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A352757 a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1)/2 for n > 0.

%I A352757 #42 Feb 16 2025 08:34:03
%S A352757 6,151,1016,3753,10090,22331,43356,76621,126158,196575,293056,421361,
%T A352757 587826,799363,1063460,1388181,1782166,2254631,2815368,3474745,
%U A352757 4243706,5133771,6157036,7326173,8654430,10155631,11844176,13735041,15843778,18186515,20779956,23641381,26788646,30240183,34015000,38132681
%N A352757 a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1)/2 for n > 0.
%C A352757 Numbers C > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference C - D is odd, C - D = 2*n - 1, and the difference between the positive cubes C^3 - D^3 is equal to a centered cube number, C^3 - D^3 = B^3 + (B+1)^3, 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 = A352756(n), C = a(n) (this sequence), and D = A352758(n).
%C A352757 There are infinitely many such numbers a(n) = C in this sequence.
%H A352757 Vladimir Pletser, <a href="/A352757/b352757.txt">Table of n, a(n) for n = 1..10000</a>
%H A352757 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 A352757 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 A352757 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>
%H A352757 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A352757 a(n)^3 - A352758(n)^3 = A352756(n)^3 + (A352756(n) + 1)^3 = A352755(n) and a(n) - A352758(n) = 2*n - 1.
%F A352757 a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1)/2.
%F A352757 For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 576*(n - 2), with a(1) = 6, a(2) = 151 and a(3) = 1016.
%F A352757 a(n) can be extended for negative n such that a(-n) = a(n+1) - (2*n + 1).
%F A352757 G.f.: x*(6 + 121*x + 321*x^2 + 123*x^3 + 5*x^4)/(1 - x)^5. - _Stefano Spezia_, Apr 08 2022
%e A352757 a(1) = 6 belongs to the sequence as 6^3 - 5^3 = 3^3 + 4^3 = 91 and 6 - 5 = 1 = 2*1 - 1.
%e A352757 a(2) = 151 belongs to the sequence as 151^3 - 148^3 = 46^3 + 47^3 = 201159 and 151 - 148 = 3 = 2*2 - 1.
%e A352757 a(3) = (3(2*3 - 1)^2*((2*3 - 1)^2 + 2) + 2*3 + 1)/2 = 1016.
%e A352757 a(4) = 3*a(3) - 3*a(2) + a(1) + 576*2 = 3*1016 - 3*151 + 6 + 576*2 = 3753.
%p A352757 restart; for n to 20 do (1/2)*(3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1); end do;
%o A352757 (Python)
%o A352757 def A352757(n): return n*(n*(n*(24*n - 48) + 48) - 23) + 5 # _Chai Wah Wu_, Jul 10 2022
%Y A352757 Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225, A352755, A352756, A352758, A352759, A352760, A352761, A352762.
%K A352757 nonn,easy
%O A352757 1,1
%A A352757 _Vladimir Pletser_, Apr 02 2022