A005269
a(n) = number of length-n sequences s with s[1]=1, s[2]=1, s[k-1] <=s[k] <= s[k-2]+s[k-1] (s is called a sub-Fibonacci sequence of length n).
Original entry on oeis.org
1, 2, 4, 10, 31, 127, 711, 5621, 64049, 1067599, 26287664, 963023487, 52766766100, 4342736509018, 538755914902622, 101067429677072459, 28751803102222498512, 12436935036300286507123, 8200693250120852291693833, 8262592110164298068793701546
Offset: 2
G.f. = x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 31*x^6 + 127*x^7 + 711*x^8 + 5621*x^9 + ...
a(4)=4 because we have (1,1,1,1), (1,1,1,2), (1,1,2,2), (1,1,2,3).
- Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement. Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Alois P. Heinz, Table of n, a(n) for n = 2..70
- Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
- Peter C. Fishburn and Fred S. Roberts, Elementary sequences, sub-Fibonacci sequences, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.
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f[0]:=1:for k from 0 to 19 do f[k+1]:=expand(sum(subs({x=y,y=z},f[k]),z=y..x+y)) od: seq(subs({x=1,y=1},f[k]),k=0..19);
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{a(n) = if(n<2, return(0)); my(c, e); forvec(s=vector(n, i, [1, fibonacci(i)]), e=0; for(k=3, n, if( s[k-1]>s[k] || s[k]>s[k-2]+s[k-1], e=1; break)); if(e, next); c++, 1); c}; /* Michael Somos, Dec 02 2016 */
A003513
Number of regular sequences of length n.
Original entry on oeis.org
1, 2, 6, 27, 192, 2280, 47097, 1735803, 115867758, 14137353466, 3172486137982, 1315460211433262, 1011773137731861712, 1448486351628212391462, 3872217739919424676743213
Offset: 2
From _Nathaniel Johnston_, Jun 29 2023: (Start)
When n = 4, there are 6 regular sequences:
1,1,1,1
1,1,1,2
1,1,1,3
1,1,2,2
1,1,2,3
1,1,2,4
(End)
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Marc Davio, Unpublished notes, 1975, from a letter to N. J. A. Sloane sent in May 1975.
- Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
- Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
- Peter C. Fishburn et al., Van Lier Sequences, Discrete Appl. Math. 27 (1990), pp. 209-220.
- Nathaniel Johnston and Sarah Plosker, Laplacian {-1,0,1}- and {-1,1}-diagonalizable graphs, arXiv:2308.15611 [math.CO], 2023.
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A003513 := proc() local a,b,n ; a := {[1,1]} ; n := 3 ; while true do b := {} ; for s in a do subsa := combinat[choose](s) ; for i in subsa do newa := add(k,k=i) ; if newa >= op(-1,s) then b := b union {[op(s),newa]} ; fi ; od; od; print(n,nops(b) ) ; a := b ; n := n+1 ; od; end: A003513() ; # R. J. Mathar, Oct 22 2007
A005268
Number of elementary sequences of length n.
Original entry on oeis.org
1, 1, 2, 4, 10, 31, 120, 578, 3422, 24504, 208833, 2086777, 24123293, 318800755, 4766262421, 79874304340, 1488227986802
Offset: 1
- Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement. Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
- Peter C. Fishburn, Fred S. Roberts, Elementary sequences, sub-Fibonacci sequences. Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.
- Sean A. Irvine, Complete set of sequences for a(11)
A005272
Number of Van Lier sequences of length n.
Original entry on oeis.org
1, 2, 6, 26, 164, 1529, 21439, 461481, 15616226, 851607867, 76555549499, 11550559504086
Offset: 2
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Peter C. Fishburn, Fred S. Roberts, Uniqueness in finite measurement, Applications of combinatorics and graph theory to the biological and social sciences, 103-137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
- Peter C. Fishburn, Fred S. Roberts, Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103-137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
- P. C. Fishburn et al., Van Lier Sequences, Discrete Appl. Math. 27 (1990), pp. 209-220.
A234595
Number of elementary sequences of length n, where permutations of the components are taken into account.
Original entry on oeis.org
1, 1, 4, 23, 256, 4647, 128262, 5128503
Offset: 1
- Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement. Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
- Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
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