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A352755 Positive centered cube numbers that can be written as the difference of two positive cubes: a(n) = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1 and n > 0.

%I A352755 #50 Feb 16 2025 08:34:03
%S A352755 91,201159,15407765,295233841,2746367559,16448122691,73287987409,
%T A352755 264133278045,811598515091,2202365761759,5410166901741,12249942682409,
%U A352755 25914353312575,51755729480091,98389720844009,179211321358741,314429627203659,533744613620855,879807401606341,1412624924155809
%N A352755 Positive centered cube numbers that can be written as the difference of two positive cubes: a(n) = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1 and n > 0.
%C A352755 Numbers A > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 2n - 1, with C > D > B > 0, and A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = a(n) (this sequence), B = A352756(n), C = A352757(n) and D = A352758(n).
%C A352755 There are infinitely many such numbers a(n) = A in this sequence.
%C A352755 Subsequence of A005898, of A352133 and of A352220.
%H A352755 Vladimir Pletser, <a href="/A352755/b352755.txt">Table of n, a(n) for n = 1..10000</a>
%H A352755 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 A352755 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 A352755 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>
%H A352755 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A352755 a(n) = A352756(n)^3 + (A352756(n) + 1)^3 = A352757(n)^3 - A352758(n)^3 and A352757(n) - A352758(n) = 2n - 1.
%F A352755 a(n) = (2*n - 1)*(3*(2*n - 1)^2 + 4)*((2*n - 1)^2*(3*(2*n - 1)^2 + 4)^2 + 3)/4.
%F A352755 a(n) can be extended for negative n such that a(-n) = -a(n+1).
%e A352755 a(1) = 91 belongs to the sequence because 91 = 3^3 + 4^3 = 6^3 - 5^3 and 6 - 5 = 1 = 2*1 - 1.
%e A352755 a(2) = 201159 belongs to the sequence because 201159 = 46^3 + 47^3 = 151^3 - 148^3 and 151 - 148 = 3 = 2*2 - 1.
%e A352755 a(3) = (2*3 - 1)*(3*(2*3 - 1)^2 + 4)*((2*3 - 1)^2*(3*(2*3 - 1)^2 + 4)^2 + 3)/4 = 15407765.
%p A352755 restart; for n to 20 do (1/4)*(2*n-1)*(3*(2*n-1)^2+4)*((2*n-1)^2*(3*(2*n-1)^2+4)^2+3) end do;
%Y A352755 Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352221, A352222, A352223, A352224, A352225, A352756, A352757, A352758, A352759, A355751, A355752, A355753.
%K A352755 nonn,easy
%O A352755 1,1
%A A352755 _Vladimir Pletser_, Apr 02 2022