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
Examples
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.
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Programs
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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
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Magma
[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021
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Mathematica
Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* 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
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, 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)
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