A023105 Number of distinct quadratic residues mod 2^n.

1, 2, 2, 3, 4, 7, 12, 23, 44, 87, 172, 343, 684, 1367, 2732, 5463, 10924, 21847, 43692, 87383, 174764, 349527, 699052, 1398103, 2796204, 5592407, 11184812, 22369623, 44739244, 89478487, 178956972, 357913943, 715827884, 1431655767, 2863311532

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Number of distinct n-digit suffixes of base 2 squares.

a(n) counts the elements of A234000 smaller than 2^n plus the zero: a(7)=23 counts the elements of {0, 1, 4, 9, ..., 113, 121}, for example. - R. J. Mathar, Oct 11 2014

Conjecture: a(n) = 2 + (the number of A004215 entries < 2^n), for n>0. - Tilman Neumann, Sep 20 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• John Greene and James A. Sellers, Extending recent parity results of Nyirenda and Mugwangwavari for partitions with initial repetitions, Integers (2025), Vol. 25, Art. No. A32. See p. 5.
• Lee Hae-hwang, Sequences of Growing Networks
• W. D. Stangl, Counting Squares in Z_n, Mathematics Magazine, pp. 285-289, Vol. 69 No. 4 (October 1996).
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A000224, A001045, A004315, A005578, A234000.

Programs

Haskell

a 0 = 1
a 1 = 2
a n | even n = 2*a(n-1)-2
a n | odd  n = 2*a(n-1)-1
-- James Spahlinger, Oct 07 2012

Magma

[Floor((2^n+10)/6): n in [0..30]]; // Vincenzo Librandi, Apr 21 2012

Mathematica

CoefficientList[Series[(1-3*x^2-x^3)/((1-x)*(1+x)*(1-2*x)),{x,0,35}],x] (* Vincenzo Librandi, Apr 21 2012 *)
LinearRecurrence[{2,1,-2},{1,2,2,3},40] (* Harvey P. Dale, Mar 05 2016 *)

PARI

a(n)=(2^n+10)\6 \\ Charles R Greathouse IV, Apr 21 2012

Python

def A023105(n): return ((1<Chai Wah Wu, Aug 22 2023

SageMath

[(2^n +9 -(-1)^n -3*bool(n==0))/6 for n in (0..30)] # G. C. Greubel, Aug 10 2022

Formula

a(n) = floor( (2^n+10)/6 ).

a(n) = (2^n + 9 - (-1)^n)/6 for n > 0. - David S. Dodson, Jan 06 2013

G.f.: (1-3*x^2-x^3)/((1-x)*(1+x)*(1-2*x)). - Colin Barker, Mar 08 2012

a(0)=1, a(1)=2. a(n) = 2*a(n-1)-2 if n is even, a(n) = 2*a(n-1)-1 if n is odd. - Vincenzo Librandi, Apr 21 2012

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 0. - Joerg Arndt, Apr 21 2012

a(0)=1, a(1)=2, a(n+2) = a(n+1) + A001045(n) for n >= 1. - Lee Hae-hwang, Jun 16 2014

a(n) = A000224(2^n). - R. J. Mathar, Oct 10 2014

a(n) = A005578(n-1) + 1, n > 0. - Carl Joshua Quines, Jul 17 2019

E.g.f.: (exp(2*x) + 9*exp(x) - 3 - exp(-x))/6. - G. C. Greubel, Aug 10 2022