A137299 - OEIS

%I #19 Nov 14 2023 17:04:58

%S 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,

%T 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,

%U 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

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

%C The sequence was suggested by _Leroy Quet_.

%H Paolo Xausa, <a href="/A137299/b137299.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened)

%H J. S. Markovitch, <a href="https://web-archive.southampton.ac.uk/cogprints.org/3667/">Coincidence, data compression and Mach's concept of "economy of thought"</a>, APRI-PH-2004-12b, June 3 2004.

%e The matrix limited to order 10 is given by matrix(10,10,m,n,contfrac(Pi^n)[m]):

%e [   3   9   31    97   306   961  3020  9488 29809 93648]

%e [   7   1  159     2    50     2     3     1    10    21]

%e [  15   6    3     2     1     1     2     1    14    15]

%e [   1   1    7     3     4     1     2     7     1     1]

%e [ 292   2    1     1    60     3     3     1     9     4]

%e [   1  47   13 16539     1     9     2     1     3     2]

%e [   1   1    2     1     4     1    10     3     1     1]

%e [   1   8    1     6    23     5     4     1     5     3]

%e [   2   1    3     7     1     1     1     1     8     2]

%e [   1   1    1     6     2     3     1     1    16     1]

%t 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 *)

%o (PARI) concat(vector(20,i,vector(i,j,contfrac(Pi^(i-j+1))[j])))

%o (PARI) T(m,n)=contfrac(Pi^n)[m]

%Y Cf. A001203, A001672, A138324.

%K nonn,easy,tabl

%O 1,1

%A _M. F. Hasler_, Mar 14 2008