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.

A129501 A103994 * A115361.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Apr 17 2007

Keywords

Comments

Row sums = A129502: (1, 3, 0, 6, 0, 0, 0, 10, 0, 0, ...).

Examples

			First few rows of the triangle are:
   1;
   2,  1;
  -1,  0,  1;
   3,  2,  0,  1;
  -1,  0,  0,  0,  1;
  -2, -1,  2,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   4,  3,  0,  2,  0,  0,  0,  1;
   0,  0, -1,  0,  0,  0,  0,  0,  1;
  ...

Links

Crossrefs

Column 1 is A317673 (Moebius transform of A129502).
Row sums are A129502.
Cf. A103994, A115361, A129360, A000217, A001511.

Programs

  • Mathematica
    b[n_] := Module[{e}, Sum[e = IntegerExponent[d, 2]; If[d == 2^e, MoebiusMu[n/d] Binomial[2 + e, 2], 0], {d, Divisors[n]}]];
T[n_, k_] := If[Divisible[n, k], b[n/k], 0];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019 *)
  • PARI
    T(n,k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

Formula

A103994 * A115361 as infinite lower triangular matrices.
T(n,k) = A317673(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

Extensions

Terms a(56) and beyond from Andrew Howroyd, Aug 03 2018

A129360 A054525 * A115361.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 10 2007

Keywords

Comments

Row sums = A209229 (1, 1, 0, 1, 0, 0, 0, 1, ...).
A129353 = the inverse Möbius transform of A115361.

Examples

			First few rows of the triangle are:
   1;
   0,  1;
  -1,  0,  1;
   0,  0,  0,  1;
  -1,  0,  0,  0,  1;
   0, -1,  0,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0,  1;
  ...

Links

Crossrefs

Column 1 is A087003 (Moebius transform of A209229).
Row sums are A209229.
Cf. A054525, A115361, A129353, A001511, A103994, A129501.

Programs

  • PARI
    tabl(nn) = {Tm = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); Tr = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = Tr^(-1); Tp = Tm*Ti; for (n=1, nn, for (k=1, n, print1(Tp[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015
  • PARI
    T(n, k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

Formula

Moebius transform of A115361.
T(n,k) = A087003(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

Extensions

More terms from Michel Marcus, Mar 28 2015
Offset changed by Andrew Howroyd, Aug 03 2018

A104117 For n=2^k, a(n) = k+1, else 0.

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gary W. Adamson, Apr 15 2007

Keywords

Comments

Row sums of A103994 (conjectured).

Examples

			a(8) = 4 = sum of row 8 terms of A103994: (1 + 1 + 0 + 1 + 0 + 0 + 0 + 1).
a(8) = 4 = 1 + log_2(8).

Crossrefs

Cf. A036987, A103994, A209229.

Programs

  • Mathematica
    a[n_] := Module[{e = IntegerExponent[n, 2]}, If[n == 2^e, e+1, 0]]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)
  • PARI
    a(n)=direuler(p=1,n,if(p==2,1/(1-X)^2,1))[n] /* Ralf Stephan, Mar 28 2015 */
  • PARI
    a(n)=if(n==2^valuation(n,2),valuation(n,2)+1,0) /* Ralf Stephan, Mar 28 2015 */

Formula

a(n) = 1 + log_2(n), for n = 1, 2, 4, 8, ... and the rest zeros.
Dirichlet g.f.: 1/(1-2^(-s))^2, i.e., Dirichlet convolution of A036987 (right-shifted, assuming offset 1 there) with itself.
Multiplicative with a(2^e) = 1+e, and a(p^e) = 0 for odd primes p and e>=1. Dirichlet convolution square of A209229. - R. J. Mathar, Mar 12 2012

Extensions

More terms and better name from Ralf Stephan, Mar 28 2015

A129502 For n=2^k, a(n) = binomial(k + 2, 2), else 0.

Original entry on oeis.org

1, 3, 0, 6, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gary W. Adamson, Apr 17 2007

Keywords

Comments

Row sums of triangle A129501.

Examples

			a(4) = 6 = sum of A129501 terms: (3 + 2 + 0 + 1).

Links

Crossrefs

Row sums of A129501.
Cf. A209229, A104117, A129360, A115361, A103994, A000217, A317673.

Programs

  • Mathematica
    Table[If[IntegerQ[Log2[n]],Binomial[Log2[n]+2,2],0],{n,100}] (* Harvey P. Dale, May 10 2022 *)
  • PARI
    a(n)={my(e=valuation(n, 2)); if(n==1<Andrew Howroyd, Aug 03 2018

Formula

From Andrew Howroyd, Aug 04 2018: (Start)
Multiplicative with a(2^e) = binomial(e + 2, 2), a(p^e) = 0 for odd prime p.
Dirichlet convolution of A104117 and A209229.
a(n) = Sum_{d|n} A104117(n/d) * A209229(d). (End)
Dirichlet g.f.: 1/(1 - 1/2^s)^3. - Amiram Eldar, Oct 28 2023

Extensions

Name changed and terms a(40) and beyond from Andrew Howroyd, Aug 03 2018
Showing 1-4 of 4 results.

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