A098832 Square array read by antidiagonals: even-numbered rows of the table are of the form n*(n+m) and odd-numbered rows are of the form n*(n+m)/2.

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

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

G. C. Greubel, Antidiagonals n = 1..50, flattened

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.

From G. C. Greubel, Jul 31 2022: (Start)

A(n, k) = (1/4)*(3 + (-1)^n)*k*(k+n) (array).

T(n, k) = (1/4)*(3 + (-1)^k)*(n+1)*(n-k+1) (antidiagonal triangle).

Sum_{k=1..n} T(n, k) = (1/8)*(n+1)*( (3*n-1)*(n+1) + (1+(-1)^n)/2 ).

T(2*n-1, n) = A181900(n).

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)*(n-k+1);
Table[A098832[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 31 2022 *)

Prog

(Magma)
A098832:= func< n, k | (1/4)*(3+(-1)^k)*(n+1)*(n-k+1) >;
[A098832(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 31 2022
(SageMath)
def A098832(n, k): return (1/4)*(3+(-1)^k)*(n+1)*(n-k+1)
flatten([[A098832(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Jul 31 2022

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).

Column m of array: A026741 (m=1), A022998 (m=2), A165351 (m=3).

Cf. A098737, A168509, A181900.

Sequence in context: A078477 A336077 A360375 * A368151 A107985 A114999

Adjacent sequences: A098829 A098830 A098831 * A098833 A098834 A098835

Keyword

easy,nonn,tabl

Author

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

Extensions

Missing terms added by G. C. Greubel, Jul 31 2022

Status

approved