A193868 Even central polygonal numbers.

2, 4, 16, 22, 46, 56, 92, 106, 154, 172, 232, 254, 326, 352, 436, 466, 562, 596, 704, 742, 862, 904, 1036, 1082, 1226, 1276, 1432, 1486, 1654, 1712, 1892, 1954, 2146, 2212, 2416, 2486, 2702, 2776, 3004, 3082, 3322, 3404, 3656, 3742, 4006, 4096, 4372

Offset

1,1

Comments

Odd triangular numbers plus 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = A000124(A042963(n-1)).

a(n) = 1 + A014493(n).

a(n) = 2*A174114(n).

G.f.: -2*x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011

From Colin Barker, Jan 27 2016: (Start)

a(n) = (3+(-1)^n-2*(2+(-1)^n)*n+4*n^2)/2.

a(n) = 2*n^2-3*n+2 for n even.

a(n) = 2*n^2-n+1 for n odd.

(End)

Mathematica

Table[(3 + (-1)^n - 2 (2 + (-1)^n) n + 4 n^2)/2, {n, 50}] (* or *)
Select[PolygonalNumber@ Range@ 100, OddQ] + 1 (* Version 10.4, or *)
Table[If[EvenQ@ n, 2 n^2 - 3 n + 2, 2 n^2 - n + 1], {n, 50}] (* or *)
Rest@ CoefficientList[Series[-2 x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {2, 4, 16, 22, 46}, 50] (* Harvey P. Dale, Sep 13 2020 *)

Prog

(Magma) [1+((2*n-1)*(2*n-1-(-1)^n)/2): n in [1..50]]; // Vincenzo Librandi, Aug 18 2011
(PARI) a(n)=(2*n-1)*(2*n-1-(-1)^n)/2+1 \\ Charles R Greathouse IV, Jun 11 2015
(PARI) Vec(2*x*(1+x+4*x^2+x^3+x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 27 2016

Crossrefs

Cf. A000124, A193867.

Sequence in context: A192150 A049441 A253143 * A256788 A324211 A059622

Adjacent sequences: A193865 A193866 A193867 * A193869 A193870 A193871

Keyword

nonn,easy

Author

Omar E. Pol, Aug 15 2011

Status

approved