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A144927 Numbers n such that there exists an integer k with (n+7)^3-n^3=k^2. 4

%I

%S 7,1162,128191,14100226,1550897047,170584575322,18762752388751,

%T 2063732178187666,226991776848254887,24967031721129850282,

%U 2746146497547435276511,302051147698496750566306,33222880100337095127017527,3654214759889381967221362042

%N Numbers n such that there exists an integer k with (n+7)^3-n^3=k^2.

%H Colin Barker, <a href="/A144927/b144927.txt">Table of n, a(n) for n = 1..450</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-111,1).

%F a(n+2) = 110*a(n+1) - a(n) + 378.

%F a(n) = -(7/2)+(21/4)*{[55+12*sqrt(21)]^n+[55-12*sqrt(21))^n}+(7/6)*sqrt(21)*{[55+12*sqrt(21)]^n-[55-12*sqrt(21)]^n}, with n>=0. [_Paolo P. Lava_, Nov 25 2008]

%F G.f.: 7*x*(-1-55*x+2*x^2) / ( (x-1)*(x^2-110*x+1) ). - _R. J. Mathar_, Nov 27 2011

%F a(n) = 7*A144929(n). - _R. J. Mathar_, Nov 27 2011

%e a(1)=7 because 14^3-7^3=49^2.

%t Last /@ Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && (n + 7)^3 - n^3 == k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 1, 14}] (* or *)

%t Rest@ CoefficientList[Series[7 x (-1 - 55 x + 2 x^2)/((x - 1) (x^2 - 110 x + 1)), {x, 0, 14}], x] (* _Michael De Vlieger_, Jul 14 2016 *)

%o (PARI) Vec(7*x*(-1-55*x+2*x^2)/((x-1)*(x^2-110*x+1)) + O(x^20)) \\ _Colin Barker_, Jul 14 2016

%Y Cf. A144928, A144930, A144929.

%K easy,nonn

%O 1,1

%A _Richard Choulet_, Sep 25 2008

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Last modified October 21 12:12 EDT 2019. Contains 328299 sequences. (Running on oeis4.)