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
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).
Crossrefs
Cf. A000225.
Programs
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Maple
seq((Stirling2(n+1, 2))^2-2, n=1..23); # Zerinvary Lajos, Dec 20 2006
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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 *) -
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
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PARI
a(n) = (2^n-1)^2-2 \\ Charles R Greathouse IV, Sep 10 2015
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Python
def A093112(n): return (2**n-1)**2-2 # Chai Wah Wu, Feb 18 2022
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
Extensions
More terms from Colin Barker, Jul 07 2014
Comments