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.

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A231960 Powers of 3 together with multiples of 6.

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 24, 27, 30, 36, 42, 48, 54, 60, 66, 72, 78, 81, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 243, 246, 252, 258
Offset: 1

Views

Author

Thomas M. Bridge, Nov 15 2013

Keywords

Comments

Union of A000244 and A008588 (without 0).
Also, 1 and 3*(even numbers (A005843) UNION powers of 3 (A008588)).
Also, numbers m such that m divides A057083(m-1), see the Smyth reference.

Links

Crossrefs

Cf. A029744.

Programs

  • Sage
    def is_in_A231960(n):
    return 6.divides(n) or n==3^valuation(n,3)

Extensions

Edited by Ralf Stephan, Feb 28 2014

A236311 Riordan array ((1-x)/(1-3*x+3*x^2), x/(1-3*x+3*x^2)).

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 3, 15, 8, 1, 0, 33, 36, 11, 1, -9, 54, 117, 66, 14, 1, -27, 54, 297, 282, 105, 17, 1, -54, -27, 594, 945, 555, 153, 20, 1, -81, -297, 864, 2583, 2295, 963, 210, 23, 1, -81, -891, 513, 5778, 7803, 4725, 1533, 276, 26, 1, 0, -1863, -1944, 10098
Offset: 0

Views

Author

Philippe Deléham, Jan 21 2014

Keywords

Comments

Row sums are 3^n = A000244(n).
Diagonals sums are 2^n = A000079(n).
T(n,n) = A000012(n).
T(n+1,n) = A016789(n).
T(n+2,n) = A062741(n+1).
T(n+3,n) = 3*A004188(n+1).
T(n,0) = A057682(n+1).

Examples

			Triangle begins :
1;
2, 1;
3, 5, 1;
3, 15, 8, 1;
0, 33, 36, 11, 1;
-9, 54, 117, 66, 14, 1;
-27, 54, 297, 282, 105, 17, 1;

Crossrefs

Cf. A057083, A057682, A000012, A016789, A062741

Formula

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) -3*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k<0 or if k>n.

A379825 a(n) = [x^n] x/(12*x^2 - 6*x + 1).

Original entry on oeis.org

0, 1, 6, 24, 72, 144, 0, -1728, -10368, -41472, -124416, -248832, 0, 2985984, 17915904, 71663616, 214990848, 429981696, 0, -5159780352, -30958682112, -123834728448, -371504185344, -743008370688, 0, 8916100448256, 53496602689536, 213986410758144, 641959232274432
Offset: 0

Views

Author

Peter Luschny, Jan 04 2025

Keywords

Links

Crossrefs

Cf. A057083, A000748, A190965.

Programs

  • Maple
    w := sqrt(-3): a := n -> (w/6)*((3 - w)^n - (3 + w)^n):
seq(simplify(a(n)), n = 0..28);
# Alternative:
a := proc(n) option remember; if n < 2 then n else 6*(a(n - 1) - 2*a(n - 2)) fi end:
seq(a(n), n = 0..28);
  • Mathematica
    LinearRecurrence[{6,-12},{0,1},29] (* James C. McMahon, Jan 05 2025 *)

Formula

a(n) = n! * [x^n] exp(3*x)*sin(sqrt(3)*x)/sqrt(3).
a(n) = (w/6)*((3 - w)^n - (3 + w)^n) where w = sqrt(-3).
a(n) = 6*a(n - 1) - 12*a(n - 2) for n >= 2.
a(n) = 2^n*3^((n - 1)/2)*sin((Pi*n)/6).
a(n) = 2^(n-1)*A057083(n-1) = 2^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, sqrt(3)/2) for n >= 1.
Previous Showing 41-43 of 43 results.

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

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