A202803 - OEIS

%I #60 Dec 26 2022 05:01:28

%S 0,6,22,48,84,130,186,252,328,414,510,616,732,858,994,1140,1296,1462,

%T 1638,1824,2020,2226,2442,2668,2904,3150,3406,3672,3948,4234,4530,

%U 4836,5152,5478,5814,6160,6516,6882,7258,7644,8040,8446,8862,9288,9724,10170

%N a(n) = n*(5*n+1).

%C First bisection of A219190. - _Bruno Berselli_, Nov 15 2012

%C 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

%H Jeremy Gardiner, <a href="/A202803/b202803.txt">Table of n, a(n) for n = 0..3000</a>

%H Kival Ngaokrajang, <a href="/A202803/a202803.pdf">Illustration of 5 points circle center spiral</a>.

%H Leo Tavares, <a href="/A202803/a202803.jpg">Illustration: Twin Trapezoids</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F a(n) = A033429(n) + n. - _Omar E. Pol_, Dec 24 2011

%F G.f.: 2*x*(3+2*x)/(1-x)^3. - _Philippe Deléham_, Mar 27 2013

%F 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

%F a(n) = A131242(10n+5). - _Philippe Deléham_, Mar 27 2013

%F a(n) = 2*A005475(n). - _Philippe Deléham_, Mar 27 2013

%F a(n) = A168668(n) - n. - _Philippe Deléham_, Mar 27 2013

%F a(n) = (n+1)^3 - (1 + n + n*(n-1) + n*(n-1)*(n-2)). - _Michael Somos_, Aug 10 2014

%F E.g.f.: x*(6+5*x)*exp(x). - _G. C. Greubel_, Aug 22 2017

%F 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

%e 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 + ...

%t CoefficientList[Series[2x(3+2x)/(1-x)^3, {x, 0, 50}] ,x] (* _Vincenzo Librandi_, Aug 11 2014 *)

%t Table[5*n^2+n, {n,0,50}] (* _G. C. Greubel_, Jul 04 2019 *)

%o (Magma) [n*(5*n+1):n in [0..50]]; // _Vincenzo Librandi_, Aug 11 2014

%o (PARI) a(n)=n*(5*n+1) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Sage) [n*(5*n+1) for n in (0..50)] # _G. C. Greubel_, Jul 04 2019

%o (GAP) List([0..50], n-> n*(5*n+1)) # _G. C. Greubel_, Jul 04 2019

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

%Y Cf. sequences listed in A254963.

%Y Cf. A001622, A005475.

%K nonn,easy

%O 0,2

%A _Jeremy Gardiner_, Dec 24 2011