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Search: id:A141759
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| A141759 |
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a(n) = (4*n+3)*(4*n+5). |
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+0
2
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| 15, 63, 143, 255, 399, 575, 783, 1023, 1295, 1599, 1935, 2303, 2703, 3135, 3599, 4095, 4623, 5183, 5775, 6399, 7055, 7743, 8463, 9215, 9999, 10815, 11663, 12543, 13455, 14399, 15375, 16383, 17423, 18495, 19599, 20735, 21903, 23103, 24335, 25599
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OFFSET
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0,1
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COMMENT
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sum(2*(-1)^n/((4*n+3)*(4*n+5)), n=0..infinity) = Pi*sqrt(2)/4-1
1/3-1/5-1/7+1/9+1/11-1/13-1/15+1/17+1/19--++... = Pi*sqrt(2)/4-1 = A093954-1.
If A=[A158487] 64*n.^2-8 (n>0, 56, 248, 568,.,); Y=[A005843] 2*n (n>0, 2, 4, 6,.,); X = [A141759] 16*n^2-1 (n>0, 15, 63, 143, .,), we have, for all terms, Pell's equation X^2-A*Y^2=1. Example: 15^2-56*2^2=1; 63^2-248*4^2=1; 143^2-568*6^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 20 2009]
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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G.f: G(x) = (15+18*x-x^2)/(1-x)^3 E.g.f: E(x) = (15+48*x+16*x^2)*exp(x)
Also: a(n-1)=16*n^2-1 (n>0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 20 2009, R. J. Mathar, Jul 07 2009]
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EXAMPLE
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a(0)=15; a(1)=63; a(2)=143 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 20 2009, R. J. Mathar, Jul 07 2009]
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CROSSREFS
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Cf. A133818.
Cf. A005843, A158487 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 20 2009]
Sequence in context: A065915 A062965 A157968 this_sequence A104473 A135972 A138104
Adjacent sequences: A141756 A141757 A141758 this_sequence A141760 A141761 A141762
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KEYWORD
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easy,nonn
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AUTHOR
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Miklos Kristof (kristmikl(AT)freemail.hu), Sep 15 2008
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EXTENSIONS
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Formula indices corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009
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