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One half of numbers representable in at least two different ways as sums of four nonvanishing cubes. See A259060 for these numbers and their representations.
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%I #22 Jan 27 2025 14:41:15

%S 3213,3950,4807,5796,6929,8218,9675,11312,13141,15174,17423,19900,

%T 22617,25586,28819,32328,36125,40222,44631,49364,54433,59850,65627,

%U 71776,78309,85238,92575,100332,108521,117154,126243,135800,145837,156366

%N One half of numbers representable in at least two different ways as sums of four nonvanishing cubes. See A259060 for these numbers and their representations.

%C See A259060. There may be other numbers with this property.

%D W. SierpiƄski, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970, Problem 227, p. 20 and p. 110.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (n+9)*(2*n^2 + 36*n + 357), n >= 0.

%F O.g.f.: (3213 - 8902*x + 8285*x^2 - 2584*x^3)/(1-x)^4.

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Vincenzo Librandi_, Aug 13 2015

%t CoefficientList[Series[(3213 - 8902 x + 8285 x^2 - 2584 x^3)/(1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 13 2015 *)

%o (Magma) [(n+9)*(2*n^2 + 36*n + 357): n in [0..50]]; // _Vincenzo Librandi_, Aug 13 2015

%o (Magma) I:=[3213,3950,4807,5796]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Aug 13 2015

%Y Cf. A259060.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Aug 12 2015