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%I #62 Feb 18 2024 02:00:49
%S 0,3,13,30,54,85,123,168,220,279,345,418,498,585,679,780,888,1003,
%T 1125,1254,1390,1533,1683,1840,2004,2175,2353,2538,2730,2929,3135,
%U 3348,3568,3795,4029,4270,4518,4773,5035,5304,5580,5863,6153,6450,6754,7065,7383
%N a(n) = n*(7*n - 1)/2.
%C Sequence found by reading the line from 0, in the direction 0, 13, ..., and the parallel line from 3, in the direction 3, 30, ..., in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - _Omar E. Pol_, Sep 09 2011
%H G. C. Greubel, <a href="/A022264/b022264.txt">Table of n, a(n) for n = 0..1000</a>
%H Leo Tavares, <a href="/A022264/a022264.jpg">Illustration: Crysta-gons</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = C(7*n,2)/7, n >= 0. - _Zerinvary Lajos_, Jan 02 2007
%F a(n) = A049450(n) + A000217(n). - _Reinhard Zumkeller_, Oct 09 2008
%F a(n) = 7*n + a(n-1) - 4 for n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 04 2010
%F a(n) = (2*n)^2 - n*(n+1)/2 = A016742(n) - A000217(n). - _Philippe Deléham_, Mar 08 2013
%F a(n) = A174738(7*n+2). - _Philippe Deléham_, Mar 26 2013
%F G.f.: x*(3 + 4*x)/(1 - x)^3. - _R. J. Mathar_, Aug 04 2016
%F a(n) = A000217(4*n-1) - A000217(3*n-1). - _Bruno Berselli_, Oct 17 2016
%F a(n) = (1/5) * Sum_{i=n..(6*n-1)} i. - _Wesley Ivan Hurt_, Dec 04 2016
%F E.g.f.: (1/2)*x*(7*x + 6)*exp(x). - _G. C. Greubel_, Aug 19 2017
%F a(n) = A005449(n) + A000384(n). See Crysta-gons illustration. - _Leo Tavares_, Nov 21 2021
%p [seq(binomial(7*n,2)/7, n=0..37)]; # _Zerinvary Lajos_, Jan 02 2007
%t Table[n (7*n - 1)/2, {n, 0, 40}] (* _Zerinvary Lajos_, Jul 10 2009 *)
%o (PARI) a(n)=n*(7*n-1)/2 \\ _Charles R Greathouse IV_, Mar 08 2013
%o (Magma) [n*(7*n-1)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 04 2016
%Y Cf. A000217, A022265.
%Y Cf. sequences listed in A254963.
%Y Cf. similar sequences listed in A022288.
%Y Cf. A016742, A049450, A174738, A195019, A195020.
%Y Cf. A005449, A000384.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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