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 18, 21, 24, 26, 28, 29, 33, 34, 39, 40, 42, 44, 47, 54, 55, 63, 65, 68, 72, 76, 77, 87, 89, 102, 104, 105, 110, 116, 123, 126, 141, 144, 165, 168, 170, 178, 188, 198, 199, 203, 228, 233, 267, 272, 273, 275, 288
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2016

Keywords

Comments

Are 3 and 21 the only numbers that are a product of two distinct Fibonacci numbers and also a product of two distinct Lucas numbers?

Links

Crossrefs

Cf. A049862, A274347, A274375.

Programs

  • Mathematica
    z = 400; f[n_] := Fibonacci[n];
s = Join[{0}, Take[Sort[Flatten[Table[f[m] f[n], {n, 2, z}, {m, 2, n - 1}]]], z]]
g[n_] := LucasL[n]; t = Take[Sort[Flatten[Table[g[u] g[v], {u, 1, z}, {v, 1, u - 1}]]], z]
Union[s, t]

A274375 Products of 2 distinct Fibonacci numbers and products of two distinct Lucas numbers (including 2), arranged in increasing order.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 21, 22, 24, 26, 28, 29, 33, 34, 36, 39, 40, 42, 44, 47, 54, 55, 58, 63, 65, 68, 72, 76, 77, 87, 89, 94, 102, 104, 105, 110, 116, 123, 126, 141, 144, 152, 165, 168, 170, 178, 188, 198, 199, 203, 228, 233
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2016

Keywords

Comments

Are 2,3,6,8,21 the only numbers that are a product of two distinct Fibonacci numbers and also a product of two distinct Lucas numbers (including 2)?

Links

Crossrefs

Cf. A049862, A274349, A274374.

Programs

  • Mathematica
    z = 400; f[n_] := Fibonacci[n];
s = Join[{0}, Take[Sort[Flatten[Table[f[m] f[n], {n, 2, z}, {m, 2, n - 1}]]], z]]
g[n_] := LucasL[n - 1]; t = Take[Sort[Flatten[Table[g[u] g[v], {u, 1, z}, {v, 1, u - 1}]]], z]
Union[s, t]

A049996 a(n) is the index k such that F(k)=d(n), where d=A049874 (difference sequence of ordered products of Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 3, 4, 3, 1, 5, 4, 3, 6, 5, 1, 3, 7, 6, 3, 4, 8, 7, 3, 1, 5, 9, 8, 4, 3, 6, 10, 9, 5, 1, 3, 7, 11, 10, 6, 3, 4, 8, 12, 11, 7, 3, 1, 5, 9, 13, 12, 8, 4, 3, 6, 10, 14, 13, 9, 5, 1, 3, 7, 11, 15, 14, 10, 6, 3, 4, 8, 12, 16, 15, 11, 7, 3, 1, 5, 9, 13, 17
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A049862, A049874.

Programs

  • Mathematica
    Block[{nn = 123, s, t}, s = Differences@ Take[#, nn] &@ Union@ Flatten[Table[Fibonacci[i]*Fibonacci[j], {i, 0, nn}, {j, i + 1, nn}]]; t = Fibonacci@ Range@ nn; Array[First@ FirstPosition[t, s[[#]] ] &, Length@ s]] (* Michael De Vlieger, May 27 2019 *)
  • PARI
    ifib(n) = if (n==1, 1, log(n*sqrt(5) + 1/2)\log((1+sqrt(5))/2));
lista(nn) = {my(out = List([0])); for (i=0, nn, for (j=i+1, nn, listput(out, fibonacci(i)*fibonacci(j)););); my(v = Vec(vecsort(select(x->(x < fibonacci(nn+1)), out), , 8))); my(w = vector(#v-1, k, v[k+1] - v[k])); vector(#w, k, ifib(w[k]));} \\ Michel Marcus, May 27 2019

Extensions

More terms from Michel Marcus, May 27 2019
Showing 1-5 of 5 results.

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