A137299 Square matrix read by antidiagonals: T(m,n) = m-th term in the continued fraction expansion of Pi^n.

3, 9, 7, 31, 1, 15, 97, 159, 6, 1, 306, 2, 3, 1, 292, 961, 50, 2, 7, 2, 1, 3020, 2, 1, 3, 1, 47, 1, 9488, 3, 1, 4, 1, 13, 1, 1, 29809, 1, 2, 1, 60, 16539, 2, 8, 2, 93648, 10, 1, 2, 3, 1, 1, 1, 1, 1, 294204, 21, 14, 7, 3, 9, 4, 6, 3, 1, 3, 924269, 55, 15, 1, 1, 2, 1, 23, 7, 1, 2, 1

Offset

1,1

Comments

The sequence was suggested by Leroy Quet.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)

J. S. Markovitch, Coincidence, data compression and Mach's concept of "economy of thought", APRI-PH-2004-12b, June 3 2004.

Example

The matrix limited to order 10 is given by matrix(10,10,m,n,contfrac(Pi^n)[m]):
[   3   9   31    97   306   961  3020  9488 29809 93648]
[   7   1  159     2    50     2     3     1    10    21]
[  15   6    3     2     1     1     2     1    14    15]
[   1   1    7     3     4     1     2     7     1     1]
[ 292   2    1     1    60     3     3     1     9     4]
[   1  47   13 16539     1     9     2     1     3     2]
[   1   1    2     1     4     1    10     3     1     1]
[   1   8    1     6    23     5     4     1     5     3]
[   2   1    3     7     1     1     1     1     8     2]
[   1   1    1     6     2     3     1     1    16     1]

Mathematica

A137299list[dmax_]:=With[{a=Array[ContinuedFraction[Pi^(dmax+1-#), #]&, dmax]}, Array[Diagonal[a, #]&, dmax, 1-dmax]]; A137299list[10] (* Generates 10 antidiagonals *) (* Paolo Xausa, Nov 14 2023 *)

Prog

(PARI) concat(vector(20, i, vector(i, j, contfrac(Pi^(i-j+1))[j])))
(PARI) T(m, n)=contfrac(Pi^n)[m]

Crossrefs

Cf. A001203, A001672, A138324.

Sequence in context: A220654 A302158 A337974 * A329238 A001226 A338402

Adjacent sequences: A137296 A137297 A137298 * A137300 A137301 A137302

Keyword

nonn,easy,tabl

Author

M. F. Hasler, Mar 14 2008

Status

approved