A001027 -id:A001027 - cp's OEIS frontend

A060221 Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).

18, 153, 1938, 26163, 377910, 5667681, 87460002, 1377481950, 22039920504, 357046533675, 5842582734474, 96402612275775, 1601766528128550, 26772383354990049, 449776041098370870, 7589970692848393200, 128583032925805678350, 2185911559727674682148, 37275544492386193492506

Offset: 1

Thomas Ward, Mar 21 2001

Number of Lyndon words (aperiodic necklaces) with n beads of 18 colors. - Andrew Howroyd, Dec 10 2017

			a(2)=153 since there are 324 points of period 2 in the full 18-shift and 18 fixed points, so there must be (324-18)/2 = 153 orbits of length 2.

G. C. Greubel, Table of n, a(n) for n = 1..792
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 18 of A074650.

Cf. A001027, A008683.

A060221:= func< n | (1/n)*(&+[MoebiusMu(d)*(18)^Floor(n/d): d in Divisors(n)]) >;
[A060221(n): n in [1..40]]; // G. C. Greubel, Sep 13 2024

A060221[n_]:= DivisorSum[n, (18)^(n/#)*MoebiusMu[#] &]/n;
Table[A060221[n], {n, 40}] (* G. C. Greubel, Sep 13 2024 *)

a001027(n) = 18^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001027(n/d)); \\ Michel Marcus, Sep 11 2017

def A060221(n): return (1/n)*sum(moebius(k)*(18)^(n/k) for k in (1..n) if (k).divides(n))
[A060221(n) for n in range(1,41)] # G. C. Greubel, Sep 13 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001027(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 18*x^k))/k. - Ilya Gutkovskiy, May 20 2019

More terms from Michel Marcus, Sep 11 2017

A218721 a(n) = (18^n-1)/17.

Original entry on oeis.org

0, 1, 19, 343, 6175, 111151, 2000719, 36012943, 648232975, 11668193551, 210027483919, 3780494710543, 68048904789775, 1224880286215951, 22047845151887119, 396861212733968143, 7143501829211426575, 128583032925805678351

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 18 (A001027), q-integers for q=18: diagonal k=1 in triangle A022182.
Partial sums are in A014901. Also, the sequence is related to A014935 by A014935(n) = n*a(n) - Sum_{i=0..n-1} a(i), for n>0. - Bruno Berselli, Nov 06 2012
From Bernard Schott, May 06 2017: (Start)
Except for 0, 1 and 19, all terms are Brazilian repunits numbers in base 18, and so belong to A125134. From n = 3 to n = 8286, all terms are composite. See link "Generalized repunit primes".
As explained in the extensions of A128164, a(25667) = (18^25667 - 1)/17 would be (is) the smallest prime in base 18. (End)

			a(3) = (18^3 - 1)/17 = 343 = 7 * 49; a(6) = (18^6 - 1)/17 = 2000719 = 931 * 2149. - _Bernard Schott_, May 01 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (19,-18).

Cf. A000225, A001027, A002275, A002450, A002452, A003462, A003463, A003464, A014901, A014935, A016123, A016125, A022182, A023000, A023001, A064108, A091030, A091045, A094028, A125134, A128164, A131865, A135518, A135519, A218722, A218724, A218733, A218743, A218752.

[n le 2 select n-1 else 19*Self(n-1)-18*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{19, -18}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[18^Range[0,20]]] (* Harvey P. Dale, Nov 08 2012 *)

A218721(n):=(18^n-1)/17$ makelist(A218721(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218721(n)=18^n\17
```

a(n) = floor(18^n/17).
G.f.: x/((1-x)*(1-18*x)). - Bruno Berselli, Nov 06 2012
a(n) = 19*a(n-1) - 18*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(x)*(exp(17*x) - 1)/17. - Stefano Spezia, Mar 11 2023

A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).

Original entry on oeis.org

2, 6, 54, 564, 6390, 76356, 948276, 12132504, 158984694, 2124923460, 28877309604, 398046897144, 5554209125556, 78328566695736, 1114923122685720, 15999482238880464, 231253045986317814, 3363838379489630916

Offset: 0

Jason Kimberley, Nov 25 2012

This sequence is s_18 in Cooper's paper.
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..830 (terms 0..254 from Jason Kimberley)
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See s18 p. 3.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Magma
```
s_18 := func where C is Binomial;
```

Table[Sum[(-1)^j*Binomial[n,j]*Binomial[2j,j]*Binomial[2n-2j, n-j]* (Binomial[2n-3j-1,n] +Binomial[2n-3j,n]), {j,0,Floor[n/3]}], {n,0,20}] (* G. C. Greubel, Oct 24 2017 *)

{a(n) = sum(j=0,floor(n/3), (-1)^j*binomial(n,j)*binomial(2*j,j)* binomial(2*n-2*j,n-j)*(binomial(2*n-3*j-1,n) +binomial(2*n-3*j,n)))}; \\ G. C. Greubel, Apr 02 2019

[sum((-1)^j*binomial(n,j)*binomial(2*j,j)*binomial(2*n-2*j,n-j)* (binomial(2*n-3*j-1,n)+binomial(2*n-3*j,n)) for j in (0..floor(n/3))) for n in (0..20)] # G. C. Greubel, Apr 02 2019

1/Pi
= 2*3^(-5/2) Sum {k>=0} (n a(n)/18^n) [Cooper, equation (42)]
= 2*3^(-5/2) Sum {k>=0} (n a(n)/A001027(n)).
G.f.: 1+hypergeom([1/8, 3/8],[1],256*x^3/(1-12*x)^2)^2/sqrt(1-12*x). - Mark van Hoeij, May 07 2013
Conjecture D-finite with recurrence: n^3*a(n) -2*(2*n-1)*(7*n^2-7*n+3)*a(n-1) +12*(4*n-5)*(n-1)* (4*n-3)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ 3 * 2^(4*n + 1/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A009980 Powers of 36.

Original entry on oeis.org

1, 36, 1296, 46656, 1679616, 60466176, 2176782336, 78364164096, 2821109907456, 101559956668416, 3656158440062976, 131621703842267136, 4738381338321616896, 170581728179578208256, 6140942214464815497216, 221073919720733357899776, 7958661109946400884391936

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 36), L(1, 36), P(1, 36), T(1, 36). Essentially same as Pisot sequences E(36, 1296), L(36, 1296), P(36, 1296), T(36, 1296). See A008776 for definitions of Pisot sequences.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 36-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
See David Applegate's comment in A000244 from Feb 20 2017 for a proof of Janjic's assertion. - Alonso del Arte, Sep 03 2017

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (36).

Cf. A000079, A000244, A000400, A001027, A005843, A008776.

[36^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

36^Range[0, 20]  (* Harvey P. Dale, Mar 04 2011 *)

a(n)=36^n \\ Charles R Greathouse IV, Nov 18 2011

G.f.: 1/(1-36*x). - Philippe Deléham, Nov 24 2008
a(n) = 36^n; a(n) = 36*a(n-1) for n > 0, a(0) = 1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(36*x).
a(n) = A000079(n)*A001027(n) = A000400(A005843(n)). (End)

A125818 a(n) = ((1 + 3sqrt(2))^n + (1 - 3sqrt(2))^n)/2.

Original entry on oeis.org

1, 1, 19, 55, 433, 1801, 10963, 52543, 291457, 1476145, 7907059, 40908583, 216237169, 1127920249, 5931872371, 31038388975, 162918608257, 853489829089, 4476595998547, 23462519091607, 123027170158513, 644917164874345, 3381296222443411, 17726184247750687

Offset: 1

Artur Jasinski, Dec 10 2006

nonn

Binomial transform of [1, 0, 18, 0, 324, 0, 5832, 0, 104976, 0, ...] =: powers of 18 (A001027) with interpolated zeros. - Philippe Deléham, Dec 02 2008

a(n-1) is the number of compositions of n when there are 1 type of 1 and 18 types of other natural numbers. - Milan Janjic, Aug 13 2010

T. D. Noe, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (2, 17).

Cf. A125817.

a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+17*a[n-2]; od; a; # G. C. Greubel, Aug 03 2019

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +17*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 03 2019

Expand[Table[((1+3*Sqrt[2])^n +(1-3*Sqrt[2])^n)/2, {n,0,30}]]
(* alternate program *)
LinearRecurrence[{2, 17}, {1, 1}, 30] (* T. D. Noe, Mar 28 2012 *)

my(x='x+O('x^30)); Vec((1-x)/(1-2*x-17*x^2)) \\ G. C. Greubel, Aug 03 2019

((1-x)/(1-2*x-17*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

From Philippe Deléham, Dec 12 2006: (Start)
a(n) = 2*a(n-1) + 17*a(n-2), with a(0)=a(1)=1.
G.f.: (1-x)/(1-2*x-17*x^2). (End)
a(n) = Sum_{k=0..n} A098158(n,k)*18^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=18, (i>1), and if A is Hessenberg matrix of order n If p[1]=1, and p[i]=18, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n+1)=det A. - Milan Janjic, Apr 29 2010

A067421 Sixth column of triangle A067417.

Original entry on oeis.org

1, 8, 144, 2592, 46656, 839808, 15116544, 272097792, 4897760256, 88159684608, 1586874322944, 28563737812992, 514147280633856, 9254651051409408, 166583718925369344, 2998506940656648192

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (18).

Cf. A067420 (fifth column), A067422 (seventh column), A001027 (powers of 18).

[Ceiling(8*(3*6)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+5, 5).
a(n) = 8*(3*6)^(n-1), n >= 1, a(0)=1.
G.f.: (1-10*x)/(1-18*x).

A013723 a(n) = 18^(2*n + 1).

Original entry on oeis.org

18, 5832, 1889568, 612220032, 198359290368, 64268410079232, 20822964865671168, 6746640616477458432, 2185911559738696531968, 708235345355337676357632, 229468251895129407139872768

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (324).

Bisection of A001027 (18^n).

Cf. A013708-A013722, A013724-A013729.

[18^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(18^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

18^(2*Range[0,20]+1) (* or *) NestList[324#&,18,20] (* Harvey P. Dale, Sep 28 2021 *)

a(n)=18^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 324*a(n-1); a(0)=18.
G.f.: 18/(1-324*x). (End)

A180703 Smallest power of 18 that begins with n.

Original entry on oeis.org

1, 20822964865671168, 324, 4130428534112329328517709824, 5832, 612220032, 708235345355337676357632, 819308872942260126404286866009182175232, 9028751479390699717312017900815782025058563653632, 104976, 11019960576

Offset: 1

Daniel Mondot, Sep 18 2010

Cf. A001027, A180701, A180704, A180705.

A038299 Triangle whose (i,j)-th entry is binomial(i,j)9^(i-j)9^j.

Original entry on oeis.org

1, 9, 9, 81, 162, 81, 729, 2187, 2187, 729, 6561, 26244, 39366, 26244, 6561, 59049, 295245, 590490, 590490, 295245, 59049, 531441, 3188646, 7971615, 10628820, 7971615, 3188646, 531441, 4782969, 33480783, 100442349, 167403915

Offset: 0

N. J. A. Sloane

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Row sums: sum_{j=0..i} T(i,j) = A001027(i). [From R. J. Mathar, Mar 30 2009]

cp's OEIS Frontend

A060221 Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).

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A218721 a(n) = (18^n-1)/17.

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A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).

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A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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A009980 Powers of 36.

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A125818 a(n) = ((1 + 3*sqrt(2))^n + (1 - 3*sqrt(2))^n)/2.

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A125818 a(n) = ((1 + 3sqrt(2))^n + (1 - 3sqrt(2))^n)/2.

A038299 Triangle whose (i,j)-th entry is binomial(i,j)9^(i-j)9^j.