A244911 - OEIS

%I #25 Nov 10 2024 02:22:07

%S 1,1,1,1,2,1,1,3,4,1,1,4,7,7,1,1,5,10,13,11,1,1,6,13,19,21,16,1,1,7,

%T 16,25,31,31,22,1,1,8,19,31,41,46,43,29,1,1,9,22,37,51,61,64,57,37,1,

%U 1,10,25,43,61,76,85,85,73,46,1,1,11,28,49,71,91,106,113,109,91

%N Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

%C T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.

%C It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.

%C The triangle under the main diagonal is A121722.

%C Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0, ...). Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - _Gary W. Adamson_, Oct 01 2015

%H Kival Ngaokrajang, <a href="/A244911/a244911.pdf">Illustration for n = 0..3, k = 1..4</a>

%F T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

%e Table begins:

%e        C0  C1  C2  C3  C4  C5

%e   n/k  0   1   2   3   4   5   ...

%e R0 0   1   1   1   1   1   1   ...

%e R1 1   1   2   3   4   5   6   ...

%e R2 2   1   4   7   10  13  16  ...

%e R3 3   1   7   13  19  25  31  ...

%e R4 4   1   11  21  31  41  51  ...

%e R5 5   1   16  31  46  61  76  ...

%e R6 6   1   22  43  64  85  106 ...

%e R7 7   1   29  57  85  113 141 ...

%e R8 8   1   37  73  109 145 181 ...

%e R9 9   1   46  91  136 181 226 ...

%e   ...  ... ... ... ... ... ... ...

%e C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12  =  A003154.

%e R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.

%o (Small Basic)

%o For k = 0 to 50

%o   a[0][k] = 1

%o   For n = 1 to 50

%o     a[n][k] = n*k + a[n-1][k]

%o   EndFor

%o Endfor

%o '==================================

%o For t = 1 to 20

%o   d = 1

%o   For nn = 0 To t-1

%o     kk =  t- d

%o     TextWindow.Write(a[nn][kk]+", ")

%o     d = d + 1

%o   EndFor

%o Endfor

%Y Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.

%K nonn,tabl,changed

%O 0,5

%A _Kival Ngaokrajang_, Jul 07 2014