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

6, 369, 2820, 10815, 29538, 65901, 128544, 227835, 375870, 586473, 875196, 1259319, 1757850, 2391525, 3182808, 4155891, 5336694, 6752865, 8433780, 10410543, 12715986, 15384669, 18452880, 21958635, 25941678, 30443481, 35507244, 41177895, 47502090, 54528213, 62306376, 70888419, 80327910, 90680145, 102002148, 114352671, 127792194

Offset

1,1

Comments

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).

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

Links

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 (5,-10,10,-5,1).

Formula

a(n)^3 - A355753(n)^3 = A355751(n)^3 + (A355751(n) + 1)^3 = A352759(n) and a(n) - A355753(n) = 3*(2*n - 1).

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

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.

a(n) can be extended for negative n such that a(-n) = a(n+1) - 3*(2*n + 1).

G.f.: -3*x*(2+113*x+345*x^2+115*x^3+x^4) / (x-1)^5 . - R. J. Mathar, Aug 03 2022

Example

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).
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).
a(3) = 3*(2*3 - 1)*( 3*(2*3 - 1)^3 + 1) / 2 = 2820.
a(4) = 3*a(3) - 3*a(2) + a(1) + 1728*2 = 3*2820 - 3*369 + 6 + 1728*2  = 10815.

Maple

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

Prog

(PARI) a(n)=3*(2*n-1)*(3*(2*n-1)^3+1)/2 \\ Charles R Greathouse IV, Oct 21 2022

Crossrefs

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225, A352755, A352756, A352757, A352758, A352759, A355751, A355753.

Sequence in context: A188954 A332136 A248384 * A099595 A158041 A233212

Adjacent sequences: A355749 A355750 A355751 * A355753 A355754 A355755

Keyword

nonn,easy

Author

Vladimir Pletser, Jul 15 2022

Status

approved