A124080 10 times triangular numbers: a(n) = 5*n*(n + 1).

0, 10, 30, 60, 100, 150, 210, 280, 360, 450, 550, 660, 780, 910, 1050, 1200, 1360, 1530, 1710, 1900, 2100, 2310, 2530, 2760, 3000, 3250, 3510, 3780, 4060, 4350, 4650, 4960, 5280, 5610, 5950, 6300, 6660, 7030, 7410, 7800, 8200, 8610, 9030, 9460, 9900, 10350

Offset: 0

Author

Zerinvary Lajos, Nov 24 2006

Keywords

• easy
• nonn

Comments

If Y is a 5-subset of an n-set X then, for n >= 5, a(n-4) is equal to the number of 5-subsets of X having exactly three elements in common with Y. Y is a 5-subset of an n-set X then, for n >= 6, a(n-6) is the number of (n-5)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007

Also sequence found by reading the line from 0, in the direction 0, 10, ... and the same line from 0, in the direction 0, 30, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Axis perpendicular to A195148 in the same spiral. - Omar E. Pol, Sep 18 2011

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A028895, A046092, A045943, A002378, A028896, A024966, A033996, A027468, A049598, A000217.

Programs

Magma

[ 5*n*(n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

Maple

[seq(10*binomial(n,2),n=1..51)];
seq(n*(n+1)*5, n=0..39); # Zerinvary Lajos, Mar 06 2007

Mathematica

10*Accumulate[Range[0,50]] (* or *) LinearRecurrence[{3,-3,1},{0,10,30},50] (* Harvey P. Dale, Jul 21 2011 *)

PARI

a(n)=5*n*(n+1) \\ Charles R Greathouse IV, Sep 28 2015

Formula

a(n) = 10*C(n,2), n >= 1.

a(n) = A049598(n) - A002378(n). - Zerinvary Lajos, Mar 06 2007

a(n) = 5*n*(n + 1), n >= 0. - Zerinvary Lajos, Mar 06 2007

a(n) = 5*n^2 + 5*n = 10*A000217(n) = 5*A002378(n) = 2*A028895(n). - Omar E. Pol, Dec 12 2008

a(n) = 10*n + a(n-1) (with a(0) = 0). - Vincenzo Librandi, Nov 12 2009

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 10, a(2) = 30. - Harvey P. Dale, Jul 21 2011

a(n) = A062786(n+1) - 1. - Omar E. Pol, Oct 03 2011

a(n) = A131242(10*n+9). - Philippe Deléham, Mar 27 2013

From G. C. Greubel, Aug 22 2017: (Start)

G.f.: 10*x/(1 - x)^3.

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

From Amiram Eldar, Sep 04 2022: (Start)

Sum_{n>=1} 1/a(n) = 1/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2)-1)/5. (End)

From Amiram Eldar, Feb 21 2023: (Start)

Product_{n>=1} (1 - 1/a(n)) = -(5/Pi)*cos(3*Pi/(2*sqrt(5))).

Product_{n>=1} (1 + 1/a(n)) = (5/Pi)*cos(Pi/(2*sqrt(5))). (End)