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a(n) = n*(5*n+1)/2.
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%I #68 Sep 10 2022 07:35:20

%S 0,3,11,24,42,65,93,126,164,207,255,308,366,429,497,570,648,731,819,

%T 912,1010,1113,1221,1334,1452,1575,1703,1836,1974,2117,2265,2418,2576,

%U 2739,2907,3080,3258,3441,3629

%N a(n) = n*(5*n+1)/2.

%C Sequence found by reading the line from 0, in the direction 0, 11, ..., and the line from 3, in the direction 3, 24, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. - _Omar E. Pol_, Sep 26 2011

%C For n >= 3, a(n) is the sum of the numbers appearing in the 3rd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - _Wesley Ivan Hurt_, May 17 2021

%H G. C. Greubel, <a href="/A005475/b005475.txt">Table of n, a(n) for n = 0..5000</a>

%H A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0409152">A product formula and combinatorial field theory</a>, arXiv:quant-ph/0409152, 2004.

%H Leo Tavares, <a href="/A005475/a005475.jpg">Illustration: Hexagonal Trapeziums</a>.

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

%F a(n) = A110449(n, 2) for n>1.

%F a(n) = a(n-1) + 5*n - 2 for n>0, a(0)=0. - _Vincenzo Librandi_, Nov 18 2010

%F a(n) = A130520(5*n+2). - _Philippe Deléham_, Mar 26 2013

%F a(n) = A202803(n)/2. - _Philippe Deléham_, Mar 27 2013

%F a(n) = A162147(n) - A162147(n-1). - _J. M. Bergot_, Jun 21 2013

%F a(n) = A000217(3*n) - A000217(2*n). - _Bruno Berselli_, Oct 13 2016

%F From _G. C. Greubel_, Aug 23 2017: (Start)

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

%F E.g.f.: (x/2)*(5*x+6)*exp(x). (End)

%F Sum_{n>=1} 1/a(n) = 10+2*gamma+2*Psi(1/5) = 0.57635... see A001620 and A200135. - _R. J. Mathar_, May 30 2022

%F Sum_{n>=1} 1/a(n) = 10 - sqrt(1+2/sqrt(5))*Pi - sqrt(5)*log(phi) - 5*log(5)/2, where phi is the golden ratio (A001622). - _Amiram Eldar_, Sep 10 2022

%p seq(binomial(5*n+1,2)/5, n=0..34); # _Zerinvary Lajos_, Jan 21 2007

%p a:=n->sum(2*n+j, j=1..n): seq(a(n), n=0..38); # _Zerinvary Lajos_, Apr 29 2007

%t Table[n (5 n + 1)/2, {n, 0, 40}] (* _Bruno Berselli_, Oct 13 2016 *)

%o (PARI) a(n)=n*(5*n+1)/2; \\ _Joerg Arndt_, Mar 27 2013

%Y Cf. A001622, A110449, A130520, A162147, A202803.

%Y Cf. similar sequences listed in A022289.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Incorrect comment deleted and minor errors corrected by _Johannes W. Meijer_, Feb 04 2010