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A356619 a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 3 for i = 2,...,k. 3

%I #23 Sep 04 2022 12:55:57

%S 0,1,4,11,25,52,103,198,374,699,1298,2401,4431,8166,15037,27676,50924,

%T 93685,172336,316999,583077,1072472,1972611,3628226,6673378,12274287,

%U 22575966,41523709,76374043,140473802,258371641,475219576,874065112,1607656425

%N a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 3 for i = 2,...,k.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,-1,1).

%F G.f.: x*(1 + x + x^2)/((-1 + x)^2*(1 - x - x^2 - x^3)).

%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-4) + a(n-5).

%F a(n) = A221949(n+2)-1 for n >= 0.

%t maxDiff = 3;

%t t = Map[Length[Select[Map[{#, Max[Differences[#]]} &,

%t Drop[Subsets[Range[#]], # + 1]], #[[2]] <= maxDiff &]] &, Range[16]]

%t FindGeneratingFunction[%, x]

%t FindLinearRecurrence[t]

%t LinearRecurrence[{3, -2, 0, -1, 1}, {0, 1, 4, 11, 25}, 45]

%Y Cf. A001891, A062544, A221949, A356620, A356621.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Aug 24 2022

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Last modified July 12 11:12 EDT 2024. Contains 374244 sequences. (Running on oeis4.)