A020706
Pisot sequences L(4,6), E(4,6).
Original entry on oeis.org
4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142
Offset: 0
- Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.
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I:=[4, 6, 9]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Apr 20 2012
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CoefficientList[Series[(4-2*x-3*x^2)/(1-x)/(1-x-x^2),{x,0,40}],x](* Vincenzo Librandi, Apr 20 2012 *)
A020717
Pisot sequences L(6,9), E(6,9).
Original entry on oeis.org
6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142
Offset: 0
- Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.
- Colin Barker, Table of n, a(n) for n = 0..1000
- Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
- Index entries for Pisot sequences
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
See
A008776 for definitions of Pisot sequences.
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Table[Fibonacci[n + 5] + 1, {n, 0, 36}] (* Michael De Vlieger, Jul 27 2016 *)
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pisotE(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
a
}
pisotE(50, 6, 9) \\ Colin Barker, Jul 27 2016
A226271
Index of 1/n in the Fibonacci (or rabbit) ordering of the positive rationals.
Original entry on oeis.org
1, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142
Offset: 1
Starting from the vector [1] and applying the map t->(1+t,1/t), we get [2,1] (but ignore the number 1 which already occurred earlier), then [3,1/2], then [4,1/3,3/2,2] (where we ignore 2), etc. This yields the sequence (1,2,3,1/2,4,1/3,3/2,5,1/4,4/3,5/2,2/3,....) The unit fractions 1=1/1, 1/2, 1/3, ... occur at positions 1,4,6,9,...
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LinearRecurrence[{2,0,-1},{1,4,6,9},40] (* Harvey P. Dale, Feb 04 2016 *)
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A226271(n)=if(n>1,fibonacci(n+2))+1
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{k=1;print1(s=1,",");U=Set(g=[1]);for(n=1,9,U=setunion(U,Set(g=select(f->!setsearch(U,f), concat(apply(t->[t+1,k/t],g))))); for(i=1,#g,numerator(g[i])==1&&print1(s+i","));s+=#g)} \\ for illustrative purpose
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Vec(-x*(2*x^3+2*x^2-2*x-1)/((x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, May 11 2016
A301791
Number of 2Xn 0..1 arrays with every element equal to 0 or 1 horizontally or antidiagonally adjacent elements, with upper left element zero.
Original entry on oeis.org
2, 5, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987
Offset: 1
Some solutions for n=5
..0..0..1..0..0. .0..1..0..0..1. .0..1..0..1..1. .0..1..0..0..1
..1..0..1..1..0. .0..1..1..0..0. .0..1..0..0..1. .0..1..1..0..1
Showing 1-4 of 4 results.
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