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A141759
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16n^2 + 32n + 15.
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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Via the partial fraction decomposition 1/((4n+3)*(4n+5)) = (1/2) *(1/(4n+3) -1/(4n+5)) we find 2*sum_{n>=0} (-1)^n/a(n) = 2*sum_{n>=0} (-1)^n/( (4*n+3)*(4*n+5) ) = 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ... = (1/1 + 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ..)-1 = sum_{n>=0} (-1)^n/A016813(n) + sum_{n>=0} (-1)^n/A004767(n) -1 = -1 + sum_{n>=0} b(n)/n^1 where b(n) = 1, 0, 1, 0, -1, 0, -1, 0 is a sequence with period length 8, one of the Dirichlet L-series modulo 8. The alternating sum becomes -1 +L(m=8,r=4,s=1) = Pi*sqrt(2)/4-1 = A093954 - 1.
Pi=4-8*sum(1/a(n)) noted by Bronstein-Semendjajew for the variant a(n)=(4n-1)*(4n+1) starting at n=1. (Frank Ellermann, Sep 18 2011)
The identity (16*n^2-1)^2-(64*n^2-8)*(2*n)^2 = 1 can be written as a(n)^2-A158487(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 09 2012
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REFERENCES
| Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.1.8.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| G.f.: (15+18*x-x^2)/(1-x)^3.
E.g.f.: (15+48*x+16*x^2)*exp(x).
a(n) = a(-n-2) = A016802(n+1) - 1. - Bruno Berselli, Sep 22 2011
a(n) = 16*(n+1)^2 - 1. - Vincenzo Librandi, Feb 09 2012
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MATHEMATICA
| LinearRecurrence[{3, -3, 1}, {15, 63, 143}, 50] (* Vincenzo Librandi, Feb 09 2012 *)
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PROG
| (MAGMA) [(4*n+3)*(4*n+5): n in [0..50]]; // Vincenzo Librandi, Sep 22 2011
(PARI) a(n)=n*(n+2)<<4+15 \\ Charles R Greathouse IV, Oct 27 2011
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CROSSREFS
| Cf. A133818, A005843, A158487.
Sequence in context: A065915 A062965 A157968 * A104473 A135972 A138104
Adjacent sequences: A141756 A141757 A141758 * A141760 A141761 A141762
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Miklos Kristof (kristmikl(AT)freemail.hu), Sep 15 2008
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EXTENSIONS
| Formula indices corrected by R. J. Mathar, Jul 07 2009
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