

A098832


Square array read by antidiagonals: evennumbered rows of the table are of the form n*(n+m) and oddnumbered rows are of the form n*(n+m)/2.


7



1, 3, 3, 6, 8, 2, 10, 15, 5, 5, 15, 24, 9, 12, 3, 21, 35, 14, 21, 7, 7, 28, 48, 20, 32, 12, 16, 4, 36, 63, 27, 45, 18, 27, 9, 9, 45, 80, 35, 60, 25, 40, 15, 20, 5, 55, 99, 44, 77, 33, 55, 22, 33, 11, 11, 66, 120, 54, 96, 42, 72, 30, 48, 18, 24, 6, 78, 143, 65, 117, 52, 91, 39, 65, 26, 39, 13, 13
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OFFSET

1,2


COMMENTS

The rows of this table and that in A098737 are related. Given a function f = n/( 1 + (1+n) mod(2) ), row n of A098737 can be derived from row n of T by multiplying the latter by f(n); row n of T can be derived from row n of A098737 by dividing the latter by f(n).


LINKS



FORMULA

Item m of row n of T is given (in infix form) by: n T m = n * (n + m) / (1 + m (mod 2)). E.g. Item 4 of row 3 of T: 3 T 4 = 14.
A(n, k) = (1/4)*(3 + (1)^n)*k*(k+n) (array).
T(n, k) = (1/4)*(3 + (1)^k)*(n+1)*(nk+1) (antidiagonal triangle).
Sum_{k=1..n} T(n, k) = (1/8)*(n+1)*( (3*n1)*(n+1) + (1+(1)^n)/2 ).
T(2*n+1, n) = 2*A168509(n+1). (End)


EXAMPLE

Array begins as:
1, 3, 6, 10, 15, 21, 28, 36, 45 ... A000217;
3, 8, 15, 24, 35, 48, 63, 80, 99 ... A005563;
2, 5, 9, 14, 20, 27, 35, 44, 54 ... A000096;
5, 12, 21, 32, 45, 60, 77, 96, 117 ... A028347;
3, 7, 12, 18, 25, 33, 42, 52, 63 ... A027379;
7, 16, 27, 40, 55, 72, 91, 112, 135 ... A028560;
4, 9, 15, 22, 30, 39, 49, 60, 72 ... A055999;
9, 20, 33, 48, 65, 84, 105, 128, 153 ... A028566;
5, 11, 18, 26, 35, 45, 56, 68, 81 ... A056000;
Antidiagonals begin as:
1;
3, 3;
6, 8, 2;
10, 15, 5, 5;
15, 24, 9, 12, 3;
21, 35, 14, 21, 7, 7;
28, 48, 20, 32, 12, 16, 4;
36, 63, 27, 45, 18, 27, 9, 9;
45, 80, 35, 60, 25, 40, 15, 20, 5;
55, 99, 44, 77, 33, 55, 22, 33, 11, 11;


MATHEMATICA

A098832[n_, k_]:= (1/4)*(3+(1)^k)*(n+1)*(nk+1);


PROG

(Magma)
A098832:= func< n, k  (1/4)*(3+(1)^k)*(n+1)*(nk+1) >;
(SageMath)
def A098832(n, k): return (1/4)*(3+(1)^k)*(n+1)*(nk+1)


CROSSREFS

Row m of array: A000217 (m=1), A005563 (m=2), A000096 (m=3), A028347 (m=4), A027379 (m=5), A028560 (m=6), A055999 (m=7), A028566 (m=8), A056000 (m=9), A098603 (m=10), A056115 (m=11), A098847 (m=12), A056119 (m=13), A098848 (m=14), A056121 (m=15), A098849 (m=16), A056126 (m=17), A098850 (m=18), A051942 (m=19).


KEYWORD



AUTHOR

Eugene McDonnell (eemcd(AT)mac.com), Nov 02 2004


EXTENSIONS



STATUS

approved



