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A163493
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Number of binary strings of length n which have the same number of 00 and 01 substrings.
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26
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1, 2, 2, 3, 6, 9, 15, 30, 54, 97, 189, 360, 675, 1304, 2522, 4835, 9358, 18193, 35269, 68568, 133737, 260802, 509132, 995801, 1948931, 3816904, 7483636, 14683721, 28827798, 56637969, 111347879, 219019294, 431043814, 848764585, 1672056525, 3295390800, 6497536449
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OFFSET
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0,2
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COMMENTS
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A variation of problem 11424 in the American Mathematical Monthly. Terms were brute-force calculated using Maple 10.
Proposed Problem 11610 in the Dec 2011 A.M.M.
Also the antidiagonal sums of the matrices counting integer compositions by length and alternating sum (A345197). So a(n) is the number of integer compositions of n + 1 of length (n - s + 3)/2, where s is the alternating sum of the composition. For example, the a(0) = 1 through a(6) = 7 compositions are:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (31) (41) (51) (61)
(121) (122) (123) (124)
(221) (222) (223)
(1112) (321) (322)
(1211) (1122) (421)
(1221) (1132)
(2112) (1231)
(2211) (2122)
(2221)
(3112)
(3211)
(11131)
(12121)
(13111)
For a bijection with the main (binary string) interpretation, take the run-lengths of each binary string of length n + 1 that satisfies the condition and starts with 1.
(End)
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LINKS
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R. Stanley, Problem 11610, Amer. Math. Monthly, 118 (2011), 937; 120 (2013), 943-944.
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FORMULA
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G.f.: 1/2/(1-x) + (1+2*x)/2/sqrt((1-x)*(1-2*x)*(1+x+2*x^2)). - Richard Stanley, corrected Apr 29 2011
G.f.: (1 + sqrt( 1 + 4*x / ((1 - x) * (1 - 2*x) * (1 + x + 2*x^2)))) / (2*(1 - x)). - Michael Somos, Jan 30 2012
a(n) = sum( binomial(2*k-1, k)*binomial(n-2*k,k) + binomial(2*k, k)*binomial(n-2*k-1, k), k=0..floor(n/3)). - Joel B. Lewis, May 21 2011
Conjecture: -n*a(n) +(2+n)*a(n-1) +(3n-12)*a(n-2) +(12-n)*a(n-3) +(2n-18)*a(n-4)+(56-12n)*a(n-5) +(8n-40)*a(n-6)=0. - R. J. Mathar, Nov 28 2011
G.f. y = A(x) satisfies x = (1 - x) * (1 - 2*x) * (1 + x + 2*x^2) * y * (y * (1 - x) - 1). - Michael Somos, Jan 30 2012
Sequence a(n) satisfies 0 = a(n) * (n^2-2*n) + a(n-1) * (-3*n^2+8*n-2) + a(n-2) * (3*n^2-10*n+2) + a(n-3) * (-5*n^2+18*n-6) + a(n-4) * (8*n^2-34*n+22) + a(n-5) * (-4*n^2+20*n-16) except if n=1 or n=2. - Michael Somos, Jan 30 2012
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EXAMPLE
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1 + 2*x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 15*x^6 + 30*x^7 + 54*x^8 + 97*x^9 + ...
The a(0) = 1 though a(6) = 15 binary strings:
() (0) (1,0) (0,0,1) (0,0,1,0) (0,0,1,1,0) (0,0,0,1,0,1)
(1) (1,1) (1,1,0) (0,0,1,1) (0,0,1,1,1) (0,0,1,0,0,1)
(1,1,1) (0,1,0,0) (0,1,1,0,0) (0,0,1,1,1,0)
(1,0,0,1) (1,0,0,1,0) (0,0,1,1,1,1)
(1,1,1,0) (1,0,0,1,1) (0,1,0,0,0,1)
(1,1,1,1) (1,0,1,0,0) (0,1,1,1,0,0)
(1,1,0,0,1) (1,0,0,1,1,0)
(1,1,1,1,0) (1,0,0,1,1,1)
(1,1,1,1,1) (1,0,1,1,0,0)
(1,1,0,0,1,0)
(1,1,0,0,1,1)
(1,1,0,1,0,0)
(1,1,1,0,0,1)
(1,1,1,1,1,0)
(1,1,1,1,1,1)
(End)
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MAPLE
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with(combinat): count := proc(n) local S, matches, A, k, i; S := subsets(\{seq(i, i=1..n)\}): matches := 0: while not S[finished] do A := S[nextvalue](): k := 0: for i from 1 to n-1 do: if not (i in A) and not (i+1 in A) then k := k + 1: fi: if not (i in A) and (i+1 in A) then k := k - 1: fi: od: if (k = 0) then matches := matches + 1: fi: end do; return(matches); end proc:
# second Maple program:
b:= proc(n, l, t) option remember; `if`(n-abs(t)<0, 0, `if`(n=0, 1,
add(b(n-1, i, t+`if`(l=0, (-1)^i, 0)), i=0..1)))
end:
a:= n-> b(n, 1, 0):
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MATHEMATICA
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a[0] = 1; a[n_] := Sum[Binomial[2*k - 1, k]*Binomial[n - 2*k, k] + Binomial[2*k, k]*Binomial[n - 2*k - 1, k], {k, 0, n/3}];
Table[Length[Select[Tuples[{0, 1}, n], Count[Partition[#, 2, 1], {0, 0}]==Count[Partition[#, 2, 1], {0, 1}]&]], {n, 0, 10}] (* Gus Wiseman, Jul 27 2021 *)
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CROSSREFS
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Antidiagonal sums of the matrices A345197.
Taking diagonal instead of antidiagonal sums gives A345908.
A011782 counts compositions (or binary strings).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
Cf. A000041, A000070, A000096, A000097, A000124, A000346, A007318, A008549, A025047, A131577, A238279.
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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