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%I #40 Feb 18 2022 19:25:45
%S -1,7,47,223,959,3967,16127,65023,261119,1046527,4190207,16769023,
%T 67092479,268402687,1073676287,4294836223,17179607039,68718952447,
%U 274876858367,1099509530623,4398042316799,17592177655807,70368727400447,281474943156223,1125899839733759
%N a(n) = (2^n-1)^2 - 2.
%C Cletus Emmanuel calls these "Carol numbers".
%H Michael De Vlieger, <a href="/A093112/b093112.txt">Table of n, a(n) for n = 1..1660</a>
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3240465">Binary Operators of the Groupoids of OEIS A093112 and A093069 Numbers(Carol and Kynea Numbers)</a>, Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences (2019) Vol. 8, No. 10.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F a(n) = (2^n-1)^2 - 2.
%F From _Colin Barker_, Jul 07 2014: (Start)
%F a(n) = 6*a(n-1) - 7*a(n-2) - 6*a(n-3) + 8*a(n-4).
%F G.f.: x*(16*x^2-14*x+1) / ((x-1)*(2*x-1)*(4*x-1)). (End)
%F E.g.f.: 2 - exp(x) - 2*exp(2*x) + exp(4*x). - _Stefano Spezia_, Dec 09 2019
%p seq((Stirling2(n+1, 2))^2-2, n=1..23); # _Zerinvary Lajos_, Dec 20 2006
%t lst={};Do[p=(2^n-1)^2-2;AppendTo[lst, p], {n, 66}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)
%t Rest@ CoefficientList[Series[x (16 x^2 - 14 x + 1)/((x - 1) (2 x - 1) (4 x - 1)), {x, 0, 25}], x] (* _Michael De Vlieger_, Dec 09 2019 *)
%o (PARI) Vec(x*(16*x^2-14*x+1)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ _Colin Barker_, Jul 07 2014
%o (PARI) a(n) = (2^n-1)^2-2 \\ _Charles R Greathouse IV_, Sep 10 2015
%o (Python)
%o def A093112(n): return (2**n-1)**2-2 # _Chai Wah Wu_, Feb 18 2022
%Y Cf. A000225.
%K sign,easy
%O 1,2
%A _Eric W. Weisstein_, Mar 20 2004
%E More terms from _Colin Barker_, Jul 07 2014
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