A176283 Triangle T(n,k) = 1 + A000537(n) - A000537(k) - A000537(n-k), read by rows.
1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 64, 83, 64, 1, 1, 125, 181, 181, 125, 1, 1, 216, 333, 370, 333, 216, 1, 1, 343, 551, 649, 649, 551, 343, 1, 1, 512, 847, 1036, 1097, 1036, 847, 512, 1, 1, 729, 1233, 1549, 1701, 1701, 1549, 1233, 729, 1, 1, 1000, 1721, 2206, 2485, 2576, 2485, 2206, 1721, 1000, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 8, 1; 1, 27, 27, 1; 1, 64, 83, 64, 1; 1, 125, 181, 181, 125, 1; 1, 216, 333, 370, 333, 216, 1; 1, 343, 551, 649, 649, 551, 343, 1; 1, 512, 847, 1036, 1097, 1036, 847, 512, 1; 1, 729, 1233, 1549, 1701, 1701, 1549, 1233, 729, 1; 1, 1000, 1721, 2206, 2485, 2576, 2485, 2206, 1721, 1000, 1;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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GAP
Flat(List([0..12], n-> List([0..n], k-> (4 +n^2*(n+1)^2 -k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4 ))); # G. C. Greubel, Nov 25 2019
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Magma
[(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4: k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 25 2019
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Maple
seq(seq(, k=0..n), n=0..12); # G. C. Greubel, Nov 25 2019
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Mathematica
(* Set of sequences q=1..10. This sequence is q=3. *) f[n_, k_, q_]:= f[n, k, q] = 1 + Sum[i^q, {i,0,n}] - Sum[i^q, {i,0,k}] - Sum[i^q, {i,0,n-k}]; Table[Flatten[Table[f[n, k, q], {n,0,10}, {k,0,n}]], {q,1,10}] (* Second program *) Table[(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4, {n,0,12}, {k,0, n} ]//Flatten (* G. C. Greubel, Nov 25 2019 *) -
PARI
T(n,k) = 1 + (n^2*(n+1)^2 - k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4; \\ G. C. Greubel, Nov 25 2019
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Sage
[[(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 25 2019
Formula
T(n,k) = T(n,n-k).
T(n, k) = (4 + n^2*(n+1)^2 - k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4. - G. C. Greubel, Nov 25 2019
Extensions
Edited by R. J. Mathar, May 03 2013
Comments