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A193868 Even central polygonal numbers. 4
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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 A059622 A131475

Adjacent sequences:  A193865 A193866 A193867 * A193869 A193870 A193871

KEYWORD

nonn,easy

AUTHOR

Omar E. Pol, Aug 15 2011

STATUS

approved

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Last modified December 9 14:23 EST 2016. Contains 278971 sequences.