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Table[Count[Flatten[Permutations/@IntegerPartitions[n], 1], _?(FreeQ[Differences[#], 0]&)], {n, 0, 20}] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Nov 23 2024 *)
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Number of compositions of n such that no two adjacent parts are equal (these are sometimes called Carlitz compositions).
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Christian G. Bower and Alois P. Heinz, <a href="/A003242/b003242.txt">Table of n, a(n) for n = 0..4100</a> (first 501 terms from Christian G. Bower)
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Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022, p. 42 and 117.
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For N=2n+k, with k even and k>n, a(n) is the number of Carlitz palindromic compositions having k as a central summand. For N=2n+1+k, with k odd and k>n, a(n) is the number of palindromic compositions having k as a central summand. - Gregory L. Simay, May 15 2022
Enumerate the Carlitz palindrome compositions of 13 for which the central summand is the greatest part: 13; 1,11,1; 2,9,2; 3,7,3 & 2,1,7,1,2 & 1,2,7,2,1; and 4,5,4 & 3,1,5,1,3 & 1,3,5,3,1 & 1,2,1,5,1,2,1. Note that in these instances, the number of Carlitz palindromes so enumerated is 1+1+1+3+4 = a(0) + a(1) + a(2) + a(3) + a(4). - Gregory L. Simay, May 15 2022
Cf. A239327, Carlitz palindromic compositions. -Gregory L. Simay, May 07 2022
nonn,nice,changed
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