A071300 - OEIS

%I #18 Jan 12 2024 14:09:09

%S 1,1,2,9,65,82,1111,12707,44127,270757,3324143,773311279,583426241,

%T 51327727127,1207458414241,251022006941731,1784247347470303,

%U 542442461832071,59337844204584969481,172785053530529793211

%N Numerator of b(n) where b(n+1) = Sum_{k=0..n} b'((n^2-k^2)/n), b(0) = b(1) = 1, and b'(x) = b(x) if x is an integer and is linearly interpolated otherwise.

%H Robert Israel, <a href="/A071300/b071300.txt">Table of n, a(n) for n = 0..350</a>

%e b(4) = b'(9/3) + b'(8/3) + b'(5/3) + b'(0/3) = b(3) + ((1/3)*b(2) + (2/3)*b(3)) + ((1/3)*b(1) + (2/3)*b(2)) + b(0) = 65/6.

%p bp:= proc(x) local t; option remember; if x::integer then b(x)

%p       else t:= frac(x); t*b(ceil(x))+(1-t)*b(floor(x))

%p   fi

%p end proc:

%p b:= proc(m) local k; option remember; add(bp(((m-1)^2-k^2)/(m-1)),k=0..m-1); end proc:

%p b(0):= 1: b(1):= 1:

%p map(numer@b, [$0..20]); # _Robert Israel_, May 20 2019

%t bp[x_] := bp[x] = Module[{t},If[IntegerQ[x], b[x],

%t     t = FractionalPart[x]; t*b[Ceiling[x]] + (1-t)*b[Floor[x]]]];

%t b[m_] := b[m] = Sum[bp[((m-1)^2 - k^2)/(m-1)], {k, 0, m-1}];

%t b[0] = 1; b[1] = 1;

%t Map[Numerator@b@#&, Range[0, 20]] (* _Jean-François Alcover_, Jan 12 2024, after _Robert Israel_ *)

%Y Cf. A071298, A071299, A071301.

%K nonn,easy,frac

%O 0,3

%A Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 10 2002

%E More terms from _David Wasserman_

%E Edited by _N. J. A. Sloane_, May 20 2019