A357178 First differences of cubes of triangular numbers.

0, 1, 26, 189, 784, 2375, 5886, 12691, 24704, 44469, 75250, 121121, 187056, 279019, 404054, 570375, 787456, 1066121, 1418634, 1858789, 2402000, 3065391, 3867886, 4830299, 5975424, 7328125, 8915426, 10766601, 12913264, 15389459, 18231750, 21479311, 25174016, 29360529

Offset: 0

Author

Kelvin Voskuijl, Sep 16 2022

Keywords

• nonn
• easy
• changed

Comments

Row sums of centered hexagonal numbers A003215 treated as a regular triangle.

Links

• Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A059827 (cubes of triangular numbers).

Cf. A000578 (for squares) and A168364 (for fourth powers) of triangular numbers.

Cf. A000217 (triangular numbers), A003215.

Programs

Mathematica

a[n_] := (n^3 + 3*n^5)/4; Array[a, 35, 0] (* Amiram Eldar, Sep 18 2022 *)

PARI

a(n) = n^3*(3*n^2+1)/4 \\ Charles R Greathouse IV, Sep 19 2022

Formula

a(n) = (n^3 + 3*n^5)/4.

G.f.: x*(1 + 20*x + 48*x^2 + 20*x^3 + x^4)/(1 - x)^6. - Stefano Spezia, Sep 19 2022