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A124174
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Sophie Germain triangular numbers tr: 2*tr+1 is also a triangular number.
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3
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0, 1, 10, 45, 351, 1540, 11935, 52326, 405450, 1777555, 13773376, 60384555, 467889345, 2051297326, 15894464365, 69683724540, 539943899076, 2367195337045, 18342198104230, 80414957735001, 623094791644755
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| a(n)=(A124124(n)^2+A124124(n)-2)/4.
From Kenneth Ramsey, Jun 25 2011: (Start)
Sophie Germain triangular numbers are one of an infinite number of triangular number sets tr where 2*tn^2*tr + tn is a triangular number: tr and tn both also being triangular numbers with tn being held constant. For the present numbers, a(n) = tr, 8*(2*tr + 1) + 1 = 16*tr + 9 is also a square, the square root of which is 2*y+1 with y being the argument of the triangular number 2*tr + 1. Now 1/2*(y^2+y) = a^2 + a +1 from the definition of Sophie Germain triangular numbers. Multiply both sides by 4 and subtract 3 to get 2*y^2 + 2*y -3 = 4*a^2 + 4*a +1 (a square). Cf. A124124: Numbers y such that 2*y^2 + 2*y - 3 is a square. The values y are the same y such that 2*y+1 = sqrt(16*tr + 9).
(End)
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index to sequences with linear recurrences with constant coefficients, signature (1,34,-34,-1,1).
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FORMULA
| a(n)=35*(a(n-2)-a(n-4))+a(n-6).
From Peter Pein, Dec 04 2006: (Start)
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))
O.g.f.: (x*(1 + 9*x + x^2))/((1 - x)*(1 - 6*x + x^2)*(1 + 6*x + x^2))
E.g.f.: (-22*exp(x) + exp(-3*x+2*x*sqrt(2))*(1-sqrt(2)) - 5*exp(3*x-2*x*sqrt(2))*(-2 + sqrt(2)) + exp(-3*x-2*x*sqrt(2))*(1+sqrt(2)) + 5*exp(3*x+2*x*sqrt(2))*(2+sqrt(2)))/64. (End)
a(n)=34*a(n-2)-a(n-4)+11 [From Kieren MacMillan (kieren(AT)alumni.rice.edu), Nov 08 2008]
a(0)=0, a(1)=1, a(2)=10, a(3)=45, a(4)=351, a(n)=a(n-1)+34*a(n-2)- 34*a(n-3)-a(n-4)+a(n-5) [From Harvey P. Dale, Sep 28 2011]
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MAPLE
| a:= n-> (Matrix([[10, 1, 0, 0, 1]]). Matrix(5, (i, j)-> if i=j-1 then 1 elif j=1 then [1, 34, -34, -1, 1][i] else 0 fi)^n)[1, 4]: seq (a(n), n=1..30); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 27 2009]
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MATHEMATICA
| LinearRecurrence[{1, 34, -34, -1, 1}, {0, 1, 10, 45, 351}, 30] (* From Harvey P. Dale, Sep 28 2011 *)
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PROG
| (MAGMA) I:=[0, 1, 10, 45]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4)+11: n in [1..30]]; // Vincenzo Librandi, Sep 29 2011
(PARI) a=[0, 1, 10, 45, 351]; for(n=5, 20, a=concat(a, a[#a]+34*a[#a-1]- 34*a[#a-2]-a[#a-3]+a[#a-4])); a \\ Charles R Greathouse IV, Sep 29 2011
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CROSSREFS
| Cf. A005384, A077442, A124124.
Sequence in context: A141499 A061772 A032165 * A188699 A044112 A073248
Adjacent sequences: A124171 A124172 A124173 * A124175 A124176 A124177
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KEYWORD
| nice,nonn,easy
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AUTHOR
| Zak Seidov (zakseidov(AT)gmail.com), Dec 04 2006
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 27 2009
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