%I #39 Jul 16 2022 00:55:21

%S 189,3587409,355957875,7354447191,70607389041,429735975669,

%T 1932670025559,7006302268875,21612640524741,58809832966521,

%U 144757538551899,328260072633759,695228576765625,1389765141771741,2643927354266751,4818621138983379,8458493032498509,14364150148238625,23685527077994691,38040743821584231

%N Centered cube numbers that are the difference of two positive cubes; a(n) = 27*t^3*(27*t^6 + 1)/4 with t = 2*n-1.

%C Numbers A > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 3*(2*n - 1) == 3 (mod 6), with (for n > 1) C > D > B > 0, and A = as 27*t^3*(27*t^6 + 1)/4 with t = 2*n-1, and where A = a(n) (this sequence), B = A355751(n), C = A355752(n) and D = A355753(n).

%C There are infinitely many such numbers a(n) = A in this sequence.

%C Subsequence of A005898, and of A352133.

%H Vladimir Pletser, <a href="/A352759/b352759.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>

%H <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 a(n) = A355751(n)^3 + (A355751(n) + 1)^3 = A355752(n)^3 - A355753(n)^3 and A355752(n) - A355753(n) = 3*(2*n - 1).

%F a(n) = 27*(2*n - 1)^3*(27*(2*n - 1)^6 + 1)/4.

%F a(n) can be extended for negative n such that a(-n) = -a(n+1).

%e a(1) = 189 belongs to the sequence because 189 = 4^3 + 5^3 = 6^3 - 3^3 and 6 - 3 = 3 = 3*(2*1 - 1).

%e a(2) = 3587409 belongs to the sequence because 3587409 = 121^3 + 122^3 = 369^3 - 360^3 and 369 - 360 = 9 = 3*(2*2 - 1).

%e a(3) = 27*(2*3 - 1)^3*(27*(2*3 - 1)^6 + 1)/4 = 355957875.

%p restart; for n from 1 to 20 do 27*(2*n-1)^3*(27*(2*n-1)^6+1)*(1/4); end do;

%Y Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352221, A352222, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A355751, A355752, A355753.

%K nonn,easy

%O 1,1

%A _Vladimir Pletser_, Apr 02 2022