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 A028560 a(n) = n*(n + 6), also numbers a(n) such that 9*(9 + a(n)) is a perfect square. 34
 0, 7, 16, 27, 40, 55, 72, 91, 112, 135, 160, 187, 216, 247, 280, 315, 352, 391, 432, 475, 520, 567, 616, 667, 720, 775, 832, 891, 952, 1015, 1080, 1147, 1216, 1287, 1360, 1435, 1512, 1591, 1672, 1755, 1840, 1927, 2016, 2107, 2200, 2295, 2392 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Sequence allows us to find X values of 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 (bhmd95(AT)yahoo.fr), Nov 12 2007 Number of units of a(n) belongs to a periodic sequence: 0, 7, 6, 7, 0, 5, 2, 1, 2, 5. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009 a(n) = A028884(n) - 1. - Reinhard Zumkeller, Apr 07 2013 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 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 P. De Geest, Palindromic Quasipronics of the form n(n+x) M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013. F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate. Wikipedia, Hydrogen spectral series Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = (n+3)^2 - 3^2 = n*(n+6). G.f.: x*(7-5*x)/(1-x)^3. a(n) = 2*n + a(n-1) + 5. - Vincenzo Librandi, Aug 05 2010 sum_{n>=1} 1/a(n) = 49/120 = 0.4083333... - R. J. Mathar, Mar 22 2011 E.g.f.: x*(x+7)*exp(x). - G. C. Greubel, Aug 19 2017 MAPLE A028560:=n->n*(n + 6); seq(A028560(n), n=0..100); # Wesley Ivan Hurt, Mar 27 2014 MATHEMATICA Table[n(n + 6), {n, 0, 65}] (* or *) Select[ Range[0, 5000], IntegerQ[ Sqrt[9(9 + #)]]& ] PROG (Sage) [lucas_number2(2, n, 4-n) for n in xrange(2, 49)] # Zerinvary Lajos, Mar 19 2009 (Haskell) a028560 n = n * (n + 6)  -- Reinhard Zumkeller, Apr 07 2013 (PARI) a(n)=n*(n+6) \\ Charles R Greathouse IV, Sep 24 2015 CROSSREFS a(n-3), n>=4, third column (used for the Paschen series of the hydrogen atom) of triangle A120070. Cf. A005563. Sequence in context: A017245 A052221 A119461 * A190530 A133694 A024627 Adjacent sequences:  A028557 A028558 A028559 * A028561 A028562 A028563 KEYWORD nonn,easy AUTHOR EXTENSIONS Edited by Robert G. Wilson v, Feb 06 2002 STATUS approved

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