A339236 - OEIS

%I #15 Dec 10 2023 17:29:50

%S 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 99,109,1,19,75,145,175,177,1,21,93,205,275,287,1,23,113,281,421,463,

%U 465,1,25,135,375,627,739,753,1,27,159,489,909,1161,1217,1219

%N 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 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 T(n, floor(n/2)) = A001595(n).

%e Triangle begins:

%e   1;

%e   1;

%e   1, 3;

%e   1, 5;

%e   1, 7, 9;

%e   1, 9, 15;

%e   1, 11, 23, 25;

%e   1, 13, 33, 41;

%e   1, 15, 45, 65, 67;

%e   1, 17, 59, 99, 109;

%e   ...

%t 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 (PARI) T(n, k) = 2*sum(j=0, k, binomial(n-j, j)) -1;

%o row(n) = vector(n\2+1, k, k--; T(n,k));

%Y Cf. A001595 (Leonardo numbers: right diagonal).

%Y Cf. A000012 (column 0), A005408 (column 1), A027688 (column 2).

%K nonn,tabf

%O 0,4

%A _Michel Marcus_, Nov 28 2020