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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

a(n) = A061037(2n+1) = (2n+3)^2-4. For A061037: a(2n+1)=(2n+1)*(2n+5) = (2n+3)^2-4. From Balmer spectrum of hydrogen. [Paul Curtz, Sep 24 2008]

From Vincenzo Librandi, Aug 08 2010: (Start)

The identity (4n^3+18n^2+24n+9)^2-(4n^2+12n+5)*(2n^2+6n+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) with a(n)=(2n+3)^2-4 (see Paul Curtz above). (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

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

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) = 8*(n+1)+a(n-1), with 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); # (Deutsch)

MATHEMATICA

Table[(2n+5)(2n+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

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 December 10 23:13 EST 2016. Contains 279021 sequences.