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%I #99 Feb 21 2023 02:15:09
%S 0,5,15,30,50,75,105,140,180,225,275,330,390,455,525,600,680,765,855,
%T 950,1050,1155,1265,1380,1500,1625,1755,1890,2030,2175,2325,2480,2640,
%U 2805,2975,3150,3330,3515,3705,3900,4100,4305,4515,4730,4950,5175,5405,5640
%N 5 times triangular numbers: a(n) = 5*n*(n+1)/2.
%C Sequence found by reading the line from 0, in the direction 0, 5, ... and the same line from 0, in the direction 0, 15, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Axis perpendicular to A195142 in the same spiral. - _Omar E. Pol_, Sep 18 2011
%C Bisection of A195014. Sequence found by reading the line from 0, in the direction 0, 5, ..., and the same line from 0, in the direction 0, 15, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is the main diagonal of the spiral. - _Omar E. Pol_, Sep 25 2011
%C a(n) = the Wiener index of the graph obtained by applying Mycielski's construction to the complete graph K(n) (n>=2). - _Emeric Deutsch_, Aug 29 2013
%C Sum of the numbers from 2*n to 3*n for n=0,1,2,... - _Wesley Ivan Hurt_, Nov 27 2015
%C Numbers k such that the concatenation k625 is a square, where also 625 is a square. - _Bruno Berselli_, Nov 07 2018
%C From _Paul Curtz_, Nov 29 2019: (Start)
%C Main column of the pentagonal spiral for n (A001477):
%C 50
%C 49 30 31
%C 48 29 15 16 32
%C 47 28 14 5 6 17 33
%C 46 27 13 4 0 1 7 18 34
%C 45 26 12 3 2 8 19 35
%C 44 25 11 10 9 20 36
%C 43 24 23 22 21 37
%C 42 41 40 39 38
%C (End)
%D D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.
%H Ivan Panchenko, <a href="/A028895/b028895.txt">Table of n, a(n) for n = 0..1000</a>
%H Rangaswami Balakrishnan and S. Francis Raj, <a href="http://dx.doi.org/10.7151/dmgt.1509">The Wiener number of powers of the Mycielskian</a>, Discussiones Math. Graph Theory, Vol. 30, No. 3 (2010), pp. 489-498 (see Theorem 2.1).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: 5*x/(1-x)^3.
%F a(n) = 5*n*(n+1)/2 = 5*A000217(n).
%F a(n+1) = 5*n+a(n). - _Vincenzo Librandi_, Aug 05 2010
%F a(n) = A005891(n) - 1. - _Omar E. Pol_, Oct 03 2011
%F a(n) = A130520(5n+4). - _Philippe Deléham_, Mar 26 2013
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - _Wesley Ivan Hurt_, Nov 27 2015
%F a(n) = Sum_{i=0..n} A001068(4i). - _Wesley Ivan Hurt_, May 06 2016
%F E.g.f.: 5*x*(2 + x)*exp(x)/2. - _Ilya Gutkovskiy_, May 06 2016
%F a(n) = A055998(3*n) - A055998(2*n). - _Bruno Berselli_, Sep 23 2016
%F From _Amiram Eldar_, Feb 26 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 2/5.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2/5)*(2*log(2) - 1). (End)
%F Product_{n>=1} (1 - 1/a(n)) = -(5/(2*Pi))*cos(sqrt(13/5)*Pi/2). - _Amiram Eldar_, Feb 21 2023
%p [seq(5*binomial(n,2), n=1..45)]; # _Zerinvary Lajos_, Nov 24 2006
%t Table[Sum[i + 2*n - 1, {i, 2, n}], {n, 45}] (* _Zerinvary Lajos_, Jul 11 2009 *)
%t Table[5 n (n + 1)/2, {n, 0, 50}] (* _Bruno Berselli_, Sep 23 2016 *)
%o (Magma) [5*n*(n+1)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Jun 09 2014
%o (PARI) a(n)=5*n*(n+1)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000217, A001068, A005891, A028896, A046092, A055998, A085787, A130520, A195013, A195014, A195142.
%Y Cf. index to numbers of the form n*(d*n+10-d)/2 in A140090.
%Y Cf. A000566, A005475, A005476, A033583, A085787, A147875, A192136, A326725 (all in the spiral).
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org), Dec 11 1999
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