login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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 (list; table; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 26 08:29 EDT 2021. Contains 347664 sequences. (Running on oeis4.)