OFFSET
0,2
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
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).
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
EXAMPLE
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 + ...
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 *)
PROG
(Magma) [n*(5*n+1):n in [0..50]]; // Vincenzo Librandi, Aug 11 2014
(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
(GAP) List([0..50], n-> n*(5*n+1)) # G. C. Greubel, Jul 04 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jeremy Gardiner, Dec 24 2011
STATUS
approved