A219069 Triangle read by rows: T(n,k) = n^4 + (n*k)^2 + k^4, 1 <= k <= n.

3, 21, 48, 91, 133, 243, 273, 336, 481, 768, 651, 741, 931, 1281, 1875, 1333, 1456, 1701, 2128, 2821, 3888, 2451, 2613, 2923, 3441, 4251, 5461, 7203, 4161, 4368, 4753, 5376, 6321, 7696, 9633, 12288, 6643, 6901, 7371, 8113, 9211, 10773, 12931, 15841, 19683

Offset: 1

Author

Reinhard Zumkeller, Nov 11 2012

Keywords

• nonn
• tabl
• look

Comments

Entry 17a from July 9, 1796 in Gauss's Mathematical Diary: "Summa trium quadratorum continue proportionalium numquam primus esse potest: conspicuum exemplum novimus et quod congruum videtur. Confidamus." Paul Bachmann explains that this note is based on Gauss's discovery of this factorization: n^4 + n^2*k^2 + k^4 = (n^2 + n*k + k^2) * (n^2 - n*k + k^2).

Examples

			The triangle begins:
.  1:      3
.  2:     21    48
.  3:     91   133   243
.  4:    273   336   481   768
.  5:    651   741   931  1281  1875
.  6:   1333  1456  1701  2128  2821  3888
.  7:   2451  2613  2923  3441  4251  5461  7203
.  8:   4161  4368  4753  5376  6321  7696  9633 12288
.  9:   6643  6901  7371  8113  9211 10773 12931 15841 19683
. 10:  10101 10416 10981 11856 13125 14896 17301 20496 24661 30000
. 11:  14763 15141 15811 16833 18291 20293 22971 26481 31003 36741 43923
		

References

• Carl Friedrich Gauss (Hans Wussing, ed.), Mathematisches Tagebuch 1796-1814, Ostwalds Klassiker der Exakten Wissenschaften, Leipzig (1976, 1979), pp. 43, 63, 90.

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
• Wikipedia, Gauss's diary
• Wikipedia, Paul Gustav Heinrich Bachmann

Crossrefs

Cf. A059826 (left edge), A219056 (right edge), A219070 (row sums).

Cf. A239426 (central terms).

Cf. A243201 (diagonal (n + 1, n)). - Mathew Englander, Jun 03 2014

Programs

Haskell

a219069 n k = a219069_tabl !! (n-1) !! (k-1)
a219069_row n = a219069_tabl !! n
a219069_tabl = zipWith (zipWith (*)) a215630_tabl a215631_tabl

Mathematica

Table[n^4+(n*k)^2+k^4,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 05 2020 *)

Formula

T(n,k) = A215630(n,k) * A215631(n,k), 1 <= k <= n.