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A171503 Number of 2 X 2 integer matrices with entries from {0,1,...,n} having determinant 1. 8
0, 3, 7, 15, 23, 39, 47, 71, 87, 111, 127, 167, 183, 231, 255, 287, 319, 383, 407, 479, 511, 559, 599, 687, 719, 799, 847, 919, 967, 1079, 1111, 1231, 1295, 1375, 1439, 1535, 1583, 1727, 1799, 1895, 1959, 2119, 2167, 2335, 2415, 2511, 2599 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of distinct solutions to k*x+h=0, where |h|<=n and k=1,2,...,n. - Giovanni Resta, Jan 08 2013.

Number of reduced rational numbers r/s with |r|<=n and 0<s<=n. - Juan M. Marquez, Apr 13 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

Recursion: a(n) = a(n - 1) + 4*phi(n) for n > 1, with phi being Euler's totient function. - Juan M. Marquez, Jan 19 2010

a(n) = 4 * A002088(n) - 1 for n >= 1. - Robert Israel, Jun 01 2014

MAPLE

with(numtheory):

a:= proc(n) option remember;

       `if`(n<2, [0, 3][n+1], a(n-1) + 4*phi(n))

    end:

seq(a(n), n=0..60);

MATHEMATICA

a[n_]:=Count[Det/@(Partition[ #, 2]&/@Tuples[Range[0, n], 4]), 1]

(* Second program: *)

a[0] = 0; a[1] = 3; a[n_] := a[n] = a[n-1] + 4*EulerPhi[n];

Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Jun 16 2018 *)

PROG

(PARI) a(n)=(n>0)+2*sum(k=1, n, moebius(k)*(n\k)^2) \\ Charles R Greathouse IV, Apr 20 2015

(Python)

from functools import lru_cache

@lru_cache(maxsize=None)

def A171503(n): # based on second formula in A018805

    if n == 0:

        return 0

    c, j = 0, 2

    k1 = n//j

    while k1 > 1:

        j2 = n//k1 + 1

        c += (j2-j)*(A171503(k1)-1)//2

        j, k1 = j2, n//j2

    return 2*(n*(n-1)-c+j) - 1 # Chai Wah Wu, Mar 25 2021

CROSSREFS

Cf. A062801, A000010, A018805. Differences are A002246.

See A326354 for an essentially identical sequence.

Sequence in context: A181106 A131753 A330319 * A326354 A283008 A283259

Adjacent sequences:  A171500 A171501 A171502 * A171504 A171505 A171506

KEYWORD

nonn

AUTHOR

Jacob A. Siehler, Dec 10 2009

EXTENSIONS

Edited by Alois P. Heinz, Jan 19 2011

STATUS

approved

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Last modified January 26 12:17 EST 2022. Contains 350598 sequences. (Running on oeis4.)