Or, more aptly titled:

BACKGROUND RESEARCH

https://www.physicsforums.com/showthread.php?t=462793

Consider the following form...

for

P_n = 0, 1, 2, 5, 12, 29, 70, 169 ...

http://oeis.org/A000129

For

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

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)),

then...

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

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

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

- RF

A000129

A001110

A029549

A077442

A096979

A001333

Also see:

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

Trivia: Sophie Germain was one of the first great female mathematicians.

http://www.sdsc.edu/ScienceWomen/germain.html

**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 SquareX_(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")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/A000129A001110

**Square Triangular Numbers: for n >= 2, a(n) = 34a(n-1) - a(n-2) + 2.**http://oeis.org/A001110A029549

**Pronic Triangular Numbers: for n >= 0, a(n+3) = 35*a(n+2) - 35*a(n+1) + a(n).**http://oeis.org/A029549A077442

**2*n^2 + 7 is a square.**http://oeis.org/A077442A096979

**Sum of the areas of the first n+1 Pell triangles**http://oeis.org/A096979A001333

**Numerators of continued fraction convergents to sqrt(2).**http://oeis.org/A001333Also 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/A124174Trivia: Sophie Germain was one of the first great female mathematicians.

**Sophie Germain: Revolutionary Mathematician**http://www.sdsc.edu/ScienceWomen/germain.html

Last edited: