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Number of compositions of n with the LMV property.
3

%I #16 May 18 2020 14:55:06

%S 0,1,1,4,6,14,26,54,105,213,423,849,1697,3399,6799,13608,27220,54451,

%T 108901,217789,435517,870892,1741467,3482322,6963512,13925078,

%U 27846979,55689150,111371677,222735709,445466058,890938357,1781916885,3563957177,7128223846

%N Number of compositions of n with the LMV property.

%C A composition has the largest missing value (LMV) property if its largest part is at least 2 and it does not contain a part one less than its largest part. - _Andrew Howroyd_, May 18 2020

%H Andrew Howroyd, <a href="/A188576/b188576.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Archibald and A. Knopfmacher, <a href="https://doi.org/10.1016/j.disc.2011.01.012">The largest missing value in a composition of an integer</a>, Discrete Math., 311 (2011), 723-731.

%F G.f.: Sum_{k>=2} 1/(1 - x*(1-x^(k-2))/(1-x) - x^k) - 1/(1 - x*(1-x^(k-2))/(1-x)). - _Andrew Howroyd_, May 18 2020

%e The a(5) = 6 compositions of 5 with the LMV property are: 5, 14, 41, 113, 131, 311. - _Andrew Howroyd_, May 18 2020

%o (PARI) seq(n)={Vec(sum(k=2, n, 1/(1 - x*(1-x^(k-2))/(1-x) - x^k) - 1/(1 - x*(1-x^(k-2))/(1-x)) + O(x*x^n)), -n)} \\ _Andrew Howroyd_, May 18 2020

%Y Cf. A188575, A188577.

%K nonn

%O 1,4

%A _N. J. A. Sloane_, Apr 04 2011

%E Terms a(21) and beyond from _Andrew Howroyd_, May 18 2020