Sophie Germain Triangular Numbers: An Explicit (Simple/r) Formula via Pell Numbers

[Raphie]

Or, more aptly titled:
Pell Numbers & [ Sophie Germain, Square & Pronic ] Triangular Numbers

BACKGROUND RESEARCH
Conjecture: Sophie Germain Triangles & x | 2y^2 + 2y - 3 = z^2 (Proven)
https://www.physicsforums.com/showthread.php?t=462793

Consider the following form...

X = (((z - 1)/2)^2 + ((z - 1)/2)^1)/2 = T_((z - 1)/2)
for T_n denotes a Triangular Number

Let P_n denote a Pell Number.
Pell Number Formula
((1 + sqrt (2))^n - (1 - sqrt (2))^n)/(2*sqrt (2))

P_n = 0, 1, 2, 5, 12, 29, 70, 169 ...
http://oeis.org/A000129

The following is well known...
For z = P_(n+2) - P_(n+1), then...

z = 1, 3, 7, 17, 41, 99, 239, 577, 1393 ... ("Half Companion" Pell Numbers)
http://oeis.org/A001333
X_n = 0, 1, 6, 36, 210, 1225, 7140, 41616...
http://oeis.org/A096979

X_(2n) --> Triangular Numbers that are twice another Triangular Number
X_(2n + 1) --> Triangular Numbers that are Square

If, on the other hand, we set...

z = .5*(P_(n + 2 + b) + P_(n + 1 + b)(-1)^b + P_(n - b)(-1)^b - P_(n - 1 - b)), for b = n (mod 2), which generates an alternating series...

z = .5*(P_(n + 2) + P_(n + 1) + P_(n - 0) - P_(n - 1)) [Even n]
z = .5*(P_(n + 3) - P_(n + 2) - P_(n - 1) - P_(n - 2)) [Odd n]

then...

The following would seem not to be so well known...

z = 1, 3, 9, 19, 53, 111, 309, 647 ... (n | 2*n^2 + 7 is a square)
http://oeis.org/A077442
X_n = 0, 1, 10, 45, 351, 1540, 11935, 52326 ...
http://oeis.org/A124174

X_(2n) --> Sophie Germain Triangular Numbers ("Even")
X_(2n+1) --> Sophie Germain Triangular Numbers ("Odd")

Of course, the above is a bit simplistic compared to the following explicit formula for Sophie Germain Triangular Numbers one can find on OEIS...

A124174
Sophie Germain triangular numbers tr: 2*tr+1 is also a triangular number
http://oeis.org/A124174
a(n)=-11/32 + (-3 - 2*sqrt(2))^n/64 + (5*(3 - 2*sqrt(2))^n)/32 + (-3 - 2*sqrt(2))^n/(32*sqrt(2)) - (5*(3 - 2*sqrt(2))^n)/(32*sqrt(2)) + (-3 + 2*sqrt(2))^n/64 - (-3 + 2*sqrt(2))^n/(32*sqrt(2)) + (5*(3 + 2*sqrt(2))^n)/32 + (5*(3 + 2*sqrt(2))^n)/(32*sqrt(2))

... but at least for me, not being a mathematician, I prefer the manner of mathematics that makes things simpler and shows how maths for one number progression relate to maths for other number progressions in a sensible, intuitive and accessible manner.

- RF

KEY TO PROGRESSIONS
A000129 Pell numbers: a(n) = 2*a(n-1) + a(n-2). http://oeis.org/A000129
A001110 Square Triangular Numbers: for n >= 2, a(n) = 34a(n-1) - a(n-2) + 2. http://oeis.org/A001110
A029549 Pronic Triangular Numbers: for n >= 0, a(n+3) = 35*a(n+2) - 35*a(n+1) + a(n). http://oeis.org/A029549
A077442 2*n^2 + 7 is a square. http://oeis.org/A077442
A096979 Sum of the areas of the first n+1 Pell triangles http://oeis.org/A096979
A001333 Numerators of continued fraction convergents to sqrt(2). http://oeis.org/A001333

Also see: Pell Number: Computations And Connections (mentions "Half Companion" Pell Numbers) (via Wikipedia)
http://en.wikipedia.org/wiki/Pell_number#Computations_and_connections

Note: Oddly enough, none of the commentary associated with the above progressions seems to mention Sophie Germain Triangular Numbers.
A124174 Sophie Germain triangular numbers: a(n)=34a(n-2)-a(n-4)+11 =35(a(n-2)-a(n-4))+a(n-6) http://oeis.org/A124174

Trivia: Sophie Germain was one of the first great female mathematicians.
Sophie Germain: Revolutionary Mathematician
http://www.sdsc.edu/ScienceWomen/germain.html

 

[fresh_42]

Interesting, but a bit numerological. Here is an AMS paper with a bit of background:
https://www.semanticscholar.org/search?q=S 0273-0979(2012)01370-8&sort=relevance