A001020 -id:A001020 - cp's OEIS frontend

A008571 Digits of powers of 11.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 6, 1, 0, 5, 1, 1, 7, 7, 1, 5, 6, 1, 1, 9, 4, 8, 7, 1, 7, 1, 2, 1, 4, 3, 5, 8, 8, 8, 1, 2, 3, 5, 7, 9, 4, 7, 6, 9, 1, 2, 5, 9, 3, 7, 4, 2, 4, 6, 0, 1, 2, 8, 5, 3, 1, 1, 6, 7, 0, 6, 1, 1, 3, 1, 3, 8, 4, 2, 8, 3, 7, 6, 7, 2, 1, 3, 4, 5, 2, 2, 7, 1, 2

Offset: 0

N. J. A. Sloane

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  1, 4, 6, 4, 1;
  1, 6, 1, 0, 5, 1;
  1, 7, 7, 1, 5, 6, 1;
  1, 9, 4, 8, 7, 1, 7, 1;
  2, 1, 4, 3, 5, 8, 8, 8, 1;
  ...

Seiichi Manyama, Rows n = 0..100, flattened
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A001020, A348553.

Cf. A000455, A008564, A008565, A008566, A008567, A008568, A008569, A008570, A008572, A008573, A010054.

IntegerDigits[11^Range[0,20]]//Flatten (* Harvey P. Dale, Aug 26 2025 *)

A214112 T(n,k)=Number of 0..3 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..3 introduced in row major order.

Original entry on oeis.org

1, 1, 4, 4, 11, 25, 10, 111, 121, 172, 31, 670, 3502, 1331, 1201, 91, 4994, 44900, 110985, 14641, 8404, 274, 34041, 825105, 3008980, 3517864, 161051, 58825, 820, 241021, 12777541, 136579852, 201647240, 111505491, 1771561, 411772, 2461, 1678940

Offset: 1

R. H. Hardin Jul 04 2012

Table starts
....1.....1.......4........10..........31............91.............274
....4....11.....111.......670........4994.........34041..........241021
...25...121....3502.....44900......825105......12777541.......214404272
..172..1331..110985...3008980...136579852....4797577911....191154162535
.1201.14641.3517864.201647240.22615881851.1801391900581.170522196557894

			Some solutions for n=4 k=1
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..1..0....1..2....2..3....1..0....1..0....1..0....1..2....1..0....1..0....1..2
..0..1....0..1....3..1....0..1....2..3....2..1....3..0....0..2....2..3....3..1
..1..2....1..0....1..0....1..0....3..2....3..0....0..1....1..3....3..1....0..2

R. H. Hardin, Table of n, a(n) for n = 1..160

Column 1 is A034494(n-1)
Column 2 is A001020(n-1)
Row 1 is A006342(n-1)

Empirical for column k:
k=1: a(n) = 8*a(n-1) -7*a(n-2)
k=2: a(n) = 11*a(n-1)
k=3: a(n) = 35*a(n-1) -107*a(n-2) +73*a(n-3)
k=4: a(n) = 68*a(n-1) -66*a(n-2)
k=5: a(n) = 200*a(n-1) -5769*a(n-2) +11744*a(n-3) +43057*a(n-4) -89856*a(n-5) +40625*a(n-6)
k=6: a(n) = 416*a(n-1) -15454*a(n-2) +89758*a(n-3) +90848*a(n-4) -438718*a(n-5) +62801*a(n-6)
k=7: (order 15)
Empirical for row n:
n=1: a(k)=3*a(k-1)+a(k-2)-3*a(k-3)
n=2: a(k)=4*a(k-1)+22*a(k-2)-4*a(k-3)-21*a(k-4)
n=3: a(k)=11*a(k-1)+123*a(k-2)-509*a(k-3)-1615*a(k-4)+7137*a(k-5)-19*a(k-6)-20571*a(k-7)+13176*a(k-8)+13932*a(k-9)-11664*a(k-10)
n=4: (order 26)
n=5: (order 71)

A024127 a(n) = 11^n-1.

Original entry on oeis.org

0, 10, 120, 1330, 14640, 161050, 1771560, 19487170, 214358880, 2357947690, 25937424600, 285311670610, 3138428376720, 34522712143930, 379749833583240, 4177248169415650, 45949729863572160, 505447028499293770

Offset: 0

N. J. A. Sloane

In base 11 these are 0, A, AA, AAA, ... - David Rabahy, Dec 12 2016

Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A001020.

LinearRecurrence[{12, -11},{0, 10},18] (* Ray Chandler, Aug 26 2015 *)

a(n)=11^n-1 \\ Charles R Greathouse IV, Dec 12 2016

G.f.: 1/(1-11*x)-1/(1-x). - Mohammad K. Azarian, Jan 14 2009
E.g.f.: e^(11*x)-e^x. - Mohammad K. Azarian, Jan 14 2009
a(n) = 11*a(n-1)+10 for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(n) = Sum_{i=1..n} 10^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
a(n) = A001020(n) - 1. - Sean A. Irvine, Jun 19 2019

A062634 Numbers k such that every divisor of k contains the digit 1.

Original entry on oeis.org

1, 11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 211, 221, 241, 251, 271, 281, 311, 313, 317, 331, 341, 361, 401, 419, 421, 431, 451, 461, 491, 521

Offset: 1

Erich Friedman, Jul 04 2001

First composite term is 121. All powers of 11 are in the sequence. - Alonso del Arte, Sep 29 2013

			143 has divisors 1, 11, 13 and 143, all of which contain the digit 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027750, subsequence of A011531; A206159 and A208270 are subsequences.

Cf. A001020 (powers of 11).

a062634 n = a062634_list !! (n-1)
a062634_list = filter
   (and . map ((elem '1') . show) . a027750_row) a011531_list
-- Reinhard Zumkeller, Feb 05 2012

q:= n-> andmap(x-> 1 in convert(x, base, 10), numtheory[divisors](n)):
select(q, [$1..1000])[];  # Alois P. Heinz, May 09 2022

fQ[n_, dgt_] := Union[ MemberQ[#, dgt] & /@ IntegerDigits@ Rest@ Divisors@ n][[1]]; Select[ Range[2, 525], fQ[#, 1] &] (* Robert G. Wilson v, Jun 11 2014 *)

isok(m) = fordiv(m, d, if (! #select(x->(x==1), digits(d)), return(0))); return(1); \\ Michel Marcus, May 09 2022

Offset corrected by Reinhard Zumkeller, Feb 05 2012

A133294 a(n) = 2a(n-1) + 10a(n-2), a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 12, 34, 188, 716, 3312, 13784, 60688, 259216, 1125312, 4842784, 20938688, 90305216, 389997312, 1683046784, 7266066688, 31362601216, 135385869312, 584397750784, 2522654194688, 10889285897216, 47005113741312

Offset: 0

Philippe Deléham, Dec 20 2007

Binomial transform of [1, 0, 11, 0, 121, 0, 1331, 0, 14641, 0, ...]=: powers of 11 (A001020) with interpolated zeros. - Philippe Deléham, Dec 02 2008

A083101 is an essentially identical sequence (with a different start). - N. J. A. Sloane, Dec 31 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,10).

Cf. A083101, A090042.

a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+10*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019

a[n_]:= Simplify[((1+Sqrt[11])^n + (1-Sqrt[11])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-10x^2), {x,0,30}], x] (* Or *) LinearRecurrence[{2, 10}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

my(x='x+O('x^30)); Vec((1-x)/(1-2*x-10*x^2)) \\ G. C. Greubel, Aug 02 2019

((1-x)/(1-2*x-10*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

a(n) = Sum_{k=0..n} A098158(n,k)*11^(n-k).
G.f.: (1-x)/(1-2*x-10*x^2).
a(n) = A083101(n-1) for n >= 1.
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013

Terms a(23) onward added by G. C. Greubel, Aug 02 2019

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293

Offset: 0

Omar E. Pol, Sep 09 2018

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.

The partial sums of column k give the column k of A319076.

			The corner of the square array is as follows:
         A000079 A000244 A000351  A000420    A001020    A001022     A001026
A000012        1,      1,      1,       1,         1,         1,          1, ...
A000040        2,      3,      5,       7,        11,        13,         17, ...
A001248        4,      9,     25,      49,       121,       169,        289, ...
A030078        8,     27,    125,     343,      1331,      2197,       4913, ...
A030514       16,     81,    625,    2401,     14641,     28561,      83521, ...
A050997       32,    243,   3125,   16807,    161051,    371293,    1419857, ...
A030516       64,    729,  15625,  117649,   1771561,   4826809,   24137569, ...
A092759      128,   2187,  78125,  823543,  19487171,  62748517,  410338673, ...
A179645      256,   6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...

Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031.
Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100).
Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Main diagonal gives A093360.
Second diagonal gives A062457.
Third diagonal gives A197987.
Removing the 1's we have A182944/ A182945.
Cf. A006093, A319074, A319076.

PARI
```
T(n, k) = prime(k)^n;
```

T(n,k) = A000040(k)^n, n >= 0, k >= 1.

A096884 a(n) = 101^n.

Original entry on oeis.org

1, 101, 10201, 1030301, 104060401, 10510100501, 1061520150601, 107213535210701, 10828567056280801, 1093685272684360901, 110462212541120451001, 11156683466653165551101, 1126825030131969720661201, 113809328043328941786781301, 11494742132376223120464911401, 1160968955369998535166956051501

Offset: 0

Paul Barry, Jul 14 2004

A185817(n) = smallest m such that in decimal representation n is a prefix of a(m).
a(n) gives the n-th row of Pascals' triangle (A007318) as long as all the binomial coefficients have at most two digits, otherwise the binomial coefficients with more than two digits overlap. - Daniel Forgues, Aug 12 2012
From Peter M. Chema, Apr 10 2016: (Start)
One percent growth applied n times increases a value by factor of a(n)/10^(2n), since 1% increases using "1.01". Therefore (a(n)/10^(2n) - 1)*100 = the percentage increase of one percent growth applied n times.
For instance, 432 increasing by 1% three times gives 445.090032 (i.e., 432*1.01^3), which is 1.030301 (a(3)/10^(2*3)) times 432 or a 3.0301% increase from the original 432 ((a(3)/10^(2*3)-1)*100 = 3.0301). (End)

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (101).

Cf. A007318, A003590, A001020, A097659.

Table[101^n, {n, 0, 15}] (* Ilya Gutkovskiy, Apr 10 2016 *)

a(n)=101^n \\ Charles R Greathouse IV, Oct 16 2015

my(x='x+O('x^20)); Vec(1/(1-101*x)) \\ Altug Alkan, Apr 10 2016

a(n) = Sum_{k=0..n} binomial(n, k)*10^(n-k).
a(n) = A096883(2n).
a(n) = 101^n. a(n) = Sum_{k=0..n,} binomial(n, k)*100^k. - Paul Barry, Aug 24 2004
G.f.: 1/(1-101*x). - Philippe Deléham, Nov 25 2008
E.g.f.: exp(101*x). - Ilya Gutkovskiy, Apr 10 2016

A239015 Exponents m such that the decimal expansion of 11^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 36, 41, 366, 488, 4357, 69137, 89371, 143907, 542116, 2431369, 5877361, 8966861, 121915452, 123793821, 221788016, 709455085, 1571200127, 2640630712, 6637360862, 64994336645, 74770246842

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A001020.

Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan Wechsler and Franklin T. Adams-Watters.

			Illustration of initial term, with the 0 enclosed in parentheses:
n, position of 0, 11^a(n)
1, 2, (0)1
2, 3, (0)11
3, 4, (0)121
4, 5, (0)1331
5, 6, (0)14641
6, 7, (0)1771561
7, 8, (0)19487171
8, 9, (0)214358881
9, 10, (0)2357947691
10, 11, (0)3138428376721
11, 12, (0)34522712143931
12, 13, (0)379749833583241
13, 14, (0)4177248169415651
14, 15, (0)45949729863572161
15, 16, (0)5559917313492231481
16, 17, 3091268053287(0)672635673352936887453361
...
- _N. J. A. Sloane_, Jan 16 2020

Cf. A001020, A030706, A020665, A031142, A239008, A239009, A239010, A239011, A239012, A239013, A239014.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[11, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 40000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(28)-a(34) from Bert Dobbelaere, Jan 22 2019

a(35)-a(36) from Chai Wah Wu, Jan 16 2020

A166149 a(n) = (5^n + 10*(-6)^n)/11.

Original entry on oeis.org

1, -5, 35, -185, 1235, -6785, 43835, -247385, 1562435, -8983985, 55857035, -325376585, 2001087635, -11762385185, 71795014235, -424666569785, 2578516996835, -15318514090385, 92674023995435, -552229446706985

Offset: 0

Philippe Deléham, Oct 08 2009

From Klaus Brockhaus, Oct 14 2009: (Start)
Fourth binomial transform of A014992.
Sixth binomial transform is A001020 preceded by 1.
Lim_{n -> infinity} a(n)/a(n-1) = -6. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (-1,30).

Cf. A166035, A166036.

Cf. A014992 (q-integers for q=-10), A001020 (powers of 11).

[(5^n+10*(-6)^n)/11: n in [0..30]]; // Vincenzo Librandi, May 02 2011

CoefficientList[Series[(1-4x)/(1+x-30x^2), {x,0,40}], x]  (* Harvey P. Dale, Mar 11 2011 *)
LinearRecurrence[{-1,30},{1,-5},20] (* Harvey P. Dale, Jan 20 2022 *)

a(n)=(5^n+10*(-6)^n)/11 \\ Charles R Greathouse IV, May 02 2016

a(n) = 30*a(n-2)-a(n-1), a(0)= 1, a(1)= -5.
G.f.: (1-4x)/(1+x-30*x^2).
a(n) = Sum_{k=0..n} A112555(n,k)*(-6)^k.
E.g.f.: (1/11)*(exp(5*x) + 10*exp(-6*x)). - G. C. Greubel, May 01 2016

A009977 Powers of 33.

Original entry on oeis.org

1, 33, 1089, 35937, 1185921, 39135393, 1291467969, 42618442977, 1406408618241, 46411484401953, 1531578985264449, 50542106513726817, 1667889514952984961, 55040353993448503713, 1816331681783800622529, 59938945498865420543457, 1977985201462558877934081, 65273511648264442971824673

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 33), L(1, 33), P(1, 33), T(1, 33). Essentially same as Pisot sequences E(33, 1089), L(33, 1089), P(33, 1089), T(33, 1089). 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 33-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 (33).

Cf. A000244, A001020, A008776, A327926.

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

33^Range[0, 20] (* Paolo Xausa, Jul 12 2025 *)

G.f.: 1/(1-33*x). - Philippe Deléham, Nov 24 2008
a(n) = 33^n; a(n) = 33*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(33*x).
a(n) = A000244(n)*A001020(n) = A327926(n)/A000244(n). (End)

cp's OEIS Frontend

A008571 Digits of powers of 11.

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A214112 T(n,k)=Number of 0..3 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..3 introduced in row major order.

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A024127 a(n) = 11^n-1.

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A062634 Numbers k such that every divisor of k contains the digit 1.

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A133294 a(n) = 2*a(n-1) + 10*a(n-2), a(0)=1, a(1)=1.

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A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

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A096884 a(n) = 101^n.

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A133294 a(n) = 2a(n-1) + 10a(n-2), a(0)=1, a(1)=1.