This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340423 #72 Mar 09 2022 01:15:36
%S A340423 1,4,8,1,15,4,1,21,8,4,1,1,33,15,8,4,4,1,1,41,21,15,8,8,4,4,1,1,1,1,
%T A340423 56,33,21,15,15,8,8,4,4,4,4,1,1,1,1,69,41,33,21,21,15,15,8,8,8,8,4,4,
%U A340423 4,4,1,1,1,1,1,1,1,87,56,41,33,33,21,21,15,15,15,15,8,8,8,8
%N A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.
%C A340423 T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the tower described in A221529 whose largest side of the base is equal to n (see example). - _Omar E. Pol_, Jan 08 2022
%F A340423 T(n,k) = A024916(A336811(n,k)).
%F A340423 T(n,k) = Sum_{j=1..n} A339278(j,k). - _Omar E. Pol_, Jan 08 2022
%e A340423 Triangle begins:
%e A340423 1;
%e A340423 4;
%e A340423 8, 1;
%e A340423 15, 4, 1;
%e A340423 21, 8, 4, 1, 1;
%e A340423 33, 15, 8, 4, 4, 1, 1;
%e A340423 41, 21, 15, 8, 8, 4, 4, 1, 1, 1, 1;
%e A340423 56, 33, 21, 15, 15, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1;
%e A340423 69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
%e A340423 ...
%e A340423 For n = 9 the length of row 9 is A000041(9-1) = 22.
%e A340423 From _Omar E. Pol_, Jan 08 2022: (Start)
%e A340423 For n = 9 the lateral view and top view of the tower described in A221529 look like as shown below:
%e A340423 _
%e A340423 22 1 | |
%e A340423 21 1 | |
%e A340423 20 1 | |
%e A340423 19 1 | |
%e A340423 18 1 | |
%e A340423 17 1 | |
%e A340423 16 1 |_|_
%e A340423 15 4 | |
%e A340423 14 4 | |
%e A340423 13 4 | |
%e A340423 12 4 |_ _|_
%e A340423 11 8 | | |
%e A340423 10 8 | | |
%e A340423 9 8 | | |
%e A340423 8 8 |_ _|_|_
%e A340423 7 15 | | |
%e A340423 6 15 |_ _ _| |_
%e A340423 5 21 | | |
%e A340423 4 21 |_ _ _|_ _|_
%e A340423 3 33 |_ _ _ _| | |_
%e A340423 2 41 |_ _ _ _|_|_ _|_ _
%e A340423 1 69 |_ _ _ _ _|_ _|_ _|
%e A340423 .
%e A340423 Level Row 9 Lateral view
%e A340423 k T(9,k) of the tower
%e A340423 .
%e A340423 _ _ _ _ _ _ _ _ _
%e A340423 |_| | | | | | | |
%e A340423 |_ _|_| | | | | |
%e A340423 |_ _| _|_| | | |
%e A340423 |_ _ _| _|_| |
%e A340423 |_ _ _| _| _ _|
%e A340423 |_ _ _ _| |
%e A340423 |_ _ _ _| _ _|
%e A340423 | |
%e A340423 |_ _ _ _ _|
%e A340423 .
%e A340423 Top view
%e A340423 of the tower
%e A340423 .
%e A340423 For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the tower, so T(9,1) = 69.
%e A340423 For n = 9 and k = 22 there is only one cubic cell in the level 22 (the top) of the tower, so T(9,22) = 1.
%e A340423 The volume of the tower (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000041 hence it's equal to A066186(9) = 270, equaling the sum of the 9th row of triangle. (End)
%o A340423 (PARI) f(n) = numbpart(n-1);
%o A340423 T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--; ); 1+s; } \\ A336811
%o A340423 g(n) = sum(k=1, n, n\k*k); \\ A024916
%o A340423 row(n) = vector(f(n), k, g(T(n,k))); \\ _Michel Marcus_, Jan 22 2022
%Y A340423 Row sums give A066186.
%Y A340423 Row lengths give A000041.
%Y A340423 The length of the m-th block in row n is A187219(m), m >= 1.
%Y A340423 Cf. A350637 (analog for the stepped pyramid described in A245092).
%Y A340423 Cf. A000203, A024916, A196020, A221529, A236104, A235791, A237270, A237271, A237593, A339278, A262626, A336811, A338156, A340035, A341149, A346533, A350333.
%K A340423 nonn,tabf
%O A340423 1,2
%A A340423 _Omar E. Pol_, Jan 07 2021