A133494 -id:A133494 - cp's OEIS frontend

A334603 Period of the fraction 1/11^n for n >= 1.

2, 22, 242, 2662, 29282, 322102, 3543122, 38974342, 428717762, 4715895382, 51874849202, 570623341222, 6276856753442, 69045424287862, 759499667166482, 8354496338831302, 91899459727144322, 1010894056998587542, 11119834626984462962, 122318180896829092582

Offset: 1

Bernard Schott, May 07 2020

Conjecture proposed by the authors in References page 205: if p is a prime with gcd(p,30) = 1 and if the period of 1/p is m then the period of 1/p^n is m*p^(n-1).

			1/121 = 0. 0082644628099173553719 0082644628099173553719 ... with periodic part {0082644628099173553719}, whose length is 22 digits, so a(2) = 22.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 346 pp. 50, 204-205, Ellipses, Paris 2004.

Cf. period of fractions: A051626 (1/n), A133494 (1/3^n), A055272 (1/7^n).

Cf. A001020 (11^n).

MultiplicativeOrder[10, 11^#] & /@ Range[20] (* Giovanni Resta, May 07 2020 *)

a(n) = znorder(Mod(10, 11^n)); \\ Michel Marcus, May 09 2020

a(n) = 2 * 11^(n-1) [conjectured, see comments].

a(n) = A051626(A001020(n)).

More terms from Giovanni Resta, May 07 2020

A358138 Difference between maximum and minimum part in the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 2, 1, 1, 0, 0, 3, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 0, 4, 2, 3, 0, 2, 2, 2, 2, 2, 0, 1, 2, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 0, 5, 3, 4, 1, 3, 3, 3, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 1, 2, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 31 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The first and last parts are A065120 and A001511, difference A358135.
This is the maximum minus minimum part in row n of A066099.
The version for Heinz numbers of partitions is A243055.
The maximum and minimum parts are A333766 and A333768.
The partial sums of standard compositions are A358134, adjusted A242628.
A011782 counts compositions.
A351014 counts distinct runs in standard compositions.
Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A329395, A357187, A358133.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Max[stc[n]]-Min[stc[n]],{n,1,100}]

a(n) = A333766(n) - A333768(n).

A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 15, 32, 31, 63, 56, 142, 101, 240, 211, 467, 297, 985, 490, 1524, 1247, 2542, 1255, 6371, 1979, 7486, 7070, 14128, 4565, 32953, 6842, 42229, 37863, 56266, 17887, 192914, 21637, 145820, 197835, 371853, 44583, 772740, 63261, 943966, 1124840

Offset: 0

Gus Wiseman, Dec 14 2022

nonn

			The a(1) = 1 through a(6) = 14 sets:
  {(1)}  {(2)}   {(3)}    {(4)}       {(5)}      {(6)}
         {(11)}  {(21)}   {(22)}      {(32)}     {(33)}
                 {(111)}  {(31)}      {(41)}     {(42)}
                          {(211)}     {(221)}    {(51)}
                          {(1111)}    {(311)}    {(222)}
                          {(2),(11)}  {(2111)}   {(321)}
                                      {(11111)}  {(411)}
                                                 {(2211)}
                                                 {(3111)}
                                                 {(21111)}
                                                 {(111111)}
                                                 {(3),(21)}
                                                 {(3),(111)}
                                                 {(21),(111)}

This is the constant-sum case of A261049, ordered A358906.
The version for all different sums is A271619, ordered A336342.
Allowing repetition gives A305551, ordered A279787.
The version for compositions instead of partitions is A358904.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A098407 counts sets of compositions, ordered A358907.
Cf. A000005, A000041, A038041, A055887, A063834, A074854, A289078, A304961, A305552, A306017.

Table[If[n==0,1,Sum[Binomial[PartitionsP[d],n/d],{d,Divisors[n]}]],{n,0,50}]

a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ Michel Marcus, Dec 14 2022

a(n) = Sum_{d|n} binomial(A000041(d),n/d).

A367631 Triangle read by rows: T(n,k) is the number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 4, 0, 0, 0, 5, 3, 0, 0, 0, 2, 14, 0, 0, 0, 0, 0, 23, 9, 0, 0, 0, 0, 0, 16, 48, 0, 0, 0, 0, 0, 0, 4, 97, 27, 0, 0, 0, 0, 0, 0, 0, 94, 162, 0, 0, 0, 0, 0, 0, 0, 0, 44, 387, 81, 0, 0, 0, 0, 0, 0, 0, 0, 8, 476, 540, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 320, 1485, 243, 0, 0, 0, 0, 0, 0

Offset: 0

Tian Han, Nov 24 2023

Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k. A descent in a permutation a(1)a(2)...a(n) is position i such that a(i) > a(i+1).

			Triangle T(n,k) begins:
  1;
  1, 0;
  1, 1,  0;
  0, 4,  0,  0;
  0, 5,  3,  0,   0;
  0, 2, 14,  0,   0,    0;
  0, 0, 23,  9,   0,    0,   0;
  0, 0, 16, 48,   0,    0,   0, 0;
  0, 0,  4, 97,  27,    0,   0, 0, 0;
  0, 0,  0, 94, 162,    0,   0, 0, 0, 0;
  0, 0,  0, 44, 387,   81,   0, 0, 0, 0, 0;
  0, 0,  0,  8, 476,  540,   0, 0, 0, 0, 0, 0;
  0, 0,  0,  0, 320, 1485, 243, 0, 0, 0, 0, 0, 0;
  ...

Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023. See formula 7 at page 7.

Row sums give A011782.
Column sums give 3*A005054.
T(2n,n) gives A133494.
T(3n+2,n) gives A000079.
T(3n+1,n) gives A053220(n+1).

G.f.: (1 + x + x^2 - 2*x^2*z - x^3*z)/(1 - 3*x^2*z - 2*x^3*z).

A034745 Dirichlet convolution of Fibonacci numbers with 3^(n-1).

Original entry on oeis.org

1, 4, 11, 33, 86, 266, 742, 2244, 6613, 19834, 59138, 177639, 531674, 1595468, 4783786, 14352225, 43048318, 129149968, 387424670, 1162288458, 3486796922, 10460430230, 31381088266, 94143408282, 282429611911, 847289262976

Offset: 1

Erich Friedman

nonn

Cf. A000045, A133494.

Table[Sum[Fibonacci[n/d]*3^(d - 1), {d, Divisors[n]}], {n, 1, 25}] (* Vaclav Kotesovec, Sep 10 2019 *)

G.f.: Sum_{k>=1} 3^(k-1) * x^k/(1 - x^k - x^(2*k)). - Ilya Gutkovskiy, Jul 24 2019

a(n) ~ 3^(n-1). - Vaclav Kotesovec, Sep 11 2019

A155734 Binomial transform of A154879.

Original entry on oeis.org

3, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329

Offset: 0

Paul Curtz, Jan 26 2009

Binomial transform of the third differences of A001045.
The binomial transform of the first differences of A001045 is in A133494.
The binomial transform of the 2nd differences of A001045 is in A133494, with the sign of A133494(0) flipped.
The binomial transform of the p-th differences of A001045 is the number A077925(p-1) followed by A000244.

Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A154879, A078008. Essentially the same as A140429 and A000244.

read("transforms") ; A001045 := proc(n) option remember ; if n <= 1 then n; else procname(n-1)+2*procname(n-2) ; fi; end:
a001045 := [seq(A001045(n),n=0..80) ] ; a154879 := DIFF(DIFF(DIFF(a001045))) ; BINOMIAL(a154879) ; # R. J. Mathar, Jul 23 2009

From Colin Barker, Apr 05 2012: (Start)
a(n) = 3*a(n-1) for n > 1.
G.f.: (3-8*x)/(1-3*x). (End)
G.f.: (1 - 2/G(0))/x where G(k) = 1 + 2^k/(1 - 2*x/(2*x + 2^k/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012

Edited and extended by R. J. Mathar, Jul 23 2009

A247936 Riordan array ((1-2x)/(1-3x), 2x).

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 9, 6, 4, 8, 27, 18, 12, 8, 16, 81, 54, 36, 24, 16, 32, 243, 162, 108, 72, 48, 32, 64, 729, 486, 324, 216, 144, 96, 64, 128, 2187, 1458, 972, 648, 432, 288, 192, 128, 256, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 512, 19683, 13122

Offset: 0

Philippe Deléham, Sep 26 2014

Mirror of A153279.

			Triangle begins:
1
1, 2
3, 2, 4
9, 6, 4, 8
27, 18, 12, 8, 16
81, 54, 36, 24, 16, 32
243, 162, 108, 72, 48, 32, 64
Production matrix begins:
1, 2
1, 0, 2
1, 0, 0, 2
1, 0, 0, 0, 2
1, 0, 0, 0, 0, 2
1, 0, 0, 0, 0, 0, 2
1, 0, 0, 0, 0, 0, 0, 2

Cf. A000079, A000244, A133494, A153279, A167747.

Sum_{k, 0<=k<=n} T(n,k) = 3^n = A000244(n).
T(n,k) = A133494(n-k)*2^k.
T(2n,n) = A167747(n).

A336129 Number of strict compositions of divisors of n.

Original entry on oeis.org

1, 2, 4, 5, 6, 16, 14, 24, 31, 64, 66, 120, 134, 208, 360, 459, 618, 894, 1178, 1622, 2768, 3364, 4758, 6432, 8767, 11440, 15634, 24526, 30462, 42296, 55742, 75334, 98112, 131428, 168444, 258403, 315974, 432244, 558464, 753132, 958266, 1280840, 1621274

Offset: 1

Gus Wiseman, Jul 11 2020

nonn

A strict composition of k is a finite sequence of distinct positive integers summing to k.

			The a(1) = 1 through a(7) = 14 compositions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (1,2)  (4)    (1,4)  (3)      (1,6)
            (2,1)  (1,3)  (2,3)  (6)      (2,5)
                   (3,1)  (3,2)  (1,2)    (3,4)
                          (4,1)  (1,5)    (4,3)
                                 (2,1)    (5,2)
                                 (2,4)    (6,1)
                                 (4,2)    (1,2,4)
                                 (5,1)    (1,4,2)
                                 (1,2,3)  (2,1,4)
                                 (1,3,2)  (2,4,1)
                                 (2,1,3)  (4,1,2)
                                 (2,3,1)  (4,2,1)
                                 (3,1,2)
                                 (3,2,1)

Compositions of divisors are A034729.
Strict partitions of divisors are A047966.
Partitions of divisors are A047968.
Cf. A000005, A001970, A006951, A133494, A318683, A336128, A336130, A336132.

Table[Sum[Length[Join@@Permutations/@Select[IntegerPartitions[d],UnsameQ@@#&]],{d,Divisors[n]}],{n,12}]

Moebius transform is A032020 (strict compositions).

cp's OEIS Frontend

A334603 Period of the fraction 1/11^n for n >= 1.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A358138 Difference between maximum and minimum part in the n-th composition in standard order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A367631 Triangle read by rows: T(n,k) is the number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A034745 Dirichlet convolution of Fibonacci numbers with 3^(n-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A155734 Binomial transform of A154879.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

Extensions

A247936 Riordan array ((1-2x)/(1-3x), 2x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A336129 Number of strict compositions of divisors of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula