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a(0) = a(1) = a(2) = 1, for n > 2, a(n) = a(n-1) + a(n-k) + k with k = 2.
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%I #7 Apr 19 2023 02:47:14

%S 1,1,1,4,7,10,16,25,37,55,82,121,178,262,385,565,829,1216,1783,2614,

%T 3832,5617,8233,12067,17686,25921,37990,55678,81601,119593,175273,

%U 256876,376471,551746,808624,1185097,1736845,2545471,3730570,5467417,8012890,11743462,17210881

%N a(0) = a(1) = a(2) = 1, for n > 2, a(n) = a(n-1) + a(n-k) + k with k = 2.

%C Called Leonardo 2-numbers in the Tan-Leung paper.

%H Elif Tan and Ho-Hon Leung, <a href="https://doi.org/10.5281/zenodo.7569221">On Leonardo p-Numbers</a>, Integers (2023) Vol. 23, #A7. See p. 2.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1).

%t LinearRecurrence[{2, -1, 1, -1}, {1, 1, 1, 4}, 40] (* or *)

%t With[{k = 2}, Nest[Append[#, #[[-1]] + #[[-k - 1]] + k] &, {1, 1, 1}, 40] ]

%Y Cf. A001595, A111314, A362256.

%K nonn,easy

%O 0,4

%A _Michael De Vlieger_, Apr 13 2023