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.

A257010 Number of sequences of positive integers with length 3 and alternant equal to n.

Original entry on oeis.org

0, 2, 2, 4, 3, 6, 2, 9, 4, 6, 5, 11, 4, 9, 6, 10, 5, 14, 2, 16, 7, 6, 9, 16, 6, 11, 8, 17, 5, 14, 4, 20, 10, 8, 9, 22, 2, 17, 10, 16, 11, 14, 6, 18, 13, 12, 5, 28, 6, 19, 9, 15, 13, 16, 8, 24, 6, 12, 11, 32, 6, 15, 16, 16, 9, 19, 8, 30, 8, 14, 9, 30, 8, 15, 12, 21, 16, 22
Offset: 3

Views

Author

Barry R. Smith, Apr 18 2015

Keywords

Comments

See A257009 for the definition of the alternant of a sequence. The number of sequences of length 1 with given alternant value n is 1, while the number of sequences of length 2 with given alternant value n is d(n), the number of divisors of n (see A000005).
There are infinitely many sequences of length 3 and alternant equal to 2. It is for this reason that the offset is 3.

Examples

			For n=6, the a(6) = 4 sequences with alternant 6 are (1,1,3), (1,3,2), (2,3,1), (3,1,1)

Links

Crossrefs

Cf. A257009, A257011, A257012, A257013, A000012, A000005.

Programs

  • Maple
    Dbm:= proc(b,m) nops(select(t -> (t-1) mod b = 0, numtheory:-divisors(m))) end proc:
seq(add(Dbm(b,b^2+n*b+1)-2, b=1..n-1), n=3..100); # Robert Israel, Jan 24 2016
  • Mathematica
    Length3Q[x_, y_] :=
Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[OddQ[Length[l]], Return[Length[l] == 3],
   If[Last[l] == 1, Return[Length[l] - 1 == 3], Return[Length[l] + 1 == 3]]]];
Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n^2 - 4]], Floor[Sqrt[n^2 - 4]]],
        Mod[# - n^2 + 4, 2] == 0 &]}, {a,
       Select[Divisors[(n^2 - 4 - k^2)/4], # > (Sqrt[n^2 - 4] - k)/2 &]}],
     UnsameQ[#, {}] &], 1], Length3Q[#, n] &]], {n, 3, 60}]
  • PARI
    a(n)={sum(b=1, n-1, sumdiv(b^2+n*b+1, d, (d-1)%b==0) - 2)} \\ Andrew Howroyd, May 01 2020

Formula

a(n) = Sum_{b=1..n-1} (Dbm (b,b^2+nb+1)-2), where Dbm(b,m) is the number of positive divisors of m that are congruent to 1 modulo b. - Barry R. Smith, Jan 24 2016

Extensions

Terms a(61) and beyond from Andrew Howroyd, May 01 2020

A257012 Number of sequences of positive integers with length 5 and alternant equal to n.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 5, 10, 8, 11, 11, 19, 15, 19, 17, 27, 17, 36, 17, 43, 27, 29, 31, 54, 30, 41, 45, 63, 29, 57, 33, 75, 49, 59, 47, 96, 39, 79, 57, 84, 61, 81, 49, 97, 81, 85, 47, 150, 64, 105, 75, 101, 69, 123, 77, 141, 81, 103, 71, 189, 75, 119, 121, 137, 82, 143, 85, 183, 101, 129, 93, 211, 89, 129, 131, 187, 116, 201
Offset: 1

Views

Author

Barry R. Smith, Apr 19 2015

Keywords

Comments

See A257009 for the definition of the alternant of a sequence. The number of sequences of length 1 with given alternant value n is 1, while the number of sequences of length 2 with given alternant value n is d(n), the number of divisors of n (see A000005).

Examples

			The a(7) = 3 sequences with length 5 and alternant 7 are (1,1,1,3,1), (1,2,1,2,1), and (1,3,1,1,1).

Links

Crossrefs

Cf. A257009, A257010, A257011, A257013, A000012, A000005

Programs

  • Mathematica
    Length5Q[x_, y_] :=
 Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[OddQ[Length[l]], Return[Length[l] == 5],
   If[Last[l] == 1, Return[Length[l] - 1 == 5], Return[Length[l] + 1 == 5]]]];
Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n^2 - 4]], Floor[Sqrt[n^2 - 4]]],
        Mod[# - n^2 + 4, 2] == 0 &]}, {a,
       Select[Divisors[(n^2 - 4 - k^2)/4], # > (Sqrt[n^2 - 4] - k)/2 &]}],
     UnsameQ[#, {}] &], 1], Length5Q[#, n] &]], {n, 3, 80}]

A257013 Number of sequences of positive integers with length 6 and alternant equal to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 5, 0, 4, 4, 9, 0, 12, 1, 13, 10, 8, 4, 33, 4, 14, 12, 21, 4, 44, 2, 33, 22, 24, 12, 62, 8, 16, 29, 63, 2, 64, 4, 57, 52, 26, 10, 111, 21, 40, 48, 45, 8, 106, 26, 94, 40, 46, 18, 164, 21, 40, 61, 97, 40, 118, 12, 87, 65, 104, 14, 221, 14, 52, 116, 88, 30, 146, 21, 157
Offset: 1

Views

Author

Barry R. Smith, Apr 19 2015

Keywords

Comments

See A257009 for the definition of the alternant of a sequence. The number of sequences of length 1 with given alternant value n is 1, while the number of sequences of length 2 with given alternant value n is d(n), the number of divisors of n (see A000005).

Examples

			For n=14, the a(14)=4 sequences with alternant 14 and length 6 are (1,1,1,1,4,1), (1,2,1,1,3,1), (1,3,1,1,2,1), and (1,4,1,1,1,1).

Links

Crossrefs

Cf. A257009, A257010, A257011, A257012, A000012, A000005

Programs

  • Mathematica
    Length6Q[x_, y_] :=
 Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[EvenQ[Length[l]], Return[Length[l] == 6],
   If[Last[l] == 1, Return[Length[l] - 1 == 6], Return[Length[l] + 1 == 6]]]]
Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n^2 + 4]], Floor[Sqrt[n^2 + 4]]],
        Mod[# - n^2 - 4, 2] == 0 &]}, {a,
       Select[Divisors[(n^2 + 4 - k^2)/4], # > (Sqrt[n^2 + 4] - k)/2 &]}],
     UnsameQ[#, {}] &], 1], Length6Q[#, n] &]], {n, 1, 80}]
Showing 1-3 of 3 results.

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