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%I #97 Feb 28 2024 05:32:22
%S 0,7,16,27,40,55,72,91,112,135,160,187,216,247,280,315,352,391,432,
%T 475,520,567,616,667,720,775,832,891,952,1015,1080,1147,1216,1287,
%U 1360,1435,1512,1591,1672,1755,1840,1927,2016,2107,2200,2295,2392
%N a(n) = n*(n + 6), also numbers j such that 9*(9 + j) is a perfect square.
%C Nonnegative X values of solutions to the equation X + (X + 3)^2 + (X + 6)^3 = Y^2. To prove that X = n^2 + 6n: Y^2 = X + (X + 3)^2 + (X + 6)^3 = X^3 + 19*X^2 + 115X + 225 = (X + 9)*(X^2 + 10X + 25) = (X + 9)*(X + 5)^2 it means: (X + 9) must be a perfect square, so X = k^2 - 9 with k>=3. we can put: k = n + 3, which gives: X = n^2 + 6n and Y = (n + 3)*(n^2 + 6n + 5). - _Mohamed Bouhamida_, Nov 12 2007
%C a(m) where m is a positive integer are the only positive integer values of t for which the Binet-de Moivre Formula of the recurrence b(n)=6*b(n-1)+t*b(n-2) with b(0)=0 and b(1)=1 has a root which is a square. In particular, sqrt(6^2+4*t) is an integer since 6^2+4*t=6^2+4*a(m)=(2*m+6)^2. Thus, the charcteristic roots are k1=6+m and k2=-m. - _Felix P. Muga II_, Mar 27 2014
%H G. C. Greubel, <a href="/A028560/b028560.txt">Table of n, a(n) for n = 0..1000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>.
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.
%H Leo Tavares, <a href="/A028560/a028560.jpg">Illustration: Diamond Pairs</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hydrogen_spectral_series">Hydrogen spectral series</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (n+3)^2 - 3^2 = n*(n+6).
%F G.f.: x*(7-5*x)/(1-x)^3.
%F a(n) = 2*n + a(n-1) + 5. - _Vincenzo Librandi_, Aug 05 2010
%F Sum_{n>=1} 1/a(n) = 49/120 = 0.4083333... - _R. J. Mathar_, Mar 22 2011
%F a(n) = A028884(n) - 1. - _Reinhard Zumkeller_, Apr 07 2013
%F E.g.f.: x*(x+7)*exp(x). - _G. C. Greubel_, Aug 19 2017
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 37/360. - _Amiram Eldar_, Nov 04 2020
%F a(n) = A056220(n+1) - A000290(n-1). - _Leo Tavares_, Sep 29 2022
%F From _Amiram Eldar_, Feb 05 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = -4*sqrt(10)*sin(sqrt(10)*Pi)/(3*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = 45*sqrt(2)*sin(2*sqrt(2)*Pi)/(7*Pi). (End)
%p A028560:=n->n*(n + 6); seq(A028560(n), n=0..100); # _Wesley Ivan Hurt_, Mar 27 2014
%t Table[n(n + 6), {n, 0, 65}]
%o (Haskell)
%o a028560 n = n * (n + 6) -- _Reinhard Zumkeller_, Apr 07 2013
%o (PARI) a(n)=n*(n+6) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y a(n-3), n>=4, third column (used for the Paschen series of the hydrogen atom) of triangle A120070.
%Y Cf. A005563.
%Y Cf. A056220, A000290.
%K nonn,easy
%O 0,2
%A _Patrick De Geest_
%E Edited by _Robert G. Wilson v_, Feb 06 2002
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