Register to reply

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

Share this thread:
Raphie
#1
Mar17-11, 06:50 PM
P: 153
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)
http://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_nu...nd_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
Phys.Org News Partner Science news on Phys.org
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye

Register to reply

Related Discussions
Conjecture: Sophie Germain Triangles & x | 2y^2 + 2y - 3 = z^2 Linear & Abstract Algebra 25
Proof of Goldbach,Polignac,Legendre,Sophie Germain conjecture.pdf Linear & Abstract Algebra 2
Explicit formula for Euler zigzag numbers(Up/down numbers) General Math 3
Help about properties of Pell numbers Linear & Abstract Algebra 1
Generalized Pell Numbers Linear & Abstract Algebra 1