A084061 Square number array read by antidiagonals.

1, 1, 1, 1, 1, 4, 1, 1, 5, 27, 1, 1, 6, 36, 256, 1, 1, 7, 45, 353, 3125, 1, 1, 8, 54, 452, 4400, 46656, 1, 1, 9, 63, 553, 5725, 66637, 823543, 1, 1, 10, 72, 656, 7100, 87704, 1188544, 16777216, 1, 1, 11, 81, 761, 8525, 109863, 1577849, 24405761, 387420489, 1, 1, 12, 90, 868, 10000, 133120, 1991752, 32618512, 567108864, 10000000000

Offset

0,6

Links

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Formula

T(n, k) = ( (n - sqrt(k))^n + (n + sqrt(k))^n )/2.

Example

Rows begin:
1 1 4 27 256 ...
1 1 5 36 353 ...
1 1 6 45 452 ...
1 1 7 54 553 ...
1 1 8 63 656 ...

Maple

seq(seq( round(((k+sqrt(n-k))^k + (k-sqrt(n-k))^k)/2), k=0..n), n=0..10); # G. C. Greubel, Jan 11 2020

Mathematica

Table[If[n==0 && k==0, 1, Round[((k-Sqrt[n-k])^k + (k+Sqrt[n-k])^k)/2]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 11 2020 *)

Prog

(PARI) T(n, k) = round( ((k+sqrt(n-k))^n + (k-sqrt(n-k))^k)/2 ); \\ G. C. Greubel, Jan 11 2020
(Magma) [Round(((k+Sqrt(n-k))^k + (k-Sqrt(n-k))^k)/2): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 11 2020
(Sage) [[round(((k+sqrt(n-k))^k + (k-sqrt(n-k))^k)/2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 11 2020
(GAP) Flat(List([0..10], n-> List([0..n], k-> ((k+Sqrt(n-k))^k + (k-Sqrt(n-k))^k)/2 ))); # G. C. Greubel, Jan 11 2020

Crossrefs

Rows include A000312, A062024, A084063, A084064, A084065.

Diagonals include A084062, A084063, A084095.

Sequence in context: A334426 A209417 A267990 * A140262 A049702 A159040

Adjacent sequences: A084058 A084059 A084060 * A084062 A084063 A084064

Keyword

nonn,tabl,easy

Author

Paul Barry, May 11 2003

Status

approved