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A355751 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 A352759(n).

%I A355751 #21 Feb 16 2025 08:34:03
%S A355751 4,121,562,1543,3280,5989,9886,15187,22108,30865,41674,54751,70312,
%T A355751 88573,109750,134059,161716,192937,227938,266935,310144,357781,410062,
%U A355751 467203,529420,596929,669946,748687,833368,924205,1021414,1125211,1235812,1353433
%N A355751 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 A352759(n).
%C A355751 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 = 3 (2n - 1) == 3 (mod 6), with C > D > B > 0, and A > 0, A = 27*t^3 * (27*t^6 + 1) /4 with t = 2*n-1, and where A = A352759(n), B = a(n) (this sequence), C = A355752(n) and D = A355753(n).
%C A355751 There are infinitely many such numbers a(n) = B in this sequence.
%C A355751 Subsequence of A352134.
%H A355751 Vladimir Pletser, <a href="/A355751/b355751.txt">Table of n, a(n) for n = 1..10000</a>
%H A355751 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 A355751 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 A355751 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>
%H A355751 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A355751 a(n)^3 + (a(n)+1)^3 = A355752(n)^3 - A355753(n)^3 and A355752(n) - A355753(n) = 3*(2*n - 1).
%F A355751 a(n) = (9*(2*n - 1)^3 - 1) / 2.
%F A355751 For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 216, with a(1) = 4, a(2) = 121 and a(3) = 562.
%F A355751 a(n) can be extended for negative n such that a(-n) = - a(n+1) - 1.
%F A355751 From _Jianing Song_, Jul 18 2022: (Start)
%F A355751 G.f.: x*(4+105*x+102*x^2+5*x^3)/(1-x)^4.
%F A355751 E.g.f.: 5 + exp(x)*(-5+9*x+54*x^2+36*x^3). (End)
%e A355751 a(1) = 4 is a term because 4^3 + 5^3 = 6^3 - 3^3 and 6 - 3 = 3 = 3*(2*1 - 1).
%e A355751 a(2) = 121 is a term because 121^3 + 122^3 = 369^3 - 360^3 and 369 - 360 = 9 = 3*(2*2 - 1).
%e A355751 a(3) = (9*(2*3 - 1)^3 - 1) / 2 = 562.
%e A355751 a(4) = 3*562 - 3*121 + 4 + 216 = 1543.
%p A355751 restart; for n to 20 do (1/2)*(9*(2*n - 1)^3-1); end do;
%Y A355751 Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A352759, A355752, A355753.
%K A355751 nonn,easy
%O A355751 1,1
%A A355751 _Vladimir Pletser_, Jul 15 2022