A193576 a(n) = T(n)^3 + n^3 where T(n) is a triangular number.

2, 35, 243, 1064, 3500, 9477, 22295, 47168, 91854, 167375, 288827, 476280, 755768, 1160369, 1731375, 2519552, 3586490, 5006043, 6865859, 9269000, 12335652, 16204925, 21036743, 27013824, 34343750, 43261127, 54029835, 66945368, 82337264, 100571625, 122053727

Offset: 1

Author

Vincenzo Librandi, Sep 08 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000217, A000578, A059827.

Programs

Magma

[(n^3*(n^3+3*n^2+3*n+9)/8): n in [1..40]];

Python

def A193576(n): return n**3*(n*(n*(n+3)+3)+9)>>3 # Chai Wah Wu, Jun 12 2025

Formula

a(n) = (n^3*(n^3+3*n^2+3*n+9)/8) = (1/8)*(n+3)*(n^2+3)*n^3.

From Chai Wah Wu, Jun 12 2025: (Start)

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.

G.f.: x*(x^5 - 28*x^3 - 40*x^2 - 21*x - 2)/(x - 1)^7. (End)

a(n) = A000578(n) + A059827(n). - Alois P. Heinz, Jun 12 2025

E.g.f.: exp(x)*x*(16 + 124*x + 192*x^2 + 98*x^3 + 18*x^4 + x^5)/8. - Stefano Spezia, Jun 13 2025