A198063
Triangle read by rows (n >= 0, 0 <= k <= n, m = 3); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).
Original entry on oeis.org
0, 1, 1, 8, 4, 8, 27, 15, 15, 27, 64, 40, 32, 40, 64, 125, 85, 65, 65, 85, 125, 216, 156, 120, 108, 120, 156, 216, 343, 259, 203, 175, 175, 203, 259, 343, 512, 400, 320, 272, 256, 272, 320, 400, 512, 729, 585, 477, 405, 369, 369, 405, 477, 585, 729
Offset: 0
[0] 0
[1] 1, 1
[2] 8, 4, 8
[3] 27, 15, 15, 27
[4] 64, 40, 32, 40, 64
[5] 125, 85, 65, 65, 85, 125
[6] 216, 156, 120, 108, 120, 156, 216
[7] 343, 259, 203, 175, 175, 203, 259, 343
From _M. F. Hasler_, Nov 22 2018: (Start)
Can also be seen as the square array A(n,k)=(n+k)*(n^2 + k^2) read by antidiagonals:
n | k: 0 1 2 3 ...
--+----------------------
0 | 0 1 8 27 ...
1 | 1 4 15 40 ...
2 | 8 15 32 65 ...
3 | 27 40 65 108 ...
... ... ...
(End)
-
[[2*k^2*n-2*k*n^2+n^3: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 23 2018
-
A198063 := (n,k) -> 2*k^2*n-2*k*n^2+n^3:
-
t[n_, k_] := 2 k^2*n - 2 k*n^2 + n^3; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)
-
A198063(n,k)=2*k^2*n-2*k*n^2+n^3 \\ See also A321500. - M. F. Hasler, Nov 22 2018
-
[[ 2*k^2*n-2*k*n^2+n^3 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Nov 23 2018
A348897
Numbers of the form (x + y)*(x^2 + y^2).
Original entry on oeis.org
0, 1, 4, 8, 15, 27, 32, 40, 64, 65, 85, 108, 120, 125, 156, 175, 203, 216, 256, 259, 272, 320, 343, 369, 400, 405, 477, 500, 512, 520, 580, 585, 671, 680, 715, 729, 803, 820, 864, 888, 935, 960, 1000, 1080, 1105, 1111, 1157, 1248, 1261, 1331, 1372, 1400, 1417
Offset: 1
1010101 is in this sequence because 1010101 = (100 + 1)*(100^2 + 1^2).
-
# Returns the terms less than or equal to b^3.
function A348897List(b)
b3 = b^3; R = [0]
for n in 1:b
for k in 0:n
a = (n + k) * (n^2 + k^2)
a > b3 && break
push!(R, a)
end end
unique!(sort!(R)) end
A348897List(12) |> println
-
# Returns the terms less than or equal to b^3.
A348897List := proc(b) local n, k, a, b3, R;
b3 := abs(b^3); R := {};
for n from 0 to b do for k from 0 to n do
a := (n + k)*(n^2 + k^2);
if a > b3 then break fi;
R := R union {a};
od od; sort(R) end:
A348897List(12);
-
max = 2000;
xmax = max^(1/3) // Ceiling;
Table[(x + y) (x^2 + y^2), {x, 0, xmax}, {y, x, xmax}] // Flatten // Union // Select[#, # <= max&]& (* Jean-François Alcover, Oct 23 2023 *)
A321490
Triangular table T[n,k] = (n+k)(n^2+k^2), 1 <= k <= n = 1, 2, 3, ...; read by rows.
Original entry on oeis.org
4, 15, 32, 40, 65, 108, 85, 120, 175, 256, 156, 203, 272, 369, 500, 259, 320, 405, 520, 671, 864, 400, 477, 580, 715, 888, 1105, 1372, 585, 680, 803, 960, 1157, 1400, 1695, 2048, 820, 935, 1080, 1261, 1484, 1755, 2080, 2465, 2916, 1111, 1248, 1417, 1624, 1875, 2176, 2533, 2952, 3439, 4000, 1464, 1625, 1820, 2055, 2336
Offset: 1
The table starts:
Row 1: 4;
Row 2: 15, 32;
Row 3: 40, 65, 108;
Row 4: 85, 120, 175, 256;
Row 5: 156, 203, 272, 369, 500;
Row 6: 259, 320, 405, 520, 671, 864;
Row 7: 400, 477, 580, 715, 888, 1105, 1372;
Row 8: 585, 680, 803, 960, 1157, 1400, 1695, 2048;
etc.
Cf.
A321491 (numbers of the form T(n,k) with n > k > 0).
Cf.
A321492 (numbers which can be written at least twice in this form).
Cf.
A198063 (read as a square array equals T(n,k) for all n, k >= 0).
Cf.
A321500 (variant of this table with additional row 0 and column 0).
-
t[n_, k_] := (n + k) (n^2 + k^2); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)
-
A321490(n,k)=(n+k)*(n^2+k^2)
A321490_row(n)=vector(n,k,(n+k)*(n^2+k^2))
A321490_list(N=12)=concat(apply(A321490_row,[1..N]))
Showing 1-3 of 3 results.
Comments