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 1-6 of 6 results.

A308077 G.f. A(x) satisfies: A(x) = x - A(x^2) + A(x^3) - A(x^4) + A(x^5) - A(x^6) + ...

Original entry on oeis.org

1, -1, 1, 0, 1, -3, 1, 0, 2, -3, 1, 2, 1, -3, 3, 0, 1, -8, 1, 2, 3, -3, 1, 0, 2, -3, 4, 2, 1, -13, 1, 0, 3, -3, 3, 10, 1, -3, 3, 0, 1, -13, 1, 2, 8, -3, 1, 0, 2, -8, 3, 2, 1, -20, 3, 0, 3, -3, 1, 18, 1, -3, 8, 0, 3, -13, 1, 2, 3, -13, 1, -4, 1, -3, 8, 2, 3, -13, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2019

Keywords

Links

Crossrefs

Cf. A006005 (positions of 1's), A067856, A307776, A347031.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, n, add(a(n/d)*
     (-1)^(d-1), d=numtheory[divisors](n) minus {1}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 30 2023
  • Mathematica
    terms = 79; A[] = 0; Do[A[x] = x + Sum[(-1)^(k + 1) A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := If[n == 1, n, Sum[If[d < n, (-1)^(n/d + 1) a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 79}]

Formula

a(1) = 1; a(n) = Sum_{d|n, d

A348956 a(0) = 1; a(n) = Sum_{d|n, d < n} (-1)^(n/d + 1) * a(d - 1).

Original entry on oeis.org

1, 0, -1, 1, -1, 1, 0, 1, -2, 0, 0, 1, 0, 1, -1, -1, -3, 1, 3, 1, 1, 0, -1, 1, -2, 0, -1, -2, -1, 1, 3, 1, -2, 0, 2, 0, 2, 1, -4, 0, -1, 1, 1, 1, 0, -4, 0, 1, -6, 1, 2, -3, 0, 1, 5, 0, 0, 3, 0, 1, 3, 1, -4, -1, -3, 0, 3, 1, 3, -1, -1, 1, 0, 1, -3, -4, -4, 1, 5, 1, -4
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A067856, A167865, A307776, A308077, A335062, A338639, A343371.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[If[d < n, (-1)^(n/d + 1) a[d - 1], 0], {d, Divisors[n]}]; Table[a[n], {n, 0, 80}]
nmax = 80; A[] = 0; Do[A[x] = 1 - Sum[(-x)^k A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    A348956(n) = if(!n,1,sumdiv(n,d,if(dA348956(d-1)*(-1)^(1 + (n/d)),0))); \\ Antti Karttunen, Nov 05 2021

Formula

G.f. A(x) satisfies: A(x) = 1 - x^2 * A(x^2) + x^3 * A(x^3) - x^4 * A(x^4) + ...

A307778 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(d+1)*a(d).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 0, 1, -2, -1, 0, 1, -2, -1, 1, 1, -3, -2, 0, 1, -2, -1, 1, 2, -5, -5, 3, 2, -2, -1, 2, 3, -6, -5, 2, 2, -4, -3, 3, 2, -5, -4, 3, 4, -4, -5, 6, 7, -13, -12, 7, 5, -3, -2, 5, 5, -8, -7, 5, 6, -7, -6, 8, 5, -11, -13, 8, 9, -8, -6, 9, 10, -17, -16, 12, 8, -6
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A003238, A281487, A307776, A307777, A307779.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[(-1)^(d + 1) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 77}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[(-1)^(k + 1) a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 77}]

Formula

G.f.: x * (1 + Sum_{n>=1} (-1)^(n+1)*a(n)*x^n/(1 - x^n)).
L.g.f.: log(Product_{n>=1} (1 - x^n)^((-1)^n*a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A307779 a(1) = 1; a(n+1) = Sum_{d|n, n/d odd} a(d).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 11, 12, 13, 17, 17, 18, 22, 23, 25, 31, 32, 33, 38, 41, 42, 49, 51, 52, 63, 64, 64, 74, 75, 82, 93, 94, 95, 108, 113, 114, 130, 131, 133, 155, 156, 157, 174, 179, 187, 206, 208, 209, 231, 242, 247, 271, 272, 273, 307, 308, 309, 345, 345
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A002033, A003238, A307776, A307777, A307778.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[Boole[OddQ[(n - 1)/d]] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 65}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[a[k] x^k/(1 - x^(2 k)), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 65}]

Formula

G.f.: x * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^(2*n))).
L.g.f.: log(Product_{n>=1} ((1 + x^n)/(1 - x^n))^(a(n)/(2*n))) = Sum_{n>=1} a(n+1)*x^n/n.

A307777 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+d)*a(d).

Original entry on oeis.org

1, 1, -2, -1, 1, 2, -2, -1, 0, -1, -2, -1, 6, 7, -7, -7, 5, 6, -8, -7, 6, 3, -3, -2, 5, 7, -15, -16, 26, 27, -22, -21, 12, 9, -16, -16, 28, 29, -23, -18, 9, 10, -23, -22, 28, 21, -20, -19, 25, 24, -31, -27, 29, 30, -23, -23, 16, 7, -35, -34, 79, 80, -60, -57, 27, 35, -50, -49, 54, 50, -41
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A003238, A281487, A307776, A307778, A307779.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[(-1)^((n - 1)/d + d) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 71}]
a[n_] := a[n] = SeriesCoefficient[x (1 - Sum[a[k] (-x)^k/(1 + x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 71}]

Formula

G.f.: x * (1 - Sum_{n>=1} a(n)*(-x)^n/(1 + x^n)).
L.g.f.: log(Product_{n>=1} (1 + x^n)^((-1)^(n+1)*a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A351407 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d) * a(d).

Original entry on oeis.org

1, -1, 2, -3, 3, -4, 8, -9, 6, -9, 14, -15, 16, -17, 27, -33, 21, -22, 36, -37, 34, -45, 61, -62, 51, -55, 73, -82, 76, -77, 124, -125, 80, -97, 120, -132, 132, -133, 171, -190, 153, -154, 221, -222, 194, -233, 296, -297, 239, -248, 313, -337, 301, -302
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 10 2022

Keywords

Crossrefs

Cf. A003238, A067856, A281487, A307776, A307778, A308077, A325144, A343370.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1,
      add((-1)^((n-1)/d)*a(d), d=numtheory[divisors](n-1)))
    end:
seq(a(n), n=1..54);  # Alois P. Heinz, Feb 10 2022
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 54}]
nmax = 54; A[] = 0; Do[A[x] = x (1 + Sum[(-1)^k A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * ( 1 - A(x) + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... ).
G.f.: x * ( 1 - Sum_{n>=1} a(n) * x^n / (1 + x^n) ).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.