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A140811
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a(n) = 6*n^2 - 1.
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12
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-1, 5, 23, 53, 95, 149, 215, 293, 383, 485, 599, 725, 863, 1013, 1175, 1349, 1535, 1733, 1943, 2165, 2399, 2645, 2903, 3173, 3455, 3749, 4055, 4373, 4703, 5045, 5399, 5765, 6143, 6533, 6935, 7349, 7775, 8213, 8663, 9125, 9599, 10085, 10583, 11093, 11615
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OFFSET
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0,2
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COMMENTS
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Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161, ... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).
a(n) is the set of all k such that 6k+6 is a perfect square. - Gary Detlefs, Mar 04 2010
Apart from first term, sequence found by reading the line from 5, in the direction 5, 23, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
Terms from center to right in the following spiral:
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65--63--61--59
/ \
67 31--29--27 57
/ / \ \
69 33 9---7 25 55
/ / / \ \ \
71 35 11 -1===5==23==53==>
/ / / / / /
37 13 1---3 21 51
\ \ / /
39 15--17--19 49
\ /
41--43--45--47 (End)
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REFERENCES
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P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969, 132 pages, pp. 28-36. CCSA, then CELAR. Now DGA Maitrise de l'Information 35131 Bruz.
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LINKS
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FORMULA
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a(n) = 2*a(n-1) - a(n-2) + 12.
First differences: a(n+1) - a(n) = A017593(n).
For n > 0, a(n) = floor(3/(cosh(1/n) - 1)) = floor(1/(n*sinh(1/n) - 1)); for similar formulas for cosine and sine, see A033581. - Clark Kimberling, Oct 19 2014, corrected by M. F. Hasler, Oct 21 2014
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(6))*cot(Pi/sqrt(6)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(6))*csc(Pi/sqrt(6)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(6))*csc(Pi/sqrt(6)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(6))*sin(Pi/sqrt(3))/sqrt(2). (End)
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MATHEMATICA
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CoefficientList[Series[(1-8*x-5*x^2)/(x-1)^3 , {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)
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PROG
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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