OFFSET
1,2
COMMENTS
Cletus Emmanuel calls these "Carol numbers".
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1660
P. Shanmuganandham and C. Deepa, Sum of Squares of n Consecutive Carol Numbers, Baghdad Science Journal, Vol. 20, No. 1 (Special Issue: ICAAM), 2023, pp. 263-267.
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
a(n) = 8*A006516(n-1) - 1. - Ivan N. Ianakiev, Jan 30 2026
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
KEYWORD
sign,easy
AUTHOR
Eric W. Weisstein, Mar 20 2004
EXTENSIONS
More terms from Colin Barker, Jul 07 2014
STATUS
approved
