Displaying 1-4 of 4 results found.
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1, 1, 2, 4, 7, 14, 27, 54, 106, 212, 420, 840, 1673, 3346, 6678, 13356, 26685, 53370, 106686, 213372, 426638, 853276, 1706340, 3412680, 6824940, 13649880
Number of binary words of length n (beginning with 0) whose autocorrelation function is the indicator of a singleton.
+10
29
1, 1, 2, 3, 6, 10, 20, 37, 74, 142, 284, 558, 1116, 2212, 4424, 8811, 17622, 35170, 70340, 140538, 281076, 561868, 1123736, 2246914, 4493828, 8986540, 17973080, 35943948, 71887896, 143771368, 287542736, 575076661, 1150153322, 2300289022, 4600578044, 9201120918
COMMENTS
The number of binary strings sharing the same autocorrelations.
The number of binary words of length n (beginning with 0) which do not start with an even palindrome (i.e. which are not of the form ss*t where s is a (nonempty) word, s* is its reverse, and t is any (possibly empty) word). - Mamuka Jibladze, Sep 30 2014
This sequence counts each of the following essentially equivalent things:
1. Sets of distinct positive integers with maximum n in which all adjacent elements have quotients > 1/2. For example, the a(1) = 1 through a(6) = 10 sets are:
{1} {2} {3} {4} {5} {6}
{2,3} {3,4} {3,5} {4,6}
{2,3,4} {4,5} {5,6}
{2,3,5} {3,4,6}
{3,4,5} {3,5,6}
{2,3,4,5} {4,5,6}
{2,3,4,6}
{2,3,5,6}
{3,4,5,6}
{2,3,4,5,6}
2. For n > 1, sets of distinct positive integers with maximum n - 1 whose first-differences are term-wise less than their decapitation (remove the maximum). For example, the set q = {2,4,5} has first-differences (2,1), which are not less than (2,4), so q is not counted under a(5). On the other hand, r = {2,3,5,6} has first-differences {1,2,1}, which are less than {2,3,5}, so r is counted under a(6).
3. Compositions of n where each part after the first is less than the sum of all preceding parts. For example, the a(1) = 1 through a(6) = 10 compositions are:
(1) (2) (3) (4) (5) (6)
(21) (31) (41) (51)
(211) (32) (42)
(311) (411)
(212) (321)
(2111) (312)
(3111)
(2121)
(2112)
(21111)
(End)
FORMULA
a(2n) = 2*a(2n-1) - a(n) for n >= 1; a(2n+1) = 2*a(2n) for n >= 1.
MAPLE
a:= proc(n) option remember; `if`(n=0, 1/2,
2*a(n-1)-`if`(n::odd, 0, a(n/2)))
end:
MATHEMATICA
a[1] = 1; a[n_] := a[n] = If[EvenQ[n], 2*a[n-1] - a[n/2], 2*a[n-1]]; Array[a, 40] (* Jean-François Alcover, Jul 17 2015 *)
Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&Min@@Divide@@@Partition[#, 2, 1]>1/2&]], {n, 8}] (* Gus Wiseman, Mar 08 2021 *)
PROG
(PARI) a(n)=if(n<2, n>0, 2*a(n-1)-(1-n%2)*a(n\2))
CROSSREFS
The version with quotients <= 1/2 is A018819.
The version with quotients < 1/2 is A040039.
A000045 counts sets containing n with all differences > 2.
A000929 counts partitions with no adjacent parts having quotient < 2.
A342094 counts partitions with no adjacent parts having quotient > 2.
AUTHOR
Torsten.Sillke(AT)uni-bielefeld.de
Narayana-Zidek-Capell numbers: a(n) = 1 for n <= 2. Otherwise a(2n) = 2a(2n-1), a(2n+1) = 2a(2n) - a(n).
(Formerly M0787 N0297)
+10
24
1, 1, 1, 2, 3, 6, 11, 22, 42, 84, 165, 330, 654, 1308, 2605, 5210, 10398, 20796, 41550, 83100, 166116, 332232, 664299, 1328598, 2656866, 5313732, 10626810, 21253620, 42505932, 85011864, 170021123, 340042246, 680079282, 1360158564
COMMENTS
Number of compositions p(1) + p(2) + ... + p(k) = n of n into positive parts p(i) with p(1)=1 and p(k) <= Sum_{j=1..k-1} p(j), see example - Claude Lenormand (claude.lenormand(AT)free.fr), Jan 29 2001 (clarified by Joerg Arndt, Feb 01 2013)
a(n) is the number of sequences (b(1),b(2),...) of unspecified length satisfying b(1) = 1, 1 <= b(i) <= 1 + Sum[b(j),{j,i-1}] for i>=2, Sum[b(i)] = n-1. For example, a(5) = 3 counts (1, 1, 1, 1), (1, 2, 1), (1, 1, 2). These sequences are generated by the Mathematica code below. - David Callan, Nov 02 2005
a(n+1) is the number of padded compositions (d_1,d_2,...,d_n) of n satisfying d_i <= i for all i. A padded composition of n is obtained from an ordinary composition (c_1,c_2,...,c_r) of n (r >= 1, each c_i >= 1, Sum_{i=1..r} c_i = n) by inserting c_i - 1 zeros immediately after each c_i. Thus (3,1,2) -> (3,0,0,1,2,0) is a padded composition of 6 and a padded composition of n has length n. For example, with n=4, a(5) counts (1,1,1,1), (1,1,2,0), (1,2,0,1). - David Callan, Feb 04 2006
a(n) is the number of ordered trees on n edges in which (i) every node (= non-root non-leaf vertex) has at least 2 children and (ii) each leaf is either the leftmost or rightmost child of its parent.
For example, a(4)=2 counts
./\.../\
/\...../\,
and a(5)=3 counts
.|.......|....../|\
/ \...../ \...../ \
../\.../\.
(End)
If we remove the condition that a(2) = 1, then the resulting sequence is A045690 minus the first term. - Chai Wah Wu, Nov 08 2022
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.28.
T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. A. Stern, 5. Aufgaben., Journal für die reine und angewandte Mathematik (Crelle's journal), volume 18, 1838, p. 100.
FORMULA
a(1)=1, else a(n) is sum of floor(n/2) previous terms.
Limit is equal to 0.633368347305411640436713144616576659... = 2*Atkinson-Negro-Santoro constant = 2* A242729, see Finch's book, chapter 2.28 (Erdős' Sum-Distinct Set Constant), pp. 189, 552. - Vaclav Kotesovec, Nov 19 2012
a(n) is the permanent of the (n-1) X (n-1) matrix with (i, j) entry = 1 if i-1 <= j <= 2*i-1 and = 0 otherwise. - David Callan, Nov 02 2005
a(n) = Sum_{k=1..n} K(n-k+1, k)*a(n-k), where K(n,k) = 1 if 0 <= k AND k <= n and K(n,k)=0 else. (Several arguments to the K-coefficient K(n,k) can lead to the same sequence. For example, we get A002083 also from a(n) = Sum_{k=1..n} K((n-k)!,k!)*a(n-k), where K(n,k) = 1 if 0 <= k <= n and 0 else. See also the comment to a similar formula for A002487.) - Thomas Wieder, Jan 13 2008
G.f. satisfies: A(x) = (1-x - x^2*A(x^2))/(1-2x). - Paul D. Hanna, Mar 17 2010
EXAMPLE
The a(7) = 11 compositions p(1) + p(2) + ... + p(k) = 7 of 7 into positive parts p(i) with p(1)=1 and p(k) <= Sum_{j=1..k-1} p(j) are
[ 1] [ 1 1 1 1 1 1 1 ]
[ 2] [ 1 1 1 1 1 2 ]
[ 3] [ 1 1 1 1 2 1 ]
[ 4] [ 1 1 1 1 3 ]
[ 5] [ 1 1 1 2 1 1 ]
[ 6] [ 1 1 1 2 2 ]
[ 7] [ 1 1 1 3 1 ]
[ 8] [ 1 1 2 1 1 1 ]
[ 9] [ 1 1 2 1 2 ]
[10] [ 1 1 2 2 1 ]
[11] [ 1 1 2 3 ]
(End)
MAPLE
A002083 := proc(n) option remember; if n<=3 then 1 elif n mod 2 = 0 then 2*procname(n-1) else 2*procname(n-1)-procname((n-1)/2); end if; end proc:
a := proc(n::integer) # A002083 Narayana-Zidek-Capell numbers: a(2n)=2a(2n-1), a(2n+1)=2a(2n)-a(n). local k; option remember; if n = 0 then 1 else add(K(n-k+1, k)*procname(n - k), k = 1 .. n); #else add(K((n-k)!, k!)*procname(n - k), k = 1 .. n); end if end proc; K := proc(n::integer, k::integer) local KC; if 0 <= k and k <= n then KC := 1 else KC := 0 end if; end proc; # Thomas Wieder, Jan 13 2008
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Sum[a[k], {k, n/2, n-1} ]; Table[ a[n], {n, 2, 70, 2} ] (* Robert G. Wilson v, Apr 22 2001 *)
bSequences[1]={ {1} }; bSequences[n_]/; n>=2 := bSequences[n] = Flatten[Table[Map[Join[ #, {i}]&, bSequences[n-i]], {i, Ceiling[n/2]}], 1] (* David Callan *)
a=ConstantArray[0, 20]; a[[1]]=1; a[[2]]=1; Do[If[EvenQ[n], a[[n]]=2a[[n-1]], a[[n]]=2a[[n-1]]-a[[(n-1)/2]]]; , {n, 3, 20}]; a (* Vaclav Kotesovec, Nov 19 2012 *)
PROG
(PARI) a(n)=if(n<3, n>0, 2*a(n-1)-(n%2)*a(n\2))
(PARI) a(n)=local(A=1+x); for(i=1, n, A=(1-x-x^2*subst(A, x, x^2+O(x^n)))/(1-2*x)); polcoeff(A, n) \\ Paul D. Hanna, Mar 17 2010
(Haskell)
a002083 n = a002083_list !! (n-1)
a002083_list = 1 : f [1] where
f xs = x : f (x:xs) where x = sum $ take (div (1 + length xs) 2) xs
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
Triangle read by rows. Signed version of A145362. Main diagonal positive, rest of the nonzero terms negative.
+10
1
1, -1, 1, 0, -1, 1, 0, -1, -1, 1, 0, 0, -1, -1, 1, 0, 0, -1, -1, -1, 1, 0, 0, 0, -1, -1, -1, 1, 0, 0, 0, -1, -1, -1, -1, 1, 0, 0, 0, 0, -1, -1, -1, -1, 1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, -1
COMMENTS
Matrix inverse of this triangle is A155092.
EXAMPLE
Table begins:
1,
-1,1,
0,-1,1,
0,-1,-1,1,
0,0,-1,-1,1,
0,0,-1,-1,-1,1,
0,0,0,-1,-1,-1,1,
0,0,0,-1,-1,-1,-1,1,
0,0,0,0,-1,-1,-1,-1,1,
0,0,0,0,-1,-1,-1,-1,-1,1,
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