A028994 Even 10-gonal (or decagonal) numbers.

0, 10, 52, 126, 232, 370, 540, 742, 976, 1242, 1540, 1870, 2232, 2626, 3052, 3510, 4000, 4522, 5076, 5662, 6280, 6930, 7612, 8326, 9072, 9850, 10660, 11502, 12376, 13282, 14220, 15190, 16192, 17226, 18292, 19390, 20520, 21682, 22876, 24102, 25360, 26650, 27972

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

a(n) (for n >= 1) is also the Wiener index of the windmill graph D(5, n). The windmill graph D(m, n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e. a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. The Wiener index of D(m, n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(3, n), D(4, n), and D(6, n) see A033991, A152743, and A180577, respectively. - Emeric Deutsch, Sep 21 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Decagonal Number.
• Eric Weisstein's World of Mathematics, Windmill Graph. - _Emeric Deutsch_, Sep 21 2010
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001107, A028993, A033991, A139273, A152743, A180577.

Programs

Magma

[2*n*(8*n - 3): n in [0..60]]; // Vincenzo Librandi, Oct 18 2013

Mathematica

CoefficientList[Series[-2 x (11 x + 5)/(x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 18 2013 *)
LinearRecurrence[{3, -3, 1}, {0, 10, 52}, 40] (* Harvey P. Dale, Dec 10 2014 *)
Table[16n^2 - 6n, {n, 0, 49}] (* Alonso del Arte, Jan 24 2017 *)

PARI

a(n)=2*n*(8*n-3) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 2*n*(8*n - 3). - Omar E. Pol, Aug 19 2011

G.f.: -2*x*(11*x+5)/(x-1)^3. - Colin Barker, Nov 18 2012

Sum_{n>=1} 1/a(n) = (8*log(2) - (sqrt(2)-1)*Pi - 2*sqrt(2)*log(1+sqrt(2)))/12. - Amiram Eldar, Feb 27 2022

From Elmo R. Oliveira, Oct 27 2024: (Start)

E.g.f.: 2*x*(5 + 8*x)*exp(x).

a(n) = 2*A139273(n) = A001107(2*n).

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