A006323 4-dimensional analog of centered polygonal numbers.

1, 10, 41, 115, 260, 511, 910, 1506, 2355, 3520, 5071, 7085, 9646, 12845, 16780, 21556, 27285, 34086, 42085, 51415, 62216, 74635, 88826, 104950, 123175, 143676, 166635, 192241, 220690, 252185, 286936, 325160, 367081, 412930

Offset: 1

Author

Albert Rich (Albert_Rich(AT)msn.com)

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Programs

Magma

[7*Binomial(n+2,4)+Binomial(n+1,2): n in [1..40]]; // Vincenzo Librandi, Sep 06 2013

Mathematica

CoefficientList[Series[(-1 - x^2 - 5 x) / (x - 1)^5, {x, 0, 40}],x] (* Vincenzo Librandi, Sep 06 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,10,41,115,260},40] (* Harvey P. Dale, Dec 27 2022 *)

PARI

a(n) = 7*binomial(n + 2, 4) + binomial(n + 1, 2); \\ Michel Marcus, Sep 05 2013

Formula

a(n) = 7*C(n + 2, 4) + C(n + 1, 2).

G.f.: x*(-1-x^2-5*x)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009; adapted to the offset by Vincenzo Librandi, Sep 06 2013

Sum_{n>=1} 1/a(n) = 30 + 4*sqrt(21/5)*Pi*tan(sqrt(15/7)*Pi/2). - Amiram Eldar, Aug 23 2022