[long version] Re: Relation between A011944 and A005246, A002531

[Ralf Stephan]

> %Y A002531 Bisections are A001075 and A001834.

Oh, and:

%Y A001075 Bisections are A011943 and A094347.

Sorry for being so terse but, as you show, your diophantine equation
and related properties are a property not of the two sequences you listed
(A002531 and A005246) but of one of their bisections (A001075). As such,
they can already be deduced from one of Mr Beedassy's comments about the
Pellian x^2 + 3y^2 = 1.

But certainly, your specific equation w(h) = sqrt(12h^2 + 1) is missing
from the OEIS so we have

%C A011943 Equals sqrt(12*A011944(n)^2 + 1).

It remains to be shown that the bisection relations hold, first

    G002531 G.f.: (1+x-2x^2+x^3)/(1-4x^2+x^4).
    G001834 G.f.: (1+x)/((1-4*x+x^2)). 
    G001075 G.f.: (1-2x)/(1-4x+x^2).
where
    A002531 1,1,2,5,7,19,26,71,97,265,362,989,1351,3691,5042,13775,18817,
    A001834 1,5,19,71,265,989,3691,13775,51409,191861,716035,2672279,
    A001075 1,2,7,26,97,362,1351,5042,18817,70226,262087,978122,3650401,
(all having offset 0)
using the fact that Sum[n>=0, x^(2n)*a(2n)] spreads out the sequence
with interspersed zeros. So G002531(x) = G001075(x^2) + x*G001834(x^2).

Secondly,
    A001075 G.f.: (1-2x)/(1-4x+x^2).
    A011943 G.f.: (1-7*x)/(1-14*x+x^2).
		A094347 G.f.: (2-26x)/(1-14x+x^2).  [missing from OEIS]
where
    A001075 1,2,7,26,97,362,1351,5042,18817,70226,262087,978122,3650401,
    A011943 1,7,97,1351,18817,262087,3650401,50843527,708158977,9863382151,
    A094347 2,2,26,362,5042,70226,978122,13623482,189750626,2642885282,
(all having offset 0)
using that (1-14x^2+x^4) = (1+x^2+4x)(1+x^2-4x) and similar arguments
not ignoring the second 2 at start of A094347.


ralf