A000930 -id:A000930 - cp's OEIS frontend

A002478 Bisection of A000930.

1, 1, 3, 6, 13, 28, 60, 129, 277, 595, 1278, 2745, 5896, 12664, 27201, 58425, 125491, 269542, 578949, 1243524, 2670964, 5736961, 12322413, 26467299, 56849086, 122106097, 262271568, 563332848, 1209982081, 2598919345, 5582216355, 11990037126, 25753389181

Offset: 0

N. J. A. Sloane

Number of ways to tile a 3 X n region with 1 X 1, 2 X 2 and 3 X 3 tiles.
Number of ternary words with subwords (0,0), (0,1) and (1,1) not allowed. - Olivier Gérard, Aug 28 2012
Diagonal sums of A063967. - Paul Barry, Nov 09 2005
Row sums of number triangle A116088. - Paul Barry, Feb 04 2006
Sequence is identical to its second differences negated, minus the first 3 terms. - Paul Curtz, Feb 10 2008
a(n) = term (3,3) in the 3 X 3 matrix [0,1,0; 0,0,1; 1,2,1]^n. - Gary W. Adamson, May 30 2008
a(n)/a(n-1) tends to 2.147899035..., an eigenvalue of the matrix and a root to x^3 - x^2 - 2x - 1 = 0. - Gary W. Adamson, May 30 2008
INVERT transform of (1, 2, 1, 0, 0, 0, ...) = (1, 3, 6, 13, 28, ...); such that (1, 2, 1, 0, 0, 0, ...) convolved with (1, 1, 3, 6, 13, 28, 0, 0, 0, ...) shifts to the left. - Gary W. Adamson, Apr 18 2010
a(n) is the top left entry in the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 0, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014

			a(3)=6 as there is one tiling of a 3 X 3 region with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 1 tiling consisting of the 3 X 3 tile.

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 322.
S. Heubach, Tiling an m X n Area with Squares of Size up to k X k (m<=5), Congressus Numerantium 140 (1999), pp. 43-64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..300
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Emeric Deutsch, Counting tilings with L-tiles and squares, Problem 10877, Amer. Math. Monthly, 110 (March 2003), 245-246.
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, arXiv:1907.06517 [math.CO], 2019.
Leonhard Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 412
Milan Janjić, Pascal Triangle and Restricted Words, arXiv:1705.02497 [math.CO], 2017.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Richard J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 19 (halved...).
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 5.
Sam Northshield, Some generalizations of a formula of Reznick, SUNY Plattsburgh (2022).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Cf. A000930, A054856, A054857, A025234, A078007, A078039, A226546, A077936 (INVERT transform), A008346 (inverse INVERT transform).

I:=[1,1,3]; [n le 3 select I[n] else Self(n-1) +2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 14 2023

f[A_]:= Module[{til = A}, AppendTo[til, A[[-1]] + 2A[[-2]] + A[[-3]]]]; NumOfTilings[ n_ ]:= Nest[ f, {1,1,3}, n-2]; NumOfTilings[30]
LinearRecurrence[{1,2,1},{1,1,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
CoefficientList[Series[1/(1-x-2x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Oct 17 2024 *)

a(n)=([0,1,0; 0,0,1; 1,2,1]^n*[1;1;3])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

@CachedFunction
def a(n): # A002478
    if (n<3): return (1,1,3)[n]
    else: return sum(binomial(2,j)*a(n-j) for j in range(1,4))
[a(n) for n in (0..40)] # G. C. Greubel, Apr 14 2023

G.f.: 1 / (1-x-2*x^2-x^3). [Simon Plouffe in his 1992 dissertation.]
a(n) = a(n-1) + 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} binomial(2*n-2*k, k). - Paul Barry, Nov 13 2004
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j, n-k-j)*C(j, k). - Paul Barry, Nov 09 2005
a(n) = Sum_{k=0..n} C(2*k,n-k) = Sum_{k=0..n} C(n,k)*C(3*k,n)/C(3*k,k). - Paul Barry, Feb 04 2006
a(n) = A000930(n) + 2*Sum_{i=0..n-2} a(i)*A000930(n-2-i). - Michael Tulskikh, Jun 07 2020

Additional comments from Silvia Heubach (silvi(AT)cine.net), Apr 21 2000

A227835 3^a(n) is the highest power of 3 dividing A000930(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 1, 3, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 1, 3, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 1, 4, 0, 0, 0, 0, 3, 0, 3, 4, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 1, 3, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 1, 3, 0

Offset: 0

N. J. A. Sloane, Aug 04 2013

nonn

This is the 3-adic valuation of A000930.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000930, A007949, A227834.

A000930:=func; [Valuation(A000930(n),3): n in [0..120]]; // Bruno Berselli, Aug 05 2013

Mathematica

A000930[n_] := Sum[Binomial[n - 2*i, i], {i, 0, Floor[n/3]}]; Table[IntegerExponent[A000930[n], 3], {n, 0, 100}] (* G. C. Greubel, Apr 26 2017 *)

A350311 Replace 2^k in the binary expansion of n with A000930(k+2), Narayana's cows sequence.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 9, 10, 11, 12, 12, 13, 14, 15, 13, 14, 15, 16, 16, 17, 18, 19, 15, 16, 17, 18, 18, 19, 20, 21, 19, 20, 21, 22, 22, 23, 24, 25, 13, 14, 15, 16, 16, 17

Offset: 0

A.H.M. Smeets, Dec 24 2021

A048715(n) = m, if and only if a(n) = m and for all k > n a(k) > m.

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Cf. A000930, A048715, A350215, A350312.

Cf. A022290 (analog for Fibonacci numbers).

b:= (n, i, j, k)->`if`(n=0, 0, k*irem(n, 2, 'q')+b(q, j, k, i+k)):
a:= n-> b(n, 1$3):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 26 2022

my(p=Mod('x,'x^3-'x^2-1)); a(n) = vecsum(Vec(lift(subst(Pol(binary(n))*'x^2,'x,p)))); \\ Kevin Ryde, Dec 26 2021

def Interpretation(n):
    f0, f1, f2, r = 1, 1, 1, 0
    while n > 0:
        if n%2 == 1:
            r = r+f0
        n, f0, f1, f2 = n//2, f0+f2, f0, f1
    return r
n = 0
while n <= 69:
    print(Interpretation(n), end = ", ")
    n += 1

A108104 Sequence A000930 with terms repeated.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 9, 9, 13, 13, 19, 19, 28, 28, 41, 41, 60, 60, 88, 88, 129, 129, 189, 189, 277, 277, 406, 406, 595, 595, 872, 872, 1278, 1278, 1873, 1873, 2745, 2745, 4023, 4023, 5896, 5896, 8641, 8641, 12664, 12664, 18560, 18560

Offset: 0

Roger L. Bagula, Jun 03 2005

nonn

The usual policy in the OEIS is not to include such "doubled" sequences. This is an exception. - N. J. A. Sloane

Based on the morphism 1->{5}, 2->{6}, 3->{4}, 4->{2}, 5->{3}, 6->{1, 6}.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sergio Falcon, Generalized (k,r)-Fibonacci Numbers, Gen. Math. Notes, 25(2), 2014, 148-158.
I. Wloch, U. Bednarz, D. Bród, A Wloch and M. Wolowiec-Musial, On a new type of distance Fibonacci numbers, Discrete Applied Math., Volume 161, Issues 16-17, November 2013, Pages 2695-2701.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1).

Cf. A000930.

I:=[1,1,1,1,1,1]; [n le 6 select I[n] else Self(n-2)+Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jan 19 2016

s[1] = {5}; s[2] = {6}; s[3] = {4}; s[4] = {2}; s[5] = {3}; s[6] = {1, 2}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] a0 = Table[Length[p[i]], {i, 0, 50}]
m = 6; For[n = 0, n < m, n++, a[n] = 1]; For[n = m, n < 51, n++, a[n] = a[n - m] + a[n - 2]]; Table[a[n], {n, 0, 50}] (* Sergio Falcon, Nov 12 2015 *)
CoefficientList[Series[(1 + x) / (1 - x^2 - x^6), {x, 0, 50}], x] (* or *) LinearRecurrence[{0, 1, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1}, 60] (* Vincenzo Librandi, Jan 19 2016 *)

x='x+O('x^55); Vec((1+x)/(1-x^2-x^6)) \\ Altug Alkan, Nov 10 2015

a(n) = a(n-2) + a(n-6), starting 1,1,1,1,1,1.

G.f.: (1+x)/(1-x^2-x^6).

Edited by N. J. A. Sloane, Dec 01 2006

A231620 a(n) = A000930(n^2), where A000930 is Narayana's cows sequence.

Original entry on oeis.org

1, 1, 3, 19, 277, 8641, 578949, 83316385, 25753389181, 17098272199297, 24382819596721629, 74684329652984094451, 491347682599497451569523, 6943240361573523613067995729, 210741152533202801182666172606913, 13738849457010997118546333815068560833, 1923823572225984354415961546862346889944243

Offset: 0

Paul D. Hanna, Nov 13 2013

nonn

G. C. Greubel, Table of n, a(n) for n = 0..75

Cf. A000930, A231621, A228647, A228648.

Table[SeriesCoefficient[1/(1 - x - x^3), {x, 0, n^2}], {n,0,25}] (* G. C. Greubel, Apr 26 2017 *)

{a(n) = polcoeff(1/(1-x-x^3 + x*O(x^(n^2))), n^2)}
for(n=0, 20, print1(a(n), ", "))

a(n) = [x^(n^2)] 1 / (1 - x - x^3) for n>=0.

A271901 Length of period of Narayana sequence A000930 modulo n-th prime.

Original entry on oeis.org

7, 8, 31, 57, 60, 168, 288, 381, 528, 840, 930, 342, 1723, 1848, 46, 468, 3541, 1240, 33, 5113, 2664, 6240, 3444, 7920, 3169, 10303, 10713, 11557, 11991, 991, 2016, 130, 6256, 1610, 148, 22800, 24807, 26733, 4648, 172, 10680, 32760, 36673, 37443, 2156, 3960, 481, 12432, 226, 26220, 54523, 8160, 9680, 63000

Offset: 1

N. J. A. Sloane, Apr 17 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000
H. T. Engstrom, On sequences defined by linear recurrence relations Trans. Am. Math. Soc. 33 (1) (1931) 210-218.
K. Kirthi, Narayana Sequences for Cryptographic Applications, arXiv preprint arXiv:1509.05745 [math.NT], 2015.
M. B. Nathanson, Linear recurrences and uniform distribution, Proc. Amer. Math. Soc. 48 (1975), 289-291.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.

Cf. A000930, A271953.

a[n_] := Module[{p = Prime[n], a = 1, b = 1, c = 2, k = 1}, While[a != 1 || b != 1 || c != 1, {a, b, c} = {b, c, Mod[a + c, p]}; k++]; k];
Array[a, 100] (* Jean-François Alcover, Jul 22 2018, after Charles R Greathouse IV *)

a(n,p=prime(n))=my(a=1,b=1,c=2,k=1); while(a!=1 || b!=1 || c!=1, [a,b,c]=[b,c,(a+c)%p]; k++); k \\ Charles R Greathouse IV, Feb 26 2017

from sympy import prime
def A271901(n):
    p = prime(n)
    i, a, b, c =  1, 1, 1, 2 % p
    while a != 1 or b != 1 or c != 1:
        i += 1
        a, b, c = b, c, (a+c) % p
    return i # Chai Wah Wu, Feb 26 2017

a(n) = A271953(prime(n)). - Joerg Arndt, Apr 17 2016

a(1) corrected by Altug Alkan, Apr 17 2016

Terms a(24) and beyond from Joerg Arndt, Apr 17 2016

A271953 a(n) is the period of A000930 modulo n.

Original entry on oeis.org

1, 7, 8, 14, 31, 56, 57, 28, 24, 217, 60, 56, 168, 399, 248, 56, 288, 168, 381, 434, 456, 420, 528, 56, 155, 168, 72, 798, 840, 1736, 930, 112, 120, 2016, 1767, 168, 342, 2667, 168, 868, 1723, 3192, 1848, 420, 744, 3696, 46, 56, 399, 1085, 288, 168, 468, 504, 1860, 1596, 3048, 840, 3541, 1736, 1240, 6510

Offset: 1

Joerg Arndt, Apr 17 2016

nonn

Joerg Arndt, Table of n, a(n) for n = 1..1000
H. T. Engstrom, On sequences defined by linear recurrence relations, Trans. Am. Math. Soc. 33 (1) (1931) 210-218.
K. Kirthi, Narayana Sequences for Cryptographic Applications, arXiv preprint arXiv:1509.05745 [math.NT], 2015.
M. B. Nathanson, Linear recurrences and uniform distribution, Proc. Amer. Math. Soc. 48 (1975), 289-291.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.

Cf. A000930, A271901 (periods mod primes), A001175 (periods of A000045 modulo n).

minlen = 100; maxlen = 2*10^4;
per[lst_] := FindTransientRepeat[lst, 2] // Last // Length;
a[n_] := Module[{p0=0, len=minlen}, While[p0 = Mod[LinearRecurrence[{1, 0, 1}, {1, 1, 1}, len], n] // per; p0<=1 && len<=maxlen, len = 2 len]; p0];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Jul 21 2018 *)

per(n, S, R) = {  \\ S[]: leading terms, R[]: recurrence
    if ( n==1, return( 1 ) );
    my ( r = #R );
    if ( r != #S , error("Mismatch in length of S[] and R[]") );
    S = vector(#S, j, Mod(S[j], n) );
    R = vector(#S, j, Mod(R[j], n) );
    my( T = S );
    my( j = 0 );
    until ( 0,  \\ forever
        j += 1;
        my( t = sum(i=1, r, R[i] * T[r+1-i] ) );  \\ next term
        for (k=1, r-1, T[k] = T[k+1] );
        T[r] = t;
        if ( T == S , return(j) );
    );
}
\\vector(66, n, per(n, [0,1], [1,1]) )  \\ A001175
\\vector(66, n, per(prime(n), [0,1], [1,1]) )  \\ A060305
vector(66, n, per(n, [0,0,1], [1,0,1]) )  \\ A271953
\\vector(66, n, per(prime(n), [0,0,1], [1,0,1]) )  \\ A271901
\\vector(66, n, per(n, [0,0,1], [0,1,1]) )  \\ A104217
/* Joerg Arndt, Apr 17 2016 */

Let the prime factorization of n be p1^e1*...*pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)) [Engstrom]. - N. J. A. Sloane, Feb 18 2017

A139040 Triangle read by rows: each row is an initial segment of the terms of A000930 followed by its reflection.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 6, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 9, 6, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 6, 9, 9, 6, 4, 3, 2, 1, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, May 31 2008

			Triangle begins:
{1},
{1, 1},
{1, 1, 1},
{1, 1, 1, 1},
{1, 1, 2, 1, 1},
{1, 1, 2, 2, 1, 1},
{1, 1, 2, 3, 2, 1, 1},
{1, 1, 2, 3, 3, 2, 1, 1},
{1, 1, 2, 3, 4, 3, 2, 1, 1},
{1, 1, 2, 3, 4, 4, 3, 2, 1, 1},
{1, 1, 2, 3, 4, 6, 4, 3, 2, 1, 1}

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A139147, A000930. Row sums are in A238383.

A000930 := proc(n) coeftayl( 1/(1-x-x^3),x=0,n) ; end: A139040 := proc(n,m) A000930(min(m,n+1-m)) ; end: for n from 1 to 16 do for m from 1 to n do printf("%d,",A139040(n,m)) ; od: od: # R. J. Mathar, Jun 08 2008

a[-2]=0;a[-1]=1;a[0]=1;a[n_]:=a[n]=a[n-1]+a[n-3];(*A000930*)
g[n_,m_]:=If[m <= Floor[n/2],a[m],a[n-m]];w=Table[Table[g[n,m],{m,0,n}],{n,0,10}]; Flatten[w]

Edited and corrected by N. J. A. Sloane, Jun 30 2008

Corrected by Philippe Deléham, Feb 25 2014

A145580 Eigentriangle, row sums = A000930.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, -1, 0, 2, 3, 0, -1, 0, 3, 4, 1, 0, -2, 0, 4, 6, 0, 1, 0, -3, 0, 6, 9, -1, 0, 2, 0, -4, 0, 9, 13, 0, -1, 0, 3, 0, -6, 0, 13, 19, 1, 0, -2, 0, 4, 0, -9, 0, 19, 28

Offset: 1

Gary W. Adamson, Oct 13 2008

Row sums = A000930 starting with offset 1: (1, 1, 2, 3, 4, 6, 9, 13, 19,...).

Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
0, 1, 2;
-1, 0, 2, 3;
0, -1, 0, 3, 4;
1, 0, -2, 0, 4, 6;
0, 1, 0, -3, 0, 6, 9;
-1, 0, 2, 0, -4, 0, 9, 13;
0, -1, 0, 3, 0, -6, 0, 13, 19;
1, 0, -2, 0, 4, 0, -9, 0, 19, 28;
...
Row 6 = (1, 0, -2, 0, 4, 6) = termwise products of (1, 0, -1, 0, 1, 1) and (1, 1, 2, 3, 4, 6).

A000930

Let M = an infinite lower triangular matrix with (1, 1, 0, -1, 0, 1, 0, -1, 0, 1,...) in every column; and X = an infinite lower triangular matrix with A000930 as the main diagonal (offset 1): (1, 1, 2, 3, 4, 6, 9, 13, 19,...) and the rest zeros. Triangle A145580 = M * X.

A170954 Indices k such that A000930(k) is prime.

Original entry on oeis.org

3, 4, 8, 9, 11, 16, 21, 25, 81, 6241, 25747

Offset: 1

N. J. A. Sloane, Feb 25 2010, following a question from Kyle Ledbetter

a(12) > 200000. - Donovan Johnson, Mar 14 2010

a(12) > 500000. - Michael S. Branicky, Nov 23 2024

Two more terms from R. J. Mathar, Mar 01 2010

a(11) from Donovan Johnson, Mar 14 2010

cp's OEIS Frontend

A002478 Bisection of A000930.

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A227835 3^a(n) is the highest power of 3 dividing A000930(n).

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A350311 Replace 2^k in the binary expansion of n with A000930(k+2), Narayana's cows sequence.

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A108104 Sequence A000930 with terms repeated.

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A231620 a(n) = A000930(n^2), where A000930 is Narayana's cows sequence.

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A271901 Length of period of Narayana sequence A000930 modulo n-th prime.

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A271953 a(n) is the period of A000930 modulo n.

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