A034111 Fractional part of square root of a(n) starts with 5: first term of runs.

21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641, 1723, 1807, 1893, 1981, 2071, 2163, 2257, 2353, 2451, 2551, 2653

Offset

1,1

Comments

Agrees with sequence A002061 (n^2-n+1 = central polygonal numbers) starting with sixth term.

Links

Table of n, a(n) for n=1..48.

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

Formula

a(n) = n^2 + 7*n + 13. - Charles R Greathouse IV, Aug 10 2010

From Elmo R. Oliveira, Apr 21 2026: (Start)

G.f.: x*(21 - 32*x + 13*x^2)/(1 - x)^3.

E.g.f.: -13 + (13 + 8*x + x^2)*exp(x).

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

Mathematica

Array[#^2 + 7 # + 13 &, 48] (* Michael De Vlieger, Aug 05 2020 *)
(* Alternative: *)
LinearRecurrence[{3, -3, 1}, {21, 31, 43}, 50] (* Harvey P. Dale, Apr 29 2022 *)

Prog

(PARI) a(n)=n^2+7*n+13 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A002061, A034101.

Sequence in context: A168000 A031889 A034101 * A077687 A113762 A032585

Adjacent sequences: A034108 A034109 A034110 * A034112 A034113 A034114

Keyword

nonn,base,easy

Author

Patrick De Geest, Sep 15 1998

Extensions

Offset 1 from Alois P. Heinz, Aug 05 2020

Status

approved