A004315 -id:A004315 - cp's OEIS frontend

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

David W. Wilson

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

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).

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

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

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

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 *)

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

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

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

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

A213351 9-quantum transitions in systems of N >= 9 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 20, 220, 11, 1760, 264, 11440, 3432, 78, 64064, 32032, 2184, 320320, 240240, 32760, 455, 1464320, 1537536, 349440, 14560, 6223360, 8712704, 2970240, 247520, 2380, 24893440, 44808192, 21385728, 2970240, 85680, 94595072, 212838912, 135442944, 28217280

Offset: 9

Stanislav Sykora, Jun 13 2012

For a general discussion, please see A213343.
This a(n) is for nonuple-quantum transitions (q = 9).
It lists the flattened triangle T(9;N,k) with rows N = 9,10,... and columns k = floor((N-9)/2).

			Starting rows of the triangle:
   N | k = 0, 1, ..., floor((N-9)/2)
  ---+------------------------------
   9 | 1
  10 | 20
  11 | 220 11
  12 | 1760 264
  13 | 11440 3432 78

See A213343

Stanislav Sykora, Table of n, a(n) for n = 9..2170
Stanislav Sykora, T(9;N,k) with rows N = 9..100 and columns k = 0..floor((N-9)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213350 (q=1 to 8), A213352 (q= 10).

Cf. A140354 (first column,with offset 9), A004315 (row sums).

With[{q = 9}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, q, q + 10}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 20 2019 *)

PARI
```
\\ See A213343; set thisq = 9
```

Set q = 9 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

cp's OEIS Frontend

A023105 Number of distinct quadratic residues mod 2^n.

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