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Search: id:A107075
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| A107075 |
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Centered square numbers which are also centered pentagonal numbers. |
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+0
2
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| 1, 181, 58141, 18721081, 6028129801, 1941039074701, 625008553923781, 201250813324382641, 64802136881897286481, 20866086825157601864101, 6718815155563865902953901, 2163437614004739663149291881
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OFFSET
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1,2
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COMMENT
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The centered square numbers are n^2+(n+1)^2 while the centered pentagonal numbers are (5*r^2+5*r+2)/2. A number has both properties iff 5*(2*r+1)^2=(4*n+2)^2+1. We solve the equation 5*Y^2-1=X^2 whose solutions in natural integers are given by A075796 and A007805 respectively. The r values are 0,8,.. i.e. A053606. The n values define A119032.
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FORMULA
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G.f.: (z*(1-142*z+z^2))/((1-z)*(1-322*z+z^2)).
a(n+2)=322*a(n+1)-a(n)-140 with a(1)=1 and a(2)=181.
a(n+1)=161*a(n)-70+18*(80*a(n)^2-70*a(n)+15)^0.5.
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MAPLE
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a := n -> (Matrix([181, 1, 1]). Matrix([[323, 1, 0], [ -323, 0, 1], [1, 0, 0]])^n)[1, 3]; seq (a(n), n=1..20); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008]
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CROSSREFS
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Cf. A119032, A075796, A007805, A053606.
Adjacent sequences: A107072 A107073 A107074 this_sequence A107076 A107077 A107078
Sequence in context: A008379 A070250 A083979 this_sequence A066626 A054985 A124185
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KEYWORD
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nonn
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AUTHOR
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Richard Choulet (richardchoulet(AT)yahoo.fr), Aug 30 2007, Sep 20 2007
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EXTENSIONS
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More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008
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