# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a355752
Showing 1-1 of 1

%I A355752 #16 Oct 21 2022 21:08:29
%S A355752 6,369,2820,10815,29538,65901,128544,227835,375870,586473,875196,
%T A355752 1259319,1757850,2391525,3182808,4155891,5336694,6752865,8433780,
%U A355752 10410543,12715986,15384669,18452880,21958635,25941678,30443481,35507244,41177895,47502090,54528213,62306376,70888419,80327910,90680145,102002148,114352671,127792194
%N A355752 a(n) = 3*(2*n - 1)*( 3*(2*n - 1)^3 + 1) / 2.
%C A355752 Numbers C > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference (C - D) == 3 (mod 6), C - D = 3 (2n - 1) for 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 = 27*t^3 *(27*t^6+1)/4 with t = 2*n-1, and where A = A352759(n), B = A355751(n), C = a(n) (this sequence), and D = A355753(n).
%C A355752 There are infinitely many such numbers a(n) = C in this sequence.
%H A355752 Vladimir Pletser, <a href="/A355752/b355752.txt">Table of n, a(n) for n = 1..10000</a>
%H A355752 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 A355752 Vladimir Pletser, <a href="https://www.researchgate.net/publication/359706361">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 A355752 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>
%H A355752 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A355752 a(n)^3 - A355753(n)^3 = A355751(n)^3 + (A355751(n) + 1)^3 = A352759(n) and a(n) - A355753(n) = 3*(2*n - 1).
%F A355752 a(n) = 3*(2*n - 1)*( 3*(2*n - 1)^3 + 1) / 2.
%F A355752 For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1728*(n - 2), with a(1) = 6, a(2) = 369 and a(3) = 2820.
%F A355752 a(n) can be extended for negative n such that a(-n) = a(n+1) - 3*(2*n + 1).
%F A355752 G.f.: -3*x*(2+113*x+345*x^2+115*x^3+x^4) / (x-1)^5 . - _R. J. Mathar_, Aug 03 2022
%e A355752 a(1) = 6 belongs to the sequence as 6^3 - 3^3 = 4^3 + 5^3 = 189 and 6 - 3 = 3 = 3 (2*1 - 1).
%e A355752 a(2) = 369 belongs to the sequence as 369^3 - 360^3 = 121^3 + 122^3 = 3587409 and 369 - 360 = 9 = 3*(2*2 - 1).
%e A355752 a(3) = 3*(2*3 - 1)*( 3*(2*3 - 1)^3 + 1) / 2 = 2820.
%e A355752 a(4) = 3*a(3) - 3*a(2) + a(1) + 1728*2 = 3*2820 - 3*369 + 6 + 1728*2  = 10815.
%p A355752 restart; for n to 20 do (1/2)* 3*(2*n - 1)*(3*(2*n - 1)^3+1); end do;
%o A355752 (PARI) a(n)=3*(2*n-1)*(3*(2*n-1)^3+1)/2 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y A355752 Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A352759, A355751, A355753.
%K A355752 nonn,easy
%O A355752 1,1
%A A355752 _Vladimir Pletser_, Jul 15 2022

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE