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.

A097083 Positive values of k such that there is exactly one permutation p of (1,2,3,...,k) such that i+p(i) is a Fibonacci number for 1<=i<=k.

Original entry on oeis.org

1, 2, 3, 5, 9, 15, 24, 39, 64, 104, 168, 272, 441, 714, 1155, 1869, 3025, 4895, 7920, 12815, 20736, 33552, 54288, 87840, 142129, 229970, 372099, 602069, 974169, 1576239, 2550408, 4126647, 6677056, 10803704, 17480760, 28284464, 45765225
Offset: 1

Views

Author

John W. Layman, Jul 23 2004

Keywords

Comments

Numbers k such that A097082(k) = 1. If f is a Fibonacci number and k < f <= 2k, then a permutation for f-k-1 may be extended to a permutation for k, with p(i) = f-i for f-k < i <= k. This explains the sparseness of this sequence. - David Wasserman, Dec 19 2007
If the formula is correct, the bisections give A059840 and A064831. - David Wasserman, Dec 19 2007

Links

Crossrefs

Cf. A073364, A000045.

Programs

  • Mathematica
    a=b=c=d=0;Table[e=a+b+d+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
CoefficientList[Series[x/((x - 1)*(x^2 + 1)*(x^2 + x - 1)), {x,0,50}], x] (* G. C. Greubel, Mar 05 2017 *)
LinearRecurrence[{2,-1,1,0,-1},{1,2,3,5,9},50] (* Harvey P. Dale, Nov 09 2024 *)
  • PARI
    x='x+O('x^50); Vec(x/((x - 1)*(x^2 + 1)*(x^2 + x - 1))) \\ G. C. Greubel, Mar 05 2017

Formula

It appears that {a(n)} satisfies a(1)=1, a(2)=2 and, for n>2, a(n) = F(n+2) - a(n-2) - 1, where {F(k)} is the sequence of Fibonacci numbers, i.e, that the sequence is the partial sums of A006498.
If the partial sum assumption is correct: a(n) = floor(phi^(n+3)/5), where phi=(1+sqrt(5))/2 = A001622, and a(n) = a(n-1) + a(n-2) + ( (n*(n+1)/2) mod 2). - Gary Detlefs, Mar 12 2011
From R. J. Mathar, Mar 13 2011: (Start)
If the partial sum assumption is correct: a(n)= +2*a(n-1) -a(n-2) +a(n-3) -a(n-5).
G.f.: x/( (x-1)*(x^2+1)*(x^2+x-1) ).
a(n) = A000032(n+3)/5 -(-1)^n*A112030(n)/10 - 1/2. (End)
Conjecture: a(n) = floor(F(n+3)/sqrt(5)), where F(n) = A000045(n) are Fibonacci numbers. - Vladimir Reshetnikov, Nov 05 2015

Extensions

a(9) from Ray Chandler, Jul 29 2004
More terms from David Wasserman, Dec 19 2007
Terms > 90000 assuming the partial sums formula by Vladimir Joseph Stephan Orlovsky, Feb 26 2011

A096680 A card-arranging problem: values of n such that there exists more than one permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a cube for every i.

Original entry on oeis.org

112, 115, 116, 117, 119, 124, 125, 126, 127, 128, 129, 130, 133, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212
Offset: 1

Views

Author

Ray Chandler, Jul 25 2004

Keywords

Examples

			117 is in the sequence with permutations
(7,6,...,2,1,117,116,...,9,8) and
(26,25,...,2,1,98,97,...28,27,117,116,...,100,99)

Crossrefs

Cf. A006063, A073364, A095986, A097082, A097083.

A096901 Number of permutations p of (1,2,3,...,n) such that k+p(k) is a triangular number for 1<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 2, 4, 3, 9, 14, 13, 52, 124, 161, 181, 715, 2338, 7073, 8624, 15466, 52858, 150365, 316543, 691771, 1681604, 5324919, 15407311, 37417775, 69725286, 155786456, 579599171, 2600274145, 10530031625, 22971756045, 47057778714, 112946192928
Offset: 0

Views

Author

Ray Chandler, Jul 26 2004

Keywords

Crossrefs

Cf. A006063, A073364, A095986, A097082, A097083, A096680.
See also A000217.

Formula

a(n) = permanent(m), where the n X n matrix m is defined by m(i,j) = 1 or 0, depending on whether i+j is a triangular number or not.

Extensions

More terms from David Wasserman, Nov 27 2007
a(0) and a(31)-a(38) from Martin Ehrenstein, Mar 03 2024

A276735 Number of permutations p of (1..n) such that k+p(k) is a semiprime for 1 <= k <= n.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 5, 12, 87, 261, 324, 1433, 5881, 37444, 196797, 708901, 2020836, 12375966, 105896734, 955344450, 11136621319, 95274505723, 590283352231, 4285001635230, 36417581252044, 272699023606314
Offset: 0

Views

Author

Gary E. Davis, Sep 24 2016

Keywords

Examples

			a(4) = 1 because the permutation (3,4,1,2) added termwise to (1,2,3,4) yields (4,6,4,6) - both semiprimes - and only that permutation does so. a(5) = 3 because exactly 3 permutations - (3,2,1,5,4), (3,4,1,2,5) & (5,4,3,2,1) - added termwise to (1,2,3,4,5) yield semiprime entries.
From _David A. Corneth_, Sep 28 2016 (Start):
The semiprimes up to 10 are 4, 6, 9 and 10. To find a(5), we add (1,2,3,4,5) to some p. Therefore, p(1) in {3, 5}, p(2) in {2, 4}, p(3) in {1, 3}, p(4) in {2, 5} and p(5) in {1, 4, 5}.
If p(1) = 3 then p(3) must be 1. Then {p(2), p(4), p(5)} = {2, 4, 5} for which there are two possibilities.
If p(1) = 5 then p(3) = 3 and p(4) = 2. Then p(2) = 4 and p(5) = 1. So there's one permutation for which p(1) = 5.
This exhausts the options for p(1) and we found 3 permutations. Therefore, a(5) = 3. (End)

Crossrefs

Cf. A073364, A095986, A096904, A097082.

Programs

  • Maple
    with(LinearAlgebra): with(numtheory):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)->
        `if`((s-> bigomega(s)=2)(i+j), 1, 0)))):
seq(a(n), n=0..16);  # Alois P. Heinz, Sep 28 2016
  • Mathematica
    perms[n0_] :=
Module[{n = n0, S, func, T, T2},
  S = Select[Range[2, 2*n], PrimeOmega[#] == 2 &];
  func[k_] := Cases[S, x_ /; 1 <= x - k <= n] - k;
  T = Tuples[Table[func[k], {k, 1, n}]];
  T2 = Cases[T, x_ /; Length[Union[x]] == Length[x]];
  Length[T2]]
Table[perms[n], {n, 0, 12}]
(* Second program (version >= 10): *)
a[0] = 1; a[n_] := Permanent[Table[Boole[PrimeOmega[i + j] == 2], {i, 1, n}, {j, 1, n}]]; Table[an = a[n]; Print[an]; an, {n, 0, 20}] (* Jean-François Alcover, Jul 25 2017 *)
  • PARI
    isok(va, vb)=my(v = vector(#va, j, va[j]+vb[j])); #select(x->(bigomega(x) == 2), v) == #v;
a(n) = my(vpo = numtoperm(n,1)); sum(k=1, n!, vp = numtoperm(n,k); isok(vp, vpo)); \\ Michel Marcus, Sep 24 2016
  • PARI
    listA001358(lim)=my(v=List()); forprime(p=2, sqrtint(lim\1), forprime(q=p, lim\p, listput(v, p*q))); Set(v)
has(v)=for(k=1,#v, if(!setsearch(semi, v[k]+k), return(0))); 1
a(n)=local(semi=listA001358(2*n)); sum(k=1,n!, has(numtoperm(n,k))) \\ Charles R Greathouse IV, Sep 28 2016
  • PARI
    matperm(M)=my(n=matsize(M)[1], innerSums=vectorv(n)); if(n==0, return(1)); sum(x=1,2^n-1, my(k=valuation(x,2), s=M[,k+1], gray=bitxor(x,x>>1)); if(bittest(gray,k), innerSums+=s,innerSums-=s); (-1)^hammingweight(gray)*factorback(innerSums))*(-1)^n
a(n)=matperm(matrix(n,n,x,y,bigomega(x+y)==2)) \\ Charles R Greathouse IV, Oct 03 2016

Extensions

More terms from Alois P. Heinz, Sep 28 2016
Showing 1-4 of 4 results.

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