A093112 a(n) = (2^n-1)^2 - 2.

-1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, 4190207, 16769023, 67092479, 268402687, 1073676287, 4294836223, 17179607039, 68718952447, 274876858367, 1099509530623, 4398042316799, 17592177655807, 70368727400447, 281474943156223, 1125899839733759

Offset

1,2

Comments

Cletus Emmanuel calls these "Carol numbers".

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1660

Amelia Carolina Sparavigna, Binary Operators of the Groupoids of  OEIS A093112 and A093069 Numbers(Carol and Kynea Numbers), Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).

Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.

Eric Weisstein's World of Mathematics, Near-Square Prime

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Formula

a(n) = (2^n-1)^2 - 2.

From Colin Barker, Jul 07 2014: (Start)

a(n) = 6*a(n-1) - 7*a(n-2) - 6*a(n-3) + 8*a(n-4).

G.f.: x*(16*x^2-14*x+1) / ((x-1)*(2*x-1)*(4*x-1)). (End)

E.g.f.: 2 - exp(x) - 2*exp(2*x) + exp(4*x). - Stefano Spezia, Dec 09 2019

Maple

seq((Stirling2(n+1, 2))^2-2, n=1..23); # Zerinvary Lajos, Dec 20 2006

Mathematica

lst={}; Do[p=(2^n-1)^2-2; AppendTo[lst, p], {n, 66}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
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 *)

Prog

(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
(PARI) a(n) = (2^n-1)^2-2 \\ Charles R Greathouse IV, Sep 10 2015
(Python)
def A093112(n): return (2**n-1)**2-2 # Chai Wah Wu, Feb 18 2022

Crossrefs

Cf. A000225.

Sequence in context: A201437 A202509 A009202 * A091516 A064385 A269520

Adjacent sequences: A093109 A093110 A093111 * A093113 A093114 A093115

Keyword

sign,easy

Author

Eric W. Weisstein, Mar 20 2004

Extensions

More terms from Colin Barker, Jul 07 2014

Status

approved