A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220

Offset

0,3

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Formula

T(n, 0) = A000217(n).

T(n, n) = A046092(n).

T(2*n, n) = A062725(n) (central terms).

Sum_{k=0..n} T(n, k) = A245301(n).

From G. C. Greubel, Apr 01 2021: (Start)

T(n, 1) = A000124(n+1) = A134869(n+1), n >= 1.

T(n, 2) = A152948(n+4), n >= 2.

T(n, 3) = A152950(n+4), n >= 3.

T(n, 4) = A145018(n+5), n >= 4.

T(n, 5) = A167499(n+4), n >= 5.

T(n, 6) = A166136(n+5), n >= 6.

T(n, 7) = A167487(n+6), n >= 7.

T(n, n-1) = A056220(n), n >= 1.

T(n, n-2) = A142463(n-1), n >= 2.

T(n, n-3) = A054000(n-1), n >= 3.

T(n, n-4) = A090288(n-3), n >= 4.

T(n, n-5) = A268581(n-4), n >= 5.

T(n, n-6) = A059993(n-4), n >= 6.

T(n, n-7) = (-1)*A147973(n), n >= 7.

T(n, n-8) = A139570(n-5), n >= 8.

T(n, n-9) = A271625(n-5), n >= 9.

T(n, n-10) = A222182(n-4), n >= 10.

T(2*n, n-1) = A081270(n-1), n >= 1.

T(2*n, n+1) = A117625(n+1), n >= 1. (End)

Example

First rows and their row sums (A245301):
   0                                                                  0;
   1,  4                                                              5;
   3,  7,  12                                                        22;
   6, 11,  17,  24                                                   58;
  10, 16,  23,  31,  40                                             120;
  15, 22,  30,  39,  49,  60                                        215;
  21, 29,  38,  48,  59,  71,  84                                   350;
  28, 37,  47,  58,  70,  83,  97, 112                              532;
  36, 46,  57,  69,  82,  96, 111, 127, 144                         768;
  45, 56,  68,  81,  95, 110, 126, 143, 161, 180                   1065;
  55, 67,  80,  94, 109, 125, 142, 160, 179, 199, 220              1430;
  66, 79,  93, 108, 124, 141, 159, 178, 198, 219, 241, 264         1870;
  78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312    2392.

Mathematica

Table[k + Binomial[n+k+1, 2], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

Prog

(Haskell)
a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl
(Magma) [k + Binomial(n+k+1, 2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021
(Sage) flatten([[k + binomial(n+k+1, 2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021

Crossrefs

Cf. A000217, A046092, A062725, A245301.

Sequence in context: A010654 A244335 A275818 * A228949 A048227 A213661

Adjacent sequences: A245297 A245298 A245299 * A245301 A245302 A245303

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Jul 17 2014

Status

approved