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.

A007482 a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.

Original entry on oeis.org

1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647, 283667, 1010295, 3598219, 12815247, 45642179, 162557031, 578955451, 2061980415, 7343852147, 26155517271, 93154256107, 331773802863, 1181629920803, 4208437368135
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The even neighbor must differ from the odd number by exactly one.
If we defined this sequence by the recurrence (a(n) = 3*a(n-1) + 2*a(n-2)) that it satisfies, we could prefix it with an initial 0.
a(n) equals term (1,2) in M^n, M = the 3 X 3 matrix [1,1,2; 1,0,1; 2,1,1]. - Gary W. Adamson, Mar 12 2009
a(n) equals term (2,2) in M^n, M = the 3 X 3 matrix [0,1,0; 1,3,1; 0,1,0]. - Paul Barry, Sep 18 2009
From Gary W. Adamson, Aug 06 2010: (Start)
Starting with "1" = INVERT transform of A002605: (1, 2, 6, 16, 44, ...).
Example: a(3) = 39 = (16, 6, 2, 1) dot (1, 1, 3, 11) = (16 + 6 + 6 + 11). (End)
Pisano periods: 1, 1, 4, 1, 24, 4, 48, 2, 12, 24, 30, 4, 12, 48, 24, 4,272, 12, 18, 24, ... . - R. J. Mathar, Aug 10 2012
A007482 is also the number of ways of tiling a 3 X n rectangle with 1 X 1 squares, 2 X 2 squares and 2 X 1 (vertical) dominoes. - R. K. Guy, May 20 2015
With offset 1 (a(0) = 0, a(1) = 1) this is a divisibility sequence. - Michael Somos, Jun 03 2015
Number of elements of size 2^(-n) in a fractal generated by the second-order reversible cellular automaton, rule 150R (see the reference and the link). - Yuriy Sibirmovsky, Oct 04 2016
a(n) is the number of compositions (ordered partitions) of n into parts 1 (of three kinds) and 2 (of two kinds). - Joerg Arndt, Oct 05 2016
a(n) equals the number of words of length n over {0,1,2,3,4} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have two '1's in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.833. - Peter Karpov, Apr 20 2017
This is the Lucas sequence U(P=3,Q=-2), and hence for n>=0, a(n+2)/a(n+1) equals the continued fraction 3 + 2/(3 + 2/(3 + 2/(3 + ... + 2/3))) with n 2's. - Greg Dresden, Oct 06 2019

Examples

			G.f. = 1 + 3*x + 11*x^2 + 39*x^3 + 139*x^4 + 495*x^5 + 1763*x^6 + ...
From _M. F. Hasler_, Jun 16 2019: (Start)
For n = 0, (1, ..., 2n) = () is the empty sequence, which is equal to its only subsequence, which satisfies the condition voidly, whence a(0) = 1.
For n = 1, (1, ..., 2n) = (1, 2); among the four subsequences {(), (1), (2), (1,2)} only (1) does not satisfy the condition, whence a(1) = 3.
For n = 2, (1, ..., 2n) = (1, 2, 3, 4); among the sixteen subsequences {(), ..., (1,2,3,4)}, the 5 subsequences (1), (3), (1,3), (2,3,4) and (1,2,3,4) do not satisfy the condition, whence a(2) = 16 - 5 = 11.
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 439.

Links

Crossrefs

Row sums of triangle A073387.
Cf. A000045, A000129, A001045, A007455, A007481, A007483, A007484, A015518, A201000 (prime subsequence), A052913 (binomial transform), A026597 (inverse binomial transform).
Cf. A206776.

Programs

  • Haskell
    a007482 n = a007482_list !! (n-1)
a007482_list = 1 : 3 : zipWith (+)
               (map (* 3) $ tail a007482_list) (map (* 2) a007482_list)
-- Reinhard Zumkeller, Oct 21 2015
  • Magma
    I:=[1,3]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018
  • Maple
    a := n -> `if`(n=0, 1, 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9)):
seq(simplify(a(n)), n = 0..23); # Peter Luschny, Jun 28 2017
  • Mathematica
    a[n_]:=(MatrixPower[{{1,4},{1,2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3,2},{1,3},30] (* Harvey P. Dale, May 25 2013 *)
a[ n_] := Module[ {m = n + 1, s = 1}, If[ m < 0, {m, s} = -{m, (-2)^m}]; s SeriesCoefficient[ x / (1 - 3 x - 2 x^2), {x, 0, m}]]; (* Michael Somos, Jun 03 2015 *)
a[ n_] := With[{m = n + 1}, If[ m < 0, (-2)^m a[ -m], Expand[((3 + Sqrt[17])/2)^m - ((3 - Sqrt[17])/2)^m ] / Sqrt[17]]]; (* Michael Somos, Oct 13 2016 *)
  • Maxima
    a(n) := if n=0 then 1 elseif n=1 then 3 else 3*a(n-1)+2*a(n-2);
makelist(a(n),n,0,12); /* Emanuele Munarini, Jun 28 2017 */
  • PARI
    {a(n) = 2*imag(( (3 + quadgen(68)) / 2)^(n+1))}; /* Michael Somos, Jun 03 2015 */
  • Sage
    [lucas_number1(n,3,-2) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

Formula

G.f.: 1/(1-3*x-2*x^2).
a(n) = 3*a(n-1) + 2*a(n-2).
a(n) = (ap^(n+1)-am^(n+1))/(ap-am), where ap = (3+sqrt(17))/2 and am = (3-sqrt(17))/2.
Let b(0) = 1, b(k) = floor(b(k-1)) + 2/b(k-1); then, for n>0, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)2^k*3^(n-2k). - Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..n} A112906(n,k). - Philippe Deléham, Nov 21 2007
a(n) = - a(-2-n) * (-2)^(n+1) for all n in Z. - Michael Somos, Jun 03 2015
If c = (3 + sqrt(17))/2, then c^n = (A206776(n) + sqrt(17)*a(n-1)) / 2. - Michael Somos, Oct 13 2016
a(n) = 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9) for n>=1. - Peter Luschny, Jun 28 2017
a(n) = round(((sqrt(17) + 3)/2)^(n+1)/sqrt(17)). The distance of the argument from the nearest integer is about 1/2^(n+3). - M. F. Hasler, Jun 16 2019
E.g.f.: (1/17)*exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 07 2019
a(n) = (sqrt(2)*i)^n * ChebyshevU(n, -3*i/(2*sqrt(2))). - G. C. Greubel, Dec 24 2021
G.f.: 1/(1 - 3*x - 2*x^2) = Sum_{n >= 0} x^n * Product_{k = 1..n} (k + 2*x + 2)/(1 + k*x) (a telescoping series). Cf. A015518. - Peter Bala, May 08 2024

A112899 A skew Pell-Pascal triangle.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 0, 4, 12, 0, 0, 1, 14, 29, 0, 0, 0, 6, 44, 70, 0, 0, 0, 1, 27, 131, 169, 0, 0, 0, 0, 8, 104, 376, 408, 0, 0, 0, 0, 1, 44, 366, 1052, 985, 0, 0, 0, 0, 0, 10, 200, 1212, 2888, 2378, 0, 0, 0, 0, 0, 1, 65, 810, 3842, 7813, 5741, 0, 0, 0, 0, 0, 0, 12, 340, 3032, 11784
Offset: 0

Views

Author

Paul Barry, Oct 05 2005

Keywords

Comments

Main diagonal is A000129. Row sums are A002605. Column sums are A006190(n+1).
A skewed version of the Riordan array (1/(1-2x-x^2), x/(1-2x-x^2)), see A054456. - Philippe Deléham, Nov 21 2007
Triangle, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,...] DELTA [2,1/2,-1/2,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 30 2010

Examples

			Rows begin:
  1;
  0,   2;
  0,   1,   5;
  0,   0,   4,  12;
  0,   0,   1,  14,  29;
  0,   0,   0,   6,  44,  70;
  0,   0,   0,   1,  27, 131, 169;
  0,   0,   0,   0,   8, 104, 376, 408;

Crossrefs

Cf. A111006, A112906. - Philippe Deléham, Jan 30 2010

Formula

G.f.: 1/(1-2*x*y*(1+x/2)-x^2*y^2).
T(n, k) = Sum_{j=0..floor((2*k-n)/2)} C(k-j, n-k)*C(2*k-n-j, j)*2^(2*k-2*j-n). [corrected by Jason Yuen, Jan 21 2025]
T(n, k) = 2*T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).

A132964 Convolution triangle of A006190.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 109, 126, 57, 12, 1, 360, 516, 306, 94, 15, 1, 1189, 2034, 1491, 600, 140, 18, 1, 3927, 7807, 6813, 3385, 1035, 195, 21, 1, 12970, 29382, 29737, 17568, 6630, 1638, 259, 24, 1, 42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1
Offset: 0

Views

Author

Philippe Deléham, Nov 24 2007

Keywords

Comments

As a Riordan array, this is (1/(1-3x-x^2),x/(1-3x-x^2)).
T(n,k) is the number of words of length n over {0,1,2,3,4} having k letters 4 and avoiding runs of odd length for the letter 0. - Milan Janjic, Jan 14 2017

Examples

			Triangle begins:
      1;
      3,      1;
     10,      6,      1;
     33,     29,      9,     1;
    109,    126,     57,    12,     1;
    360,    516,    306,    94,    15,     1;
   1189,   2034,   1491,   600,   140,    18,    1;
   3927,   7807,   6813,  3385,  1035,   195,   21,   1;
  12970,  29382,  29737, 17568,  6630,  1638,  259,  24,  1;
  42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1;
  ...

Links

Crossrefs

Cf. A006190, A037027, A054456, A112906.

Formula

Sum_{k=0..n} T(n,k) = A001076(n+1).
Sum_{k=0..floor(n/2)} T(n-k,k) = A007482(n).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) + T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or k>n. - Philippe Deléham, Dec 08 2013
Showing 1-3 of 3 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.