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.

A055895 Inverse Moebius transform of powers of 2.

Original entry on oeis.org

1, 2, 6, 10, 22, 34, 78, 130, 278, 522, 1062, 2050, 4190, 8194, 16518, 32810, 65814, 131074, 262734, 524290, 1049654, 2097290, 4196358, 8388610, 16781662, 33554466, 67117062, 134218250, 268451990, 536870914, 1073775726, 2147483650, 4295033110, 8589936650
Offset: 0

Views

Author

Christian G. Bower, Jun 09 2000

Keywords

Comments

Row sums of A055894.

Examples

			G.f. = 1 + 2*x + 6*x^2 + 10*x^3 + 22*x^4 + 34*x^5 + 78*x^6 + 130*x^7 + ...

Links

Crossrefs

Cf. A034729, A113705 (binary), A339916.
Cf. A055894.

Programs

  • Mathematica
    Table[Plus @@ Map[Function[d, 2^d], Divisors[n]], {n, 0, 30}] (* Olivier Gérard, Jan 01 2012 *)
a[0]=1; a[n_] := DivisorSum[n, 2^#&]; Array[a, 40, 0] (* Jean-François Alcover, Dec 01 2015 *)
  • PARI
    a(n)=if(n<1,1,polcoeff(sum(k=1,n,1/(1-2*x^k),x*O(x^n)),n))
  • PARI
    a(n)=if(n<1,1,sumdiv(n,d,2^d)); /* Joerg Arndt, Aug 14 2012 */

Formula

G.f.: 1 + Sum_{k>=1} 2^k*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
a(n) = Sum_{d divides n} 2^d. - Olivier Gérard, Jan 01 2012
a(n) = 2 * A034729(n) for n >= 1. - Joerg Arndt, Aug 14 2012
G.f.: 1 + Sum_{k>=1} 2*x^k/(1-2*x^k). - Joerg Arndt, Mar 28 2013

A114001 Rows of A114000 expressed as decimals (a sequence related to the number of divisors of 2n-1).

Original entry on oeis.org

1, 3, 5, 9, 25, 33, 65, 225, 257, 513, 1665, 2049, 5121, 12801, 16385, 32769, 100353, 180225, 262145, 794625, 1048577, 2097153, 7634945, 8388609, 18874369, 50462721, 67108865, 171966465, 403177473, 536870913, 1073741825
Offset: 0

Views

Author

Paul Barry, Nov 12 2005

Keywords

Examples

			a(5)=25 converts to 11001 in binary, which has sum of digits equal to 3, and 9=2*5-1 has 3 divisors.

Links

Crossrefs

Cf. A114000, A099774 (binary weight), A339916 (bit reversal).

Programs

  • Mathematica
    A114001[n_]:=FromDigits[Boole[Divisible[2n+1,2Range[0,n]+1]],2];
Array[A114001,50,0] (* Paolo Xausa, Dec 04 2023 *)

Extensions

Edited by N. J. A. Sloane, Dec 23 2020
Offset changed to 0 by Paolo Xausa, Dec 04 2023

A339891 Number of fundamentally different graceful labelings of the complete tripartite graph K_{1,1,n}.

Original entry on oeis.org

1, 4, 7, 12, 20, 34, 74, 131, 260, 524, 1030, 2054, 4118, 8196, 16389, 32804, 65554, 131074, 262216, 524292, 1048580, 2097304, 4194312, 8388619, 16777478, 33554436, 67108906, 134218244, 268435464, 536870914, 1073742880, 2147483720, 4294967300, 8589936646, 17179869193
Offset: 1

Views

Author

Don Knuth, Dec 21 2020

Keywords

Comments

The difference between "fundamentally different graceful labelings" of a graph and "graceful labelings" of a graph is that the latter is the former multiplied by twice the number of automorphisms. (The extra factor of 2 comes from complementation.)
When n>1, the graph K_{1,1,n} has 2n! automorphisms.

References

  • D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3, in preparation.

Links

Crossrefs

If n>1, A334307(n) = 4*a(n)*n!.

Programs

  • Mathematica
    A339891[n_]:=If[n==1,1,DivisorSum[2n+1,2^((#-1)/2)&]+DivisorSigma[0,n+1]-2^(n-1)-1];Array[A339891,50] (* Paolo Xausa, Dec 04 2023 *)

Formula

a(n) = A339916(n) + A000005(n+1) - 2^(n-1) - 1 - 2*[n=1].
Showing 1-3 of 3 results.

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

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