A349682 a(n) = A000292(6*n + 1) where A000292 are the tetrahedral numbers.

1, 84, 455, 1330, 2925, 5456, 9139, 14190, 20825, 29260, 39711, 52394, 67525, 85320, 105995, 129766, 156849, 187460, 221815, 260130, 302621, 349504, 400995, 457310, 518665, 585276, 657359, 735130, 818805, 908600, 1004731, 1107414, 1216865, 1333300, 1456935, 1587986

Offset

0,2

Links

Table of n, a(n) for n=0..35.

Euclid of Alexandria, Elements, VII Def. 17, p. 194, 300 BCE.

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

Formula

a(n) = 1 + 11*n + 36*n^2 + 36*n^3 = (1 + 2*n)*(1 + 3*n)*(1 + 6*n).

G.f.: (1 + 80*x + 125*x^2 + 10*x^3)/(1 - x)^4. - Stefano Spezia, Nov 29 2021

From Elmo R. Oliveira, Aug 22 2025: (Start)

E.g.f.: exp(x)*(1 + 83*x + 144*x^2 + 36*x^3).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

From Amiram Eldar, Aug 31 2025: (Start)

Sum_{n>=0} 1/a(n) = Pi/(4*sqrt(3)) + 2*log(2) - 3*log(3)/4.

Sum_{n>=0} (-1)^n/a(n) = (3/4 - 1/sqrt(3))*Pi + sqrt(3)*log(2 + sqrt(3))/2 - log(2). (End)

Mathematica

nterms=50; Table[36n^3+36n^2+11n+1, {n, 0, nterms-1}] (* Paolo Xausa, Nov 25 2021 *)

Prog

(PARI) a(n) = subst(m*(m+1)*(m+2)/6, 'm, 6*n+1); \\ Michel Marcus, Dec 16 2021
(Python)
def A349682(n): return n*(n*(36*n + 36) + 11) + 1 # Chai Wah Wu, Dec 27 2021

Crossrefs

Cf. A000292, A000447, A002492, A016921.

Sequence in context: A251300 A251293 A251254 * A064198 A329935 A233055

Adjacent sequences: A349679 A349680 A349681 * A349683 A349684 A349685

Keyword

easy,nonn

Author

Ralf Steiner, Nov 25 2021

Status

approved