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%I A352756 #37 Jul 19 2022 11:51:23
%S A352756 3,46,197,528,1111,2018,3321,5092,7403,10326,13933,18296,23487,29578,
%T A352756 36641,44748,53971,64382,76053,89056,103463,119346,136777,155828,
%U A352756 176571,199078,223421,249672,277903,308186,340593,375196,412067,451278,492901,537008,583671,632962,684953,739716,797323,857846,921357
%N A352756 Positive numbers k such that the centered cube number k^3 + (k+1)^3 is equal to the difference of two positive cubes and to A352755(n).
%C A352756 Numbers B > 0 such that the centered cube number B^3 + (B+1)^3 is equal to the difference of two positive cubes, i.e., A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 2n - 1, with C > D > B > 0, and A > 0, A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = A352755(n), B = a(n) (this sequence), C = A352757(n) and D = A352758(n).
%C A352756 There are infinitely many such numbers a(n) = B in this sequence.
%C A352756 Subsequence of A352134 and of A352221.
%H A352756 Vladimir Pletser, <a href="/A352756/b352756.txt">Table of n, a(n) for n = 1..10000</a>
%H A352756 A. Grinstein, <a href="https://web.archive.org/web/20040320144821/http://zadok.org/mattandloraine/1729.html">Ramanujan and 1729</a>, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
%H A352756 Vladimir Pletser, <a href="https://www.researchgate.net/publication/359706361_EULER&#39;S_AND_THE_TAXI_CAB_RELATIONS_AND_OTHER_NUMBERS_THAT_CAN_BE_WRITTEN_TWICE_AS_SUMS_OF_TWO_CUBED_INTEGERS">Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers</a>, submitted. Preprint available on ResearchGate, 2022.
%H A352756 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>
%H A352756 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A352756 a(n)^3 + (a(n)+1)^3 = A352757(n)^3 - A352758(n)^3 and A352757(n) - A352758(n) = 2*n - 1.
%F A352756 a(n) = ((2*n - 1)*(3*(2*n - 1)^2 + 4) - 1)/2.
%F A352756 For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 72, with a(1) = 3, a(2) = 46 and a(3) = 197.
%F A352756 a(n) can be extended for negative n such that a(-n) = -a(n+1) - 1.
%F A352756 G.f.: x*(3 + 34*x + 31*x^2 + 4*x^3)/(1 - x)^4. - _Stefano Spezia_, Apr 08 2022
%e A352756 a(1) = 3 is a term because 3^3 + 4^3 = 6^3 - 5^3 and 6 - 5 = 1 = 2*1 - 1.
%e A352756 a(2) = 46 is a term because 46^3 + 47^3 = 151^3 - 148^3 and 151 - 148 = 3 = 2*2 - 1.
%e A352756 a(3) = ((2*3 - 1)*(3*(2*3 - 1)^2 + 4) - 1)/2 = 197.
%e A352756 a(4) = 3*197 - 3*46 + 3 + 72 = 528.
%p A352756 restart; for n to 20 do (1/2)* ((2*n - 1)*(3*(2*n - 1)^2 + 4) - 1);  end do;
%Y A352756 Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225, A352755, A352757, A352758, A352759, A352760, A352761, A352762.
%K A352756 nonn,easy
%O A352756 1,1
%A A352756 _Vladimir Pletser_, Apr 02 2022

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