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

 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

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