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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A264018 Number of (2+1) X (n+1) arrays of permutations of 0..n*3+2 with each element having index change +-(.,.) 0,0 1,2 or 2,2.

Original entry on oeis.org

1, 5, 25, 105, 441, 1869, 7921, 33553, 142129, 602069, 2550409, 10803705, 45765225, 193864605, 821223649, 3478759201, 14736260449, 62423800997, 264431464441, 1120149658761, 4745030099481, 20100270056685, 85146110326225
Offset: 1

Views

Author

R. H. Hardin, Nov 01 2015

Keywords

Examples

			Some solutions for n=4:
..7..1..2..3..4....0.13..9..3..4....0..8..9..3..4....7..1..9..3..4
.12..6..0..8..9...12..6..7..8..2...12..6.14..1..2...12.13..0..8..2
.10.11..5.13.14...10.11..5..1.14...10.11..5.13..7...10.11..5..6.14

Links

Crossrefs

Row 2 of A264017.

Formula

Empirical: a(n) = 4*a(n-1) + 4*a(n-3) + a(n-4).
Empirical g.f.: x*(1 + x + 5*x^2 + x^3) / ((1 + x^2)*(1 - 4*x - x^2)). - Colin Barker, Jan 03 2019
Empirical: 5*a(n) = 2*A228826(n) + A048876(n). - R. J. Mathar, Sep 09 2020

A120775 a(n) = 3*a(n-1) + 5*a(n-2) + a(n-3).

Original entry on oeis.org

1, 6, 23, 100, 421, 1786, 7563, 32040, 135721, 574926, 2435423, 10316620, 43701901, 185124226, 784198803, 3321919440, 14071876561, 59609425686, 252509579303, 1069647742900, 4531100550901, 19194049946506, 81307300336923, 344423251294200, 1459000305513721
Offset: 1

Views

Author

Gary W. Adamson and Roger L. Bagula, Jul 04 2006

Keywords

Links

Crossrefs

Cf. A120758, A120757.

Programs

  • Mathematica
    LinearRecurrence[{3,5,1},{1,6,23},25] (* James C. McMahon, Oct 09 2024 *)

Formula

a(n) = 3*a(n-1) + 5*a(n-2) + a(n-3).
G.f.: -x*(1+3*x) / ( (1+x)*(x^2+4*x-1) ). a(n) + a(n+1) = A048876(n). - R. J. Mathar, Oct 22 2013
a(n) = (Lucas(3n-1) + (-1)^n)/2. - Greg Dresden, Oct 09 2020

Extensions

Edited by N. J. A. Sloane, Dec 03 2006
a(24)-a(25) from James C. McMahon, Oct 09 2024

A323013 Form of Zorach additive triangle T(n,k) (see A035312) where each number is sum of west and northwest numbers, with the additional condition that the first element T(n,1) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 8, 13, 20, 30, 21, 29, 42, 62, 92, 34, 55, 84, 126, 188, 280, 89, 123, 178, 262, 388, 576, 856, 144, 233, 356, 534, 796, 1184, 1760, 2616, 377, 521, 754, 1110, 1644, 2440, 3624, 5384, 8000, 610, 987, 1508, 2262, 3372, 5016, 7456, 11080, 16464, 24464
Offset: 1

Views

Author

Michel Lagneau, Jan 02 2019

Keywords

Comments

Conjecture: Let F(i) be the i-th Fibonacci number. Each number of T(n, k), k = 1, 2, 3 is the difference between two Fibonacci numbers F(i) - F(j) for some i, j, where F(i) is the smallest Fibonacci number greater than T(n, k). The case T(n, 1) is trivial. Examples: 10 = 13 - 3, 29 = 34 - 5, 20 = 21 - 1, 42 = 55 - 13, 84 = 89 - 5, ...
We observe interesting properties:
T(n,1) = A117647(n) = 1, 2, 5, 8, 21, ... where n = 1, 2, ...
T(2n,2) = A033887(n) = 3, 13, 55, ... (Fibonacci(3n+1)), and T(2n+1,2) = A048876(n) = 7, 29, 123, ... (Generalized Pell equation with second term of 7) where n = 1, 2, ...
T(3n,3) = 10, 84, 754, 6388,... If n = 2m - 1, T(6m - 3, 3) = F(9m - 2) - F(9m - 5) and if n = 2m, T(6m, 3) = F(9m + 2) - F(9m - 4).
T(3n+1,3) = 20, 178, 1508, 13530, ... If n = 2m - 1, T(6m - 2, 3) = F(9m - 1) - F(9m - 7) and if n = 2m, T(6m+1, 3) = F(9m + 4) - F(9m + 1).
T(3n+2,3) = 42, 356, 3194, 27060, ... If n = 2m - 1, T(6m - 1, 3) = F(9m + 1) - F(9m - 2) and if n = 2m, T(6m + 2, 3) = F(9m + 5) - F(9m - 1).
Other property:
T(2m, 1) + T(2m, 2) = T(2m +1, 1) with T(2m, 1)= F(3m), T(2m, 2) = F(3m + 1) and T(2m + 1, 1) = F(3m + 2).
T(2m + 1, 1) + T(2m + 1, 2) = F(3m + 4) - F(3m - 1).

Examples

			The start of the sequence as a triangular array T(n, k) read by rows:
   1;
   2,   3;
   5,   7,  10;
   8,  13,  20,   30;
  21,  29,  42,   62,   92;
  34,  55,  84,  126,  188,  280;
  ...

Crossrefs

Cf. A000045, A002878, A033887, A035312, A036561, A117647.

Programs

  • Maple
    with(combinat,fibonacci):
lst:={1}:lst2:=lst:
for n from 2 to 15 do :
lst1:={}:ii:=0:
  for j from 1 to 1000 while(ii=0) do:
     i:=fibonacci(j):
     if {i} intersect lst2 = {} and {i+lst[1]} intersect lst2 = {}
      then
      lst1:=lst1 union {i}:ii:=1:
      else
     fi:
   od:
    for k from 1 to n-1 do:
      lst1:=lst1 union {lst1[k]+lst[k]}:
    od:
    lst:=lst1:lst2:=lst2 union lst:
    print(lst1):
   od:
Previous Showing 11-13 of 13 results.

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

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