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.

Previous Showing 101-102 of 102 results.

A099139 a(n) = (18^n - (-6)^n)/24.

Original entry on oeis.org

0, 1, 12, 252, 4320, 79056, 1415232, 25520832, 459095040, 8265390336, 148766948352, 2677865536512, 48201216860160, 867624080265216, 15617220384079872, 281110045277601792, 5059980344811847680, 91079649027723165696, 1639433665572357537792, 29509806081862392348672
Offset: 0

Views

Author

Paul Barry, Sep 29 2004

Keywords

Comments

In general k^(n-1)*A015518(n) is given by ((3k)^n-(-k)^n)/(4k) with g.f. x/((1+kx)(1-3kx)).

Links

Crossrefs

Cf. A053524, A053535, A053536, A053537.

Programs

  • Mathematica
    LinearRecurrence[{12,108},{0,1},20] (* Harvey P. Dale, May 24 2015 *)

Formula

G.f.: x/((1+6x)*(1-18x)).
a(n) = 12a(n-1)+108a(n-2). a(n) = 6^(n-1)*A015518(n).

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2*n))^k - (1 - sqrt(2*n))^k)/(2*sqrt(2*n)).

Original entry on oeis.org

5, 7, 101, 11, 13, 269, 17, 19, 509, 23, 709, 821, 29, 31, 46957, 55399, 37, 168846239, 41, 43, 9177868096974864412935432937651459122761, 47, 485329129, 2789, 53, 3229, 3461, 59, 61, 1563353111, 139237612541, 67, 5021, 71, 73, 484639, 6221, 79, 6869, 83, 7549
Offset: 1

Views

Author

XU Pingya, Sep 24 2017

Keywords

Examples

			For k = {1, 2, 3, 4, 5}, ((1 + sqrt(6))^k - (1 - sqrt(6))^k)/(2*sqrt(6)) = {1, 2, 9, 28, 101}. 101 is odd prime, so a(3) = 101.

Crossrefs

Cf. A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914, A079773, A161007, A099134.

Programs

  • Mathematica
    g[n_, k_] := ((1 + Sqrt[n])^k - (1 - Sqrt[n])^k)/(2Sqrt[n]);
Table[k = 3; While[! PrimeQ[Expand@g[2n, k]], k++]; Expand@g[2n, k], {n, 41}]
  • PARI
    g(n,k) = ([0,1;2*n-1,2]^k*[0;1])[1,1]
a(n) = for(k=3,oo,if(ispseudoprime(g(n,k)),return(g(n,k)))) \\ Jason Yuen, Apr 12 2025

Formula

When 2*n + 3 = p is prime, a(n) = p.
Previous Showing 101-102 of 102 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.