A080860 a(n) = 10*n^2 + 5*n + 1.

1, 16, 51, 106, 181, 276, 391, 526, 681, 856, 1051, 1266, 1501, 1756, 2031, 2326, 2641, 2976, 3331, 3706, 4101, 4516, 4951, 5406, 5881, 6376, 6891, 7426, 7981, 8556, 9151, 9766, 10401, 11056, 11731, 12426, 13141, 13876, 14631, 15406, 16201, 17016, 17851

Offset: 0

Author

Paul Barry, Feb 23 2003

Keywords

• nonn
• easy

Comments

The old definition of this sequence was "Generalized polygonal numbers".

Column T(n,5) of A080853.

Sequence found by reading the line from 1, in the direction 1, 16, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A080853, A085787.

Programs

Mathematica

Table[10n^2+5n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,16,51},50] (* Harvey P. Dale, Aug 05 2014 *)

PARI

a(n)=10*n^2+5*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = C(5,0) + C(5,1)*n + C(5,2)*n^2.

G.f.: (C(4,0) + (C(6,2) - 2)*x + C(4,2)*x^2)/(1-x)^3 = (1 + 13*x + 6*x^2)/(1-x)^3.

a(n) = 20*n + a(n-1) - 5 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

From Elmo R. Oliveira, Oct 31 2024: (Start)

E.g.f.: exp(x)*(1 + 15*x + 10*x^2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Sum_{n>=0} a(n)/n! = 26*e. - Davide Rotondo, Feb 15 2025

Extensions

Definition replaced with the closed form by Bruno Berselli, Jan 16 2013