0,1

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. [From Paul Curtz (bpcrtz(AT)free.fr), Sep 24 2008]

Except for the two terms of [A141530] and the first term of [A046092[, if X=[A141530], A=[A078371], Y=[A046092], we have, for all others terms, Pell's equation: X^2-A*Y^2=1. Example: 9^2-5*4^2=1; 55^2-21*12^2=1; 161^2-45*24^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 13 2009]

Numbers n such that (1/4)*(n^3+6*n^2+9*n+4) is square [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Sep 28 2009]

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+a(n-1), (with a(1)=5) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 24 2009]

For n=2, a(2)=8*2+5=21; n=3, a(3)=8*3+21=45; n=4, a(4)=8*4+45=77 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 24 2009]

seq((2*n+5)*(2*n+1), n=0..48); (Deutsch)

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.

Cf. A046092, A141530 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 13 2009]

Sequence in context: A054286 A031292 A147331 this_sequence A049741 A166010 A146846

Adjacent sequences: A078368 A078369 A078370 this_sequence A078372 A078373 A078374

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 24 2005