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Triangle read by rows: T(n,k) in which row n lists the first n terms of A024916 in reverse order, 1 <= k <= n.
1

%I #39 Mar 10 2022 04:56:22

%S 1,4,1,8,4,1,15,8,4,1,21,15,8,4,1,33,21,15,8,4,1,41,33,21,15,8,4,1,56,

%T 41,33,21,15,8,4,1,69,56,41,33,21,15,8,4,1,87,69,56,41,33,21,15,8,4,1,

%U 99,87,69,56,41,33,21,15,8,4,1,127,99,87,69,56,41,33,21,15,8,4,1

%N Triangle read by rows: T(n,k) in which row n lists the first n terms of A024916 in reverse order, 1 <= k <= n.

%C T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the stepped pyramid with n levels described in A245092 (see example).

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>

%F T(n,k) = A024916(A004736(n,k)).

%F T(n,k) = T(n,k) = A024916(n-k+1).

%F T(n,k) = Sum_{j=1..n} A272172(j,k).

%e Triangle begins:

%e 1;

%e 4, 1;

%e 8, 4, 1;

%e 15, 8, 4, 1;

%e 21, 15, 8, 4, 1;

%e 33, 21, 15, 8, 4, 1;

%e 41, 33, 21, 15, 8, 4, 1;

%e 56, 41, 33, 21, 15, 8, 4, 1;

%e 69, 56, 41, 33, 21, 15, 8, 4, 1;

%e 87, 69, 56, 41, 33, 21, 15, 8, 4, 1;

%e 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1;

%e 127, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1;

%e ...

%e For n = 9 the lateral view and top view of the stepped pyramid described in A245092 look as shown below:

%e _

%e 9 1 |_|_

%e 8 4 |_ _|_

%e 7 8 |_ _|_|_

%e 6 15 |_ _ _| |_

%e 5 21 |_ _ _|_ _|_

%e 4 33 |_ _ _ _| | |_

%e 3 41 |_ _ _ _|_|_ _|_

%e 2 56 |_ _ _ _ _|_|_ |_

%e 1 69 |_ _ _ _ _|_ _|_ _|

%e .

%e Level Row 9 Lateral view of

%e k T(9,k) the stepped pyramid

%e .

%e _ _ _ _ _ _ _ _ _

%e |_| | | | | | | | |

%e |_ _|_| | | | | | |

%e |_ _| _|_| | | | |

%e |_ _ _| _|_| | |

%e |_ _ _| _| _ _|_|

%e |_ _ _ _| _| |

%e |_ _ _ _| |_ _|

%e |_ _ _ _ _|

%e |_ _ _ _ _|

%e .

%e Top view of

%e the stepped pyramid

%e .

%e For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the stepped pyramid, so T(9,1) = 69.

%e For n = 9 and k = 9 there is only one cubic cell in the level k = 9 (the top) of the stepped pyramid, so T(9,9) = 1.

%e The volume of the stepped pyramid (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000027 hence it's equal to A175254(9) = 248, equaling the sum of the 9th row of triangle.

%t Join@@Array[Reverse@Array[Sum[#-Mod[#,m],{m,#}]&,#]&,12] (* _Giorgos Kalogeropoulos_, Jan 12 2022 *)

%o (PARI) row(n) = Vecrev(vector(n, k, sum(i=1, k, k\i*i))); \\ _Michel Marcus_, Jan 22 2022

%Y Column k gives A024916 starting in row k.

%Y Row sums give A175254.

%Y Cf. A340423 (analog for the tower described in A221529).

%Y Cf. A000027, A000203, A004736, A196020, A235791, A236104, A237591, A237593, A245092, A262626, A272172.

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Jan 09 2022