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 A078371 a(n) = (2*n+5)*(2*n+1). 19
 5, 21, 45, 77, 117, 165, 221, 285, 357, 437, 525, 621, 725, 837, 957, 1085, 1221, 1365, 1517, 1677, 1845, 2021, 2205, 2397, 2597, 2805, 3021, 3245, 3477, 3717, 3965, 4221, 4485, 4757, 5037, 5325, 5621, 5925, 6237, 6557, 6885, 7221, 7565, 7917, 8277, 8645 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS This is the generic form of D in the (nontrivially) solvable Pell equation x^2 - D*y^2 = +4. See A077428 and A078355. Consider all primitive Pythagorean triples (a,b,c) with c-a=8, sequence gives values of a. (Corresponding values for b are A017113(n), while c follows A078370(n).) - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004 From Vincenzo Librandi, Aug 08 2010: (Start) The identity (4*n^3+18*n^2+24*n+9)^2-(4*n^2+12*n+5)*(2*n^2+6*n+4)^2 = 1 (see Ramasamy's paper in link) can be written as A141530(n+2)^2 - a(n)*A046092(n+1)^2 = 1. a(n)^3 + 6*a(n)^2 + 9*a(n) + 4 is a square: in fact a(n)^3 + 6*a(n)^2 + 9*a(n) + 4 = (a(n) + 1)^2*(a(n) + 4), where a(n) + 4 = (2*n+3)^2. (End) Products of two positive odd integers with difference 4, (ex. 1*5, 3*7, 5*9, 7*11, 9*13, ...). - Wesley Ivan Hurt, Nov 19 2013 Starting with stage 1, the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 675", based on the 5-celled von Neumann neighborhood. - Robert Price, May 21 2016 The continued fraction expansion of (sqrt(a(n))-1)/2 is [n; {1,2*n+1}] with periodic part of length 2: repeat {1,2*n+1}. - Ron Knott, May 11 2017 a(n) is the sum of 2*n+5 consecutive integers starting from n-1. - Bruno Berselli, Jan 16 2018 LINKS Bruno Berselli, Table of n, a(n) for n = 0..1000 Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Example 6.6.5 p. 34. A. M. S. Ramasamy, Polynomial solutions for the Pell's equation, Indian Journal of Pure and Applied Mathematics 25 (1994), p. 578. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = (2*n+5)*(2*n+1) = 8*(binomial(n+2, 2)-1)+5, hence subsequence of A004770 (5 (mod 8) numbers). G.f.: (5+6*x-3*x^2)/(1-x)^3. a(n) = A061037(2*n+1) = (2*n+3)^2-4. For A061037: a(2*n+1) = (2*n+1)*(2*n+5) = (2*n+3)^2-4. - Paul Curtz, Sep 24 2008 a(n) = 8*(n+1) + a(n-1) for n>0, a(0)=5. - Vincenzo Librandi, Aug 08 2010 From Ilya Gutkovskiy, May 22 2016: (Start) E.g.f.: (5 + 4*x*(4 + x))*exp(x). Sum_{n>=0} 1/a(n) = 1/3. (End) MAPLE seq((2*n+5)*(2*n+1), n=0..48); # Emeric Deutsch, Feb 24 2005 MATHEMATICA Table[(2 n + 5) (2 n + 1), {n, 0, 100}] (* Wesley Ivan Hurt, Nov 19 2013 *) PROG (PARI) lista(nn) = {for (n=0, nn, print1((2*n+1)*(2*n+5), ", ")); } \\ Michel Marcus, Nov 21 2013 (MAGMA) [(2*n+5)*(2*n+1): n in [0..100]]; // G. C. Greubel, Sep 19 2018 CROSSREFS Subsequence of A077425 (D values (not a square) for which Pell x^2 - D*y^2 = +4 is solvable in positive integers). Cf. A017113, A078370.  Supersequence of A143206. Sequence in context: A054286 A031292 A147331 * A265056 A049741 A166010 Adjacent sequences:  A078368 A078369 A078370 * A078372 A078373 A078374 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 29 2002 EXTENSIONS More terms from Emeric Deutsch, Feb 24 2005 STATUS approved

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Last modified October 14 17:39 EDT 2019. Contains 328022 sequences. (Running on oeis4.)