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A339236 Irregular triangle of incomplete Leonardo numbers read by rows. T(n, k) = 2*(Sum_{j=0..k} binomial(n-j, j)) - 1, for n>=0 and 0<=k<=floor(n/2).

%I A339236 #15 Dec 10 2023 17:29:50
%S A339236 1,1,1,3,1,5,1,7,9,1,9,15,1,11,23,25,1,13,33,41,1,15,45,65,67,1,17,59,
%T A339236 99,109,1,19,75,145,175,177,1,21,93,205,275,287,1,23,113,281,421,463,
%U A339236 465,1,25,135,375,627,739,753,1,27,159,489,909,1161,1217,1219
%N A339236 Irregular triangle of incomplete Leonardo numbers read by rows. T(n, k) = 2*(Sum_{j=0..k} binomial(n-j, j)) - 1, for n>=0 and 0<=k<=floor(n/2).
%H A339236 P. Catarino and A. Borges, <a href="http://math.colgate.edu/~integers/u43/u43.pdf">A Note on Incomplete Leonardo Numbers</a>, INTEGERS 20A (2020) A43.
%F A339236 T(n, floor(n/2)) = A001595(n).
%e A339236 Triangle begins:
%e A339236   1;
%e A339236   1;
%e A339236   1, 3;
%e A339236   1, 5;
%e A339236   1, 7, 9;
%e A339236   1, 9, 15;
%e A339236   1, 11, 23, 25;
%e A339236   1, 13, 33, 41;
%e A339236   1, 15, 45, 65, 67;
%e A339236   1, 17, 59, 99, 109;
%e A339236   ...
%t A339236 T[n_, k_] := 2 * Sum[Binomial[n - j, j], {j, 0, k}] - 1; Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/2]}] // Flatten (* _Amiram Eldar_, Nov 28 2020 *)
%o A339236 (PARI) T(n, k) = 2*sum(j=0, k, binomial(n-j, j)) -1;
%o A339236 row(n) = vector(n\2+1, k, k--; T(n,k));
%Y A339236 Cf. A001595 (Leonardo numbers: right diagonal).
%Y A339236 Cf. A000012 (column 0), A005408 (column 1), A027688 (column 2).
%K A339236 nonn,tabf
%O A339236 0,4
%A A339236 _Michel Marcus_, Nov 28 2020