A008776 -id:A008776 - cp's OEIS frontend

A009966 Powers of 22.

1, 22, 484, 10648, 234256, 5153632, 113379904, 2494357888, 54875873536, 1207269217792, 26559922791424, 584318301411328, 12855002631049216, 282810057883082752, 6221821273427820544, 136880068015412051968, 3011361496339065143296, 66249952919459433152512, 1457498964228107529355264, 32064977213018365645815808, 705429498686404044207947776

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 22), L(1, 22), P(1, 22), T(1, 22). Essentially same as Pisot sequences E(22, 484), L(22, 484), P(22, 484), T(22, 484). 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 22-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

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 (22).

Cf. A000079, A001020, A008776, A009988.

[22^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

NestList[22#&,1,20] (* Harvey P. Dale, Apr 22 2022 *)

a(n)=22^n \\ Charles R Greathouse IV, Oct 07 2015

[lucas_number1(n,22,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-22*x). - Philippe Deléham, Nov 23 2008
a(n) = 22^n; a(n) = 22*a(n-1) n>0 a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(22*x).
a(n) = A000079(n)*A001020(n) = A009988(n)/A000079(n). (End)

A009981 Powers of 37.

Original entry on oeis.org

1, 37, 1369, 50653, 1874161, 69343957, 2565726409, 94931877133, 3512479453921, 129961739795077, 4808584372417849, 177917621779460413, 6582952005840035281, 243569224216081305397, 9012061295995008299689

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 37), L(1, 37), P(1, 37), T(1, 37). Essentially same as Pisot sequences E(37, 1369), L(37, 1369), P(37, 1369), T(37, 1369). 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 37-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(37*n) = 37*n + sigma(n). - Jahangeer Kholdi, Nov 23 2013

C. W. Trigg, The Powers of 37, Journal of Recreational Mathematics, Vol. 12:3 (1979-80), 186-191.

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 (37).

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

37^Range[0, 20] (* or *)
NestList[37*# &, 1, 20] (* Paolo Xausa, Feb 21 2024 *)

a(n)=37^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 1/(1-37*x). - Philippe Deléham, Nov 24 2008

a(n)=37^n; a(n)=37*a(n-1) n>0 a(0)=1. - Vincenzo Librandi, Nov 21 2010

Reference added by William Rex Marshall, Nov 13 2010

A168607 a(n) = 3^n + 2.

Original entry on oeis.org

3, 5, 11, 29, 83, 245, 731, 2189, 6563, 19685, 59051, 177149, 531443, 1594325, 4782971, 14348909, 43046723, 129140165, 387420491, 1162261469, 3486784403, 10460353205, 31381059611, 94143178829, 282429536483, 847288609445

Offset: 0

Vincenzo Librandi, Dec 01 2009

Second bisection is A134752.
It appears that if s(n) is a first order rational sequence of the form s(1)=5, s(n)= (2*s(n-1)+1)/(s(n-1)+2),n>1, then s(n)= a(n)/(a(n)-4), n>1. - Gary Detlefs, Nov 16 2010
Mahler exhibits this sequence with n>=1 as a proof that there exists an infinite number of x coprime to 3, such that x belongs to A125293 and x^2 belongs to A005836. - Michel Marcus, Nov 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.
Kurt Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A008776 (2*3^n), A005051 (8*3^n), A034472 (3^n+1), A000244 (powers of 3), A024023 (3^n-1), A168609 (3^n+4), A168610 (3^n+5), A134752 (3^(2*n-1)+2).

Magma
```
[3^n+2: n in [0..30]];
```

A168607:=n->3^n + 2; seq(A168607(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

CoefficientList[Series[(3 - 7 x)/((1-x) (1-3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
NestList[3 # - 4 & , 3, 25] (* Bruno Berselli, Feb 06 2013 *)

a(n)=3^n+2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*a(n-1) - 4, a(0) = 3.
a(n+1) - a(n) = A008776(n).
a(n+2) - a(n) = A005051(n).
a(n) = A034472(n)+1 = A000244(n)+2 = A024023(n)+3 = A168609(n)-2 = A168610(n)-3.
G.f.: (3 - 7*x)/((1 - x)*(1 - 3*x)).
a(n) = 4*a(n-1) - 3*a(n-2), a(0) = 3, a(1) = 5. - Vincenzo Librandi, Feb 06 2013
E.g.f.: exp(3*x) + 2*exp(x). - Elmo R. Oliveira, Nov 09 2023

Edited by Klaus Brockhaus, Apr 13 2010

Further edited by N. J. A. Sloane, Aug 10 2010

A020707 Pisot sequences E(4,8), L(4,8), P(4,8), T(4,8).

Original entry on oeis.org

4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

David W. Wilson

Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..238
Tanya Khovanova, Recursive Sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Subsequence of A000079. See A008776 for definitions of Pisot sequences.

Cf. A051916.

[2^(n+2): n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

2^(Range[0, 50] + 2) (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

a(n)=4<Charles R Greathouse IV, Apr 08 2012

def A020707(n): return 1<Chai Wah Wu, Apr 02 2025

a(n) = 2^(n+2).
a(n) = 2*a(n-1).
G.f.: 4/(1-2*x). - Philippe Deléham, Nov 23 2008
E.g.f.: 4*exp(2*x). - Stefano Spezia, May 15 2021

A009985 Powers of 41.

Original entry on oeis.org

1, 41, 1681, 68921, 2825761, 115856201, 4750104241, 194754273881, 7984925229121, 327381934393961, 13422659310152401, 550329031716248441, 22563490300366186081, 925103102315013629321, 37929227194915558802161

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 41), L(1, 41), P(1, 41), T(1, 41). Essentially same as Pisot sequences E(41, 1681), L(41, 1681), P(41, 1681), T(41, 1681). 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 41-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(41*n) = 41*n + sigma(n). - Jahangeer Kholdi, Nov 23 2013

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 (41).

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

41^Range[0, 14] (* Alonso del Arte, Sep 04 2015 *)
NestList[41#&,1,20] (* Harvey P. Dale, Mar 07 2017 *)

first(m)=vector(m,i,i--;41^i) \\ Anders Hellström, Sep 12 2015

G.f.: 1/(1-41*x). - Philippe Deléham, Nov 24 2008

a(n) = 41^n; a(n) = 41*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010

A009992 Powers of 48: a(n) = 48^n.

Original entry on oeis.org

1, 48, 2304, 110592, 5308416, 254803968, 12230590464, 587068342272, 28179280429056, 1352605460594688, 64925062108545024, 3116402981210161152, 149587343098087735296, 7180192468708211294208, 344649238497994142121984, 16543163447903718821855232

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 48), L(1, 48), P(1, 48), T(1, 48). Essentially same as Pisot sequences E(48, 2304), L(48, 2304), P(48, 2304), T(48, 2304). See A008776 for definitions of Pisot sequences.
If X_1, X_2, ..., X_n is a partition of the set {1,2,...,2*n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1,2,..., 2*n}->{1,2,3,4,5,6,7} such that for fixed y_1,y_2,...,y_n in {1,2,3,4,5,6,7} we have f(X_i)<>{y_i}, (i=1,2,...,n). - Milan Janjic, May 24 2007
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 48-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (48).

Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009991 (powers of 47), A087752 (powers of 49).

Cf. A000079 (2^n), A000244 (3^n), A000302 (4^n), A000400 (6^n), A001018 (8^n), A001021 (12^n), A001025 (16^n), A009968 (24^n).

List([0..20],n->48^n); # Muniru A Asiru, Nov 21 2018

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

A009992 := n -> 48^n: seq(A009992(n), n=0..20); # M. F. Hasler, Apr 19 2015

48^Range[0, 15] (* Michael De Vlieger, Jan 13 2018 *)

A009992(n)=48^n \\ M. F. Hasler, Apr 19 2015

for n in range(0,20): print(48**n, end=', ') # Stefano Spezia, Nov 21 2018

[(48)^n for n in range(20)] # G. C. Greubel, Nov 21 2018

G.f.: 1/(1-48*x). - Philippe Deléham, Nov 24 2008
a(n) = 48^n; a(n) = 48*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
E.g.f.: exp(48*x). - Muniru A Asiru, Nov 21 2018

Edited by M. F. Hasler, Apr 19 2015

A020701 Pisot sequences E(3,5), P(3,5).

Original entry on oeis.org

3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141

Offset: 0

David W. Wilson

Number of meaningful differential operations of the (n+1)-th order on the space R^3. - Branko Malesevic, Feb 29 2004

Pisano period lengths: A001175. - R. J. Mathar, Aug 10 2012

			Meaningful second-order differential operations appear in the form of five compositions as follows: 1. div grad f 2. curl curl F 3. grad div F 4. div curl F (=0) 5. curl grad f (=0)
Meaningful third-order differential operations appear in the form of eight compositions as follows: 1. grad div grad f 2. curl curl curl F 3. div grad div F 4. div curl curl F (=0) 5. div curl grad f (=0) 6. curl curl grad f (=0) 7. curl grad div F (=0) 8. grad div curl F (=0)

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
Branko Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A020695 and hence A000045. See A008776 for definitions of Pisot sequences.

Cf. A039834, A020695, A071679, A116183.

[Fibonacci(n-4): n in [8..80]]; // Vincenzo Librandi, Jul 12 2015

CoefficientList[Series[(-2 z - 3)/(z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1,1},{3,5},40] (* Harvey P. Dale, Apr 22 2013 *)

a(n)=fibonacci(n+4) \\ Charles R Greathouse IV, Jan 17 2012

a(n) = Fib(n+4). a(n) = a(n-1) + a(n-2).
a(n) = A020695(n+1). - R. J. Mathar, May 28 2008
G.f.: (3+2*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-7+3*sqrt(5))+(1+sqrt(5))^n*(7+3*sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016
E.g.f.: (7*sqrt(5)*sinh(sqrt(5)*x/2) + 15*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, Jun 05 2016

A087752 Powers of 49.

Original entry on oeis.org

1, 49, 2401, 117649, 5764801, 282475249, 13841287201, 678223072849, 33232930569601, 1628413597910449, 79792266297612001, 3909821048582988049, 191581231380566414401, 9387480337647754305649, 459986536544739960976801, 22539340290692258087863249, 1104427674243920646305299201

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 02 2003

Same as Pisot sequences E(1, 49), L(1, 49), P(1, 49), T(1, 49). Essentially same as Pisot sequences E(49, 2401), L(49, 2401), P(49, 2401), T(49, 2401). 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 49-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

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 (49).

Bisection of A000420.
Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48).
Cf. A005843, A008776.

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

49^Range[0,20] (* or *) Join[{1},NestList[49#&,49,20]] (* Harvey P. Dale, May 10 2019 *)

makelist(49^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=49^n \\ M. F. Hasler, Apr 19 2015
```

G.f.: 1/(1-49*x). - Philippe Deléham, Nov 24 2008
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 49^n.
a(n) = 49*a(n-1), a(0)=1. (End)
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(49*x).
a(n) = A000420(A005843(n)). (End)

Edited by M. F. Hasler, Apr 19 2015

A275037 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (0,-1) (-1,-2) or (-2,0) and new values introduced in order 0..2.

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 4, 16, 18, 6, 8, 48, 46, 54, 12, 16, 144, 137, 146, 162, 24, 32, 432, 401, 570, 450, 486, 48, 64, 1296, 1152, 2185, 2334, 1428, 1458, 96, 128, 3888, 3336, 8273, 11624, 9774, 4463, 4374, 192, 256, 11664, 9720, 31495, 57984, 64045, 40572, 14086

Offset: 1

R. H. Hardin, Jul 14 2016

Table starts
...1.....1......2.......4........8.........16..........32...........64
...2.....6.....16......48......144........432........1296.........3888
...3....18.....46.....137......401.......1152........3336.........9720
...6....54....146.....570.....2185.......8273.......31495.......120845
..12...162....450....2334....11624......57984......290927......1465435
..24...486...1428....9774....64045.....421793.....2788663.....18527905
..48..1458...4463...40572...351197....3056856....26631796....233894638
..96..4374..14086..169348..1931619...22207559...255796423...2976710318
.192.13122..44195..704704.10581148..160767947..2451632155..37779884891
.384.39366.139130.2937764.58059378.1166126087.23548292980.480461388679

			Some solutions for n=4 k=4
..0..1..2..1. .0..1..2..1. .0..1..0..1. .0..1..0..2. .0..1..0..2
..1..0..1..2. .2..0..1..0. .2..0..2..0. .0..1..2..0. .0..2..1..0
..1..2..0..2. .1..0..1..2. .2..0..1..2. .1..2..1..0. .1..0..2..0
..0..1..2..0. .1..2..0..1. .1..2..1..2. .1..2..0..1. .2..0..2..1

R. H. Hardin, Table of n, a(n) for n = 1..338

Column 1 is A003945(n-2).
Column 2 is A008776(n-1).
Row 1 is A000079(n-2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: a(n) = 3*a(n-1) for n>2
k=3: [order 16] for n>17
k=4: [order 35] for n>37
k=5: [order 70] for n>74
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 3*a(n-1) for n>3
n=3: [order 9] for n>10
n=4: [order 17] for n>19
n=5: [order 28] for n>32
n=6: [order 67] for n>72

A005051 a(n) = 8*3^n.

Original entry on oeis.org

8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616, 2259436291848, 6778308875544, 20334926626632

Offset: 0

N. J. A. Sloane

For n>=3, a(n-3) is equal to the number of functions f:{1,2,...,n}->{1,2,3} such that for fixed, different x_1, x_2, x_3 in {1,2,...,n} and fixed y_1, y_2, y_3 in {1,2,3} we have f(x_i)<>y_i, (i=1,2,3). - Milan Janjic, May 13 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A000244, A008776.

[8*3^n: n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

8*3^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

a(n)=8*3^n \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 3*a(n-1). G.f.: 8/(1-3*x). - Colin Barker, Jul 02 2012
From Elmo R. Oliveira, Aug 16 2024: (Start)
E.g.f.: 8*exp(3*x).
a(n) = 8*A000244(n) = 4*A008776(n). (End)

cp's OEIS Frontend

A009966 Powers of 22.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A009981 Powers of 37.

Views

Author

Keywords

Comments

References

Links

Programs

Magma

Mathematica

PARI

Formula

Extensions

A168607 a(n) = 3^n + 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A020707 Pisot sequences E(4,8), L(4,8), P(4,8), T(4,8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A009985 Powers of 41.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

A009992 Powers of 48: a(n) = 48^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage