A352758 - OEIS

%I #43 Feb 16 2025 08:34:03

%S 5,148,1011,3746,10081,22320,43343,76606,126141,196556,293035,421338,

%T 587801,799336,1063431,1388150,1782133,2254596,2815331,3474706,

%U 4243665,5133728,6156991,7326126,8654381,10155580,11844123,13734986,15843721,18186456,20779895,23641318,26788581,30240116,34014931,38132610

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

%C Numbers D > 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 of the positive cubes C^3 - D^3 is equal to centered cube numbers, 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 = A352757(n), and D = a(n) (this sequence).

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

%C Subsequence of A352136 and of A352223.

%H Vladimir Pletser, <a href="/A352758/b352758.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="https://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F A352757(n)^3 - a(n)^3 = A352756(n)^3 + (A352756(n) + 1)^3 = A352755(n) and A352757(n) - a(n) = 2*n - 1.

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

%F For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 576*(n - 2), with a(1) = 5, a(2) = 148 and a(3) = 1011.

%F a(n) can be extended for negative n such that a(-n) = a(n+1) + (2n + 1).

%F G.f.: x*(5 + 123*x + 321*x^2 + 121*x^3 + 6*x^4)/(1 - x)^5. - _Stefano Spezia_, Apr 08 2022

%e a(1) = 5 belongs to the sequence as 6^3 - 5^3 = 3^3 + 4^3 = 91 and 6 - 5 = 1 = 2*1 - 1.

%e a(2) = 148 belongs to the sequence as 151^3 - 148^3 = 46^3 + 47^3 = 201159 and 151 - 148 = 3 = 2*2 - 1.

%e a(3) = (3*(2*3 - 1)^2*((2*3 - 1)^2 + 2) - 2*3 + 3)/2 = 1011.

%e a(4) = 3*a(3) - 3*a(2) + a(1) + 576*2 = 3*1011 - 3*148 + 5 + 576*2  = 3746.

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

%o (Python)

%o def A352758(n): return n*(n*(n*(24*n - 48) + 48) - 25) + 6 # _Chai Wah Wu_, Jul 11 2022

%Y Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352224, A352225, A352755, A352756, A352757, A352759, A355751, A355752, A355753.

%K nonn,easy

%O 1,1

%A _Vladimir Pletser_, Apr 02 2022