A202803 a(n) = n*(5*n+1).

0, 6, 22, 48, 84, 130, 186, 252, 328, 414, 510, 616, 732, 858, 994, 1140, 1296, 1462, 1638, 1824, 2020, 2226, 2442, 2668, 2904, 3150, 3406, 3672, 3948, 4234, 4530, 4836, 5152, 5478, 5814, 6160, 6516, 6882, 7258, 7644, 8040, 8446, 8862, 9288, 9724, 10170

Offset: 0

Author

Jeremy Gardiner, Dec 24 2011

Keywords

• nonn
• easy

Comments

First bisection of A219190. - Bruno Berselli, Nov 15 2012

a(n)*Pi is the total length of 5 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A017341. The spiral length ratio rounded down [floor(L(n)/L(1))] is A032793. See illustration in links. - Kival Ngaokrajang, Dec 27 2013

Examples

			G.f. = 6*x + 22*x^2 + 48*x^3 + 84*x^4 + 130*x^5 +186*x^6 + 252*x^7 + 328*x^8 + ...
		

Links

• Jeremy Gardiner, Table of n, a(n) for n = 0..3000
• Kival Ngaokrajang, Illustration of 5 points circle center spiral.
• Leo Tavares, Illustration: Twin Trapezoids.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A033429, A168668, A126264, A135705, A124080.

Cf. sequences listed in A254963.

Cf. A001622, A005475.

Programs

GAP

List([0..50], n-> n*(5*n+1)); # G. C. Greubel, Jul 04 2019

Magma

[n*(5*n+1):n in [0..50]]; // Vincenzo Librandi, Aug 11 2014

Mathematica

CoefficientList[Series[2x(3+2x)/(1-x)^3, {x, 0, 50}] ,x] (* Vincenzo Librandi, Aug 11 2014 *)
Table[5*n^2+n, {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)

PARI

a(n)=n*(5*n+1) \\ Charles R Greathouse IV, Jun 17 2017

Sage

[n*(5*n+1) for n in (0..50)] # G. C. Greubel, Jul 04 2019

Formula

a(n) = 5*n^2 + n.

a(n) = A033429(n) + n. - Omar E. Pol, Dec 24 2011

G.f.: 2*x*(3+2*x)/(1-x)^3. - Philippe Deléham, Mar 27 2013

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 6, a(2) = 22. - Philippe Deléham, Mar 27 2013

a(n) = A131242(10n+5). - Philippe Deléham, Mar 27 2013

a(n) = 2*A005475(n). - Philippe Deléham, Mar 27 2013

a(n) = A168668(n) - n. - Philippe Deléham, Mar 27 2013

a(n) = (n+1)^3 - (1 + n + n*(n-1) + n*(n-1)*(n-2)). - Michael Somos, Aug 10 2014

E.g.f.: x*(6+5*x)*exp(x). - G. C. Greubel, Aug 22 2017

Sum_{n>=1} 1/a(n) = 5*(1-log(5)/4) - sqrt(1+2/sqrt(5))*Pi/2 -sqrt(5)*log(phi)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Jul 19 2022