A166994 Triangle, read by rows, where T(n,k) = T(n,k-1)^2 - T(k-1,k-1)^2 for n>=k>1, with T(n,1) = n for n>=1.

1, 2, 3, 3, 8, 55, 4, 15, 216, 43631, 5, 24, 567, 318464, 99515655135, 6, 35, 1216, 1475631, 2175583184000, 4723258824886629604131775, 7, 48, 2295, 5264000, 27707792335839, 767711852760361479511965696

Offset

1,2

Links

Table of n, a(n) for n=1..27.

Formula

Main diagonal is A083869, which obeys an interesting recursion of nested radicals.

Example

Triangle begins:
1;
2, 3;
3, 8, 55;
4, 15, 216, 43631;
5, 24, 567, 318464, 99515655135;
6, 35, 1216, 1475631, 2175583184000, 4723258824886629604131775;
7, 48, 2295, 5264000, 27707792335839, 767711852760361479511965696, 589359179694820074404152604620573424809709490316113791; ...
ILLUSTRATE THE RECURRENCE.
For row 4, start with 4, then continue with the rule:
"obtain the next term in the row by squaring the current term and subtracting the square of the first term in the current column":
4^2 - 1^2 = 15; 15^2 - 3^2 = 216; 216^2 - 55^2 = 43631.
Likewise for row 5:
5^2 - 1^2 = 24; 24^2 - 3^2 = 567; 567^2 - 55^2 = 318464; 318464^2 - 43631^2 = 99515655135.
Continuing in this way generates all rows of this triangle.
ILLUSTRATE GENERATING METHOD USING NESTED RADICALS.
Let a(n) = A083869(n), then row n equals the resulting integers at each stage in the successive nested radicals:
sqrt(a(1)^2+sqrt(a(2)^2+sqrt(a(3)^2+(....+sqrt(a(n)^2)))...).
For example, the terms in row n=3 are:
3 = sqrt(1^2 + sqrt(3^2 + sqrt(55^2))),
8 = sqrt(3^2 + sqrt(55^2)),
55 = sqrt(55^2).
And the terms in row 4 are:
4 = sqrt(1^2 + sqrt(3^2 + sqrt(55^2 + sqrt(43631^2)))),
15 = sqrt(3^2 + sqrt(55^2 + sqrt(43631^2))),
216 = sqrt(55^2 + sqrt(43631^2)),
43631 = sqrt(43631^2).

Mathematica

A[n_, 1] := n; A[n_, k_] := A[n, k - 1]^2 - A[n - 1, k - 1]^2; Flatten[Table[A[n, k], {n, 10}, {k, n}]] (* G. C. Greubel, May 30 2016 *)

Prog

(PARI) T(n, k)=if(k==1, n, T(n, k-1)^2-T(k-1, k-1)^2)

Crossrefs

Cf. A083869.

Sequence in context: A297703 A263464 A267563 * A059083 A207626 A232324

Adjacent sequences: A166991 A166992 A166993 * A166995 A166996 A166997

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 18 2009

Status

approved