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Total number of positive integers below 10^n requiring 19 positive biquadrates in their representation as sum of biquadrates.
20

%I #36 Aug 05 2024 18:20:24

%S 0,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,

%T 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,

%U 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7

%N Total number of positive integers below 10^n requiring 19 positive biquadrates in their representation as sum of biquadrates.

%C A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + a(n) = A002283(n).

%D J.-M. Deshouillers, K. Kawada, and T. D. Wooley, On sums of sixteen biquadrates, Mem. Soc. Math. Fr. 100 (2005), p. 120.

%H J.-M. Deshouillers, F. Hennecart and B. Landreau, <a href="http://www.math.ethz.ch/EMIS/journals/JTNB/2000-2/Dhl.ps">Waring's Problem for sixteen biquadrates - numerical results</a>, Journal de Théorie des Nombres de Bordeaux 12 (2000), pp. 411-422.

%H L. E. Dickson, <a href="http://www.ams.org/journals/bull/1933-39-10/S0002-9904-1933-05719-1/S0002-9904-1933-05719-1.pdf">Recent progress on Waring's theorem and its generalizations</a>, Bull. Amer. Math. Soc. 39:10 (1933), pp. 701-727.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem.</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F a(n) = 7 for n >= 3. - _Nathaniel Johnston_, May 09 2011

%F From _Elmo R. Oliveira_, Aug 05 2024: (Start)

%F G.f.: x^2*(1 + 6*x)/(1 - x).

%F E.g.f.: 7*(exp(x) - 1 - x) - 3*x^2. (End)

%t PadRight[{0, 1}, 100, 7] (* _Paolo Xausa_, Jul 30 2024 *)

%Y Cf. A010727, A046050.

%K nonn,easy

%O 1,3

%A _Martin Renner_, Feb 25 2011

%E a(5)-a(6) from _Lars Blomberg_, May 08 2011

%E Terms after a(6) from _Nathaniel Johnston_, May 09 2011