A135706 a(n) = n*(5*n-3).

0, 2, 14, 36, 68, 110, 162, 224, 296, 378, 470, 572, 684, 806, 938, 1080, 1232, 1394, 1566, 1748, 1940, 2142, 2354, 2576, 2808, 3050, 3302, 3564, 3836, 4118, 4410, 4712, 5024, 5346, 5678, 6020, 6372, 6734, 7106, 7488, 7880, 8282, 8694, 9116, 9548, 9990, 10442

Offset: 0

Author

N. J. A. Sloane, Mar 04 2008, Mar 05 2008

Keywords

• nonn
• easy

Comments

Twice heptagonal numbers. - Omar E. Pol, May 14 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000566, A001622, A131242.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=10). - Bruno Berselli, Jun 10 2013

Programs

GAP

List([0..50], n-> n*(5*n-3)); # G. C. Greubel, Jul 05 2019

Magma

[n*(5*n-3): n in [0..50]]; // G. C. Greubel, Jul 05 2019

Mathematica

LinearRecurrence[{3,-3,1}, {0,2,14}, 50] (* or *) Table[n*(5*n-3), {n, 0, 50}] (* G. C. Greubel, Oct 29 2016 *)

PARI

a(n)=n*(5*n-3) \\ Charles R Greathouse IV, Oct 07 2015

Sage

[n*(5*n-3) for n in (0..50)] # G. C. Greubel, Jul 05 2019

Formula

Binomial transform of [2, 12, 10, 0, 0, 0, ...]. - Gary W. Adamson, Mar 05 2008

a(n) = 2*A000566(n). - Omar E. Pol, May 14 2008

a(n) = a(n-1) + 10*n - 8 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010

a(n) = A131242(10n+1). - Philippe Deléham, Mar 27 2013

From G. C. Greubel, Oct 29 2016: (Start)

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

G.f.: 2*x*(1 + 4*x)/(1 - x)^3.

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

Sum_{n>=1} 1/a(n) = tan(Pi/10)*Pi/6 - sqrt(5)*log(phi)/6 + 5*log(5)/12, where phi is the golden ratio (A001622). - Amiram Eldar, Jul 19 2022