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.

A171498 a(n) = 4*3^n-1.

Original entry on oeis.org

3, 11, 35, 107, 323, 971, 2915, 8747, 26243, 78731, 236195, 708587, 2125763, 6377291, 19131875, 57395627, 172186883, 516560651, 1549681955, 4649045867, 13947137603, 41841412811, 125524238435, 376572715307, 1129718145923, 3389154437771, 10167463313315, 30502389939947
Offset: 0

Views

Author

Klaus Brockhaus, Dec 10 2009

Keywords

Comments

Binomial transform of A171497.
Inverse binomial transform of A171499.

Links

Crossrefs

Cf. A010703, A036543, A171497, A171499.

Programs

  • Mathematica
    NestList[3#+2&,3,30]  (* Harvey P. Dale, Feb 25 2011 *)
  • PARI
    {m=25; v=concat([3], vector(m-1)); for(n=2, m, v[n]=3*v[n-1]+2); v}

Formula

a(n) = 3*a(n-1)+2 for n > 0; a(0) = 3.
G.f.: (3-x)/((1-x)*(1-3*x)).
a(n) = A036543(n) + 2. - Philippe Deléham, Apr 13 2013
E.g.f.: exp(x)*(4*exp(2*x) - 1). - Stefano Spezia, Aug 04 2024

Extensions

a(25)-a(27) from Stefano Spezia, Aug 04 2024

A048471 Array T read by diagonals: T(k,n) = 2^(k-1) * (3^n - 1) + 1.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 9, 5, 1, 41, 27, 17, 9, 1, 122, 81, 53, 33, 17, 1, 365, 243, 161, 105, 65, 33, 1, 1094, 729, 485, 321, 209, 129, 65, 1, 3281, 2187, 1457, 969, 641, 417, 257, 129, 1, 9842, 6561, 4373, 2913, 1937, 1281, 833, 513
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Diagonals (each starting on row 1): {1}; {2,1}; {5,3,1}; ...

Crossrefs

Row 1 = (1, 2, 5, 14, 41, ...) = A007051.
Row 2 = (1, 3, 9, 27, 81, ...) = A000244.
Other rows: A048473 (k=2), A036543 (k=3), A036545 (k=4), A036546 (k=5), A036547 (k=6), A036548 (k=7), A036549 (k=8).
Diagonal is A036551, antidiagonal sums are A036550.

Formula

n-th difference of (T(k, n), T(k, n-1), ..., T(k, 0)) is 2^(n+k-1), for n=1, 2, 3, ...; k=0, 1, 2, ...

Extensions

Simpler definition from Ralf Stephan, Feb 17 2004

A303749 First differences of A302774; Number of terms in A303762 that have prime(n) as their largest prime factor (A006530).

Original entry on oeis.org

1, 2, 4, 7, 16, 19, 52, 55, 160, 163, 484, 487, 1456, 1459, 4372, 4375, 13120, 13123, 39364, 39367, 118096, 118099, 354292, 354295
Offset: 1

Views

Author

Antti Karttunen, May 05 2018

Keywords

Comments

For n >= 1, the difference A000079(n-1) - a(n): 0, 0, 0, 1, 0, 13, 12, 73, 96, 349, 540, 1561, 2640, 6733, 12012, 28393, 52416, 117949, 222780, 484921, 930480, 1979053, 3840012, ..., indicates how many squarefree numbers A303762 misses in each round. The first of these is 70 missed at the round 4.
The first differences of these terms is: 1, 2, 3, 9, 3, 33, 3, 105, 3, 321, 3, 969, 3, 2913, 3, 8745, 3, 26241, 3, 78729, 3, 236193, 3, ... which after the first two initial terms seem to be an interleaving of sequences A010701 and A036543.

Crossrefs

Cf. A010701, A036543, A302774, A303762.

Formula

a(n) = A302774(n+1) - A302774(n).

A220944 Expansion of (1+3*x+5*x^2-x^3)/((1-x^2)*(1-3*x^2)).

Original entry on oeis.org

1, 3, 9, 11, 33, 35, 105, 107, 321, 323, 969, 971, 2913, 2915, 8745, 8747, 26241, 26243, 78729, 78731, 236193, 236195, 708585, 708587, 2125761, 2125763, 6377289, 6377291, 19131873, 19131875, 57395625, 57395627, 172186881, 172186883, 516560649, 516560651
Offset: 0

Views

Author

Philippe Deléham, Apr 13 2013

Keywords

Links

Crossrefs

Cf. A036543, A171498.

Programs

  • Mathematica
    LinearRecurrence[{0, 4, 0, -3}, {1, 3, 9, 11}, 40] (* T. D. Noe, Apr 17 2013 *)

Formula

a(n) = a(n-1)+2 if n odd.
a(n) = a(n-1)*3 if n even.
a(2n) = 4*3^n-3 = A036543(n), a(2n+1) = 4*3^n-1 = A171498(n).
a(n) = 4*a(n-2) - 3*a(n-4) with a(0)=1, a(1)=3, a(2)=9, a(3)=11.
Showing 1-4 of 4 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.