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A078370
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4*(n+1)*n + 5.
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43
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5, 13, 29, 53, 85, 125, 173, 229, 293, 365, 445, 533, 629, 733, 845, 965, 1093, 1229, 1373, 1525, 1685, 1853, 2029, 2213, 2405, 2605, 2813, 3029, 3253, 3485, 3725, 3973, 4229, 4493, 4765, 5045, 5333, 5629, 5933, 6245, 6565, 6893, 7229, 7573, 7925, 8285
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| This is the generic form of D in the (nontrivially) solvable Pell equation x^2 - D*y^2 = -4. See A078356-7.
1/5 + 1/13 + 1/29 +...= (Pi/8)*tanh Pi [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2006
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
Appears in A054413 and A086902 in relation to sequences related to the numerators and denominators of continued fractions convergents to sqrt((2*n+1)^2 + 4), n = 1, 2, 3, ... .
(End)
(2*n+1 +sqrt(a(n)))/2 = [2*n+1;2*n+1,2*n+1,...], n>=0, with the regular continued fraction with period length 1. This is the odd case. See A087475 for the the general case with the Schroeder reference and comments. For the even case see A002522.
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REFERENCES
| L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| a(n)=4*(n+1)*n =8*binomial(n+1, 2)+5, hence subsequence of A004770 (5 (mod 8) numbers).
G.f.: (5-2*x+5*x^2)/(1-x)^3.
a(n)=8*n+a(n-1) (with a(0)=5) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 08 2010]
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EXAMPLE
| a(1)=8*1+5=13; a(2)=8*2+13=29; a(3)=8*3+29=53 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 08 2010]
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CROSSREFS
| Subsequence of A077426 (D values (not a square) for which Pell x^2 - D*y^2 = -4 is solvable in positive integers).
Sequence in context: A194700 A130230 A106931 * A005473 A086732 A162329
Adjacent sequences: A078367 A078368 A078369 * A078371 A078372 A078373
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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EXTENSIONS
| More terms from Max Alekseyev (maxale(AT)gmail.com), Mar 03 2010
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