A093069 a(n) = (2^n + 1)^2 - 2.

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879, 281475010265087, 1125899973951487

Offset: 1

Author

Eric W. Weisstein, Mar 17 2004

Keywords

• nonn
• easy
• changed

Comments

Cletus Emmanuel calls these "Kynea numbers".

Difference between the smallest digitally balanced number with 2n+4 binary digits and the largest digitally balanced number with 2n+2 binary digits (see A031443): 7 = 9-2 = 1001-10, 23 = 35-12 = 100011-1100, 79 = 135-56 = 10000111-111000 etc. - Juri-Stepan Gerasimov, Jun 01 2011

Examples

			G.f. = 7*x + 23*x^2 + 79*x^3 + 287*x^4 + 1087*x^5 + 4223*x^6 + 16639*x^7 + ...
		

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).

Crossrefs

Cf. A091514 (primes of the form (2^n + 1)^2 - 2).

Cf. A244663.

Programs

Magma

[(2^n+1)^2-2 : n in [1..30]]; // Wesley Ivan Hurt, Jul 08 2014

Maple

A093069:=n->(2^n+1)^2-2: seq(A093069(n), n=1..30);

Mathematica

a[ n_] := If[ n < 1, 0, 4^n + 2^(n + 1) - 1]; (* Michael Somos, Jul 08 2014 *)
CoefficientList[Series[(7 - 26*x + 16*x^2)/((1 - x)*(2*x - 1)*(4*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 08 2014 *)
LinearRecurrence[{7,-14,8},{7,23,79},30] (* Harvey P. Dale, Aug 25 2025 *)

PARI

vector(100, n, (2^n+1)^2-2) \\ Colin Barker, Jul 08 2014

PARI

Vec(-(16*x^2-26*x+7)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014

Formula

a(n) = 4^n+2^(n+1)-1.

G.f.: -x*(7-26*x+16*x^2) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011

a(n) = A092431(n+2) - A020522(n+1). - R. J. Mathar, Jun 01 2011

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

Extensions

More terms from Colin Barker, Jul 08 2014