1,1

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).

There are infinitely many such numbers a(n) = B in this sequence.

Subsequence of A352134.

Vladimir Pletser, Table of n, a(n) for n = 1..10000

A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.

Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.

Eric Weisstein's World of Mathematics, Centered Cube Number

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

a(n) = (9*(2*n - 1)^3 - 1) / 2.

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.

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

From Jianing Song, Jul 18 2022: (Start)

G.f.: x*(4+105*x+102*x^2+5*x^3)/(1-x)^4.

E.g.f.: 5 + exp(x)*(-5+9*x+54*x^2+36*x^3). (End)

a(1) = 4 is a term because 4^3 + 5^3 = 6^3 - 3^3 and 6 - 3 = 3 = 3*(2*1 - 1).

a(2) = 121 is a term because 121^3 + 122^3 = 369^3 - 360^3 and 369 - 360 = 9 = 3*(2*2 - 1).

a(3) = (9*(2*3 - 1)^3 - 1) / 2 = 562.

a(4) = 3*562 - 3*121 + 4 + 216 = 1543.

restart; for n to 20 do (1/2)*(9*(2*n - 1)^3-1); end do;

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A352759, A355752, A355753.

Sequence in context: A053881 A075114 A017186 * A098839 A227525 A071129

Adjacent sequences: A355748 A355749 A355750 * A355752 A355753 A355754

nonn,easy

Vladimir Pletser, Jul 15 2022

approved