Search: a234595 -id:a234595
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A005269
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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).
(Formerly M1234)
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+10
9
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1, 2, 4, 10, 31, 127, 711, 5621, 64049, 1067599, 26287664, 963023487, 52766766100, 4342736509018, 538755914902622, 101067429677072459, 28751803102222498512, 12436935036300286507123, 8200693250120852291693833, 8262592110164298068793701546
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listen;
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internal format)
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OFFSET
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2,2
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REFERENCES
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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).
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LINKS
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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]
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FORMULA
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See the Maple program; f[k](x, y) is the number of sequences s[1], s[2], ..., s[k+2] such that s[1]=x, s[2]=y, s[j-1] <=s[j] <= s[j-2]+s[j-1]. - Emeric Deutsch and Don Reble, Feb 07 2005
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EXAMPLE
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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).
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MAPLE
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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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PROG
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(PARI) {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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A003513
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Number of regular sequences of length n.
(Formerly M1685)
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+10
8
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1, 2, 6, 27, 192, 2280, 47097, 1735803, 115867758, 14137353466, 3172486137982, 1315460211433262, 1011773137731861712, 1448486351628212391462, 3872217739919424676743213
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OFFSET
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2,2
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COMMENTS
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A sequence x_1, ..., x_n is regular if 1 = x_1 <= x_2 <= ... <= x_n and x_j <= Sum_{i=1..j-1} x_i for all j >= 2. It is immediate from this definition that x_2 = 1 and x_j <= 2^(j-2) for all j >= 2.
A sequence x_1, x_2, ..., x_n is regular if and only if (x_2, ..., x_n) is a complete partition of x_2+...+x_n (see A126796 for the definition of a complete partition). As a result, the number of regular sequences with sum equal to n is given by A126796(n-1).
(End)
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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EXAMPLE
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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)
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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn,nice,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A005268
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Number of elementary sequences of length n.
(Formerly M1233)
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+10
6
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1, 1, 2, 4, 10, 31, 120, 578, 3422, 24504, 208833, 2086777, 24123293, 318800755, 4766262421, 79874304340, 1488227986802
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graph;
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listen;
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text;
internal format)
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OFFSET
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1,3
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COMMENTS
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In Fishburn-Roberts (1989) it is stated that no recurrence is known. - N. J. A. Sloane, Jan 04 2014
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REFERENCES
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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).
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LINKS
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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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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A005272
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Number of Van Lier sequences of length n.
(Formerly M1682)
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+10
6
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1, 2, 6, 26, 164, 1529, 21439, 461481, 15616226, 851607867, 76555549499, 11550559504086
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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2,2
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COMMENTS
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From Fishburn et al.'s abstract (from the 1990 article): "We study two types of sequences of positive integers which arise from problems in the measurement of comparative judgements of probability. The first type consists of the Van Lier sequences, which are nondecreasing sequences x_1, x_2, ..., x_n of positive integers that start with two 1's and have the property that, whenever j < k <= n, x_k - x_j can be expressed as a sum of terms from the sequence other than x_j. The second type consists of the regular sequences, which are nondecreasing sequences of positive integers that start with two 1's and have the property that each subsequent term is a partial sum of preceding terms. ... We also study one-term extensions of Van Lier sequences and obtain some asymptotic results on the number of Van Lier sequences." - Jonathan Vos Post, Apr 16 2011
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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]
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CROSSREFS
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KEYWORD
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nonn,nice,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A008926
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Number of uniquely agreeing sequences.
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+10
6
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OFFSET
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1,3
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REFERENCES
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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)
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LINKS
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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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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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