A096884 -id:A096884 - cp's OEIS frontend

A001020 Powers of 11: a(n) = 11^n.

1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, 379749833583241, 4177248169415651, 45949729863572161, 505447028499293771, 5559917313492231481, 61159090448414546291

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 11), L(1, 11), P(1, 11), T(1, 11). Essentially same as Pisot sequences E(11, 121), L(11, 121), P(11, 121), T(11, 121). 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 11-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n), for n <= 4, gives the n-th row of Pascal's triangle (A007318); a(n), n >= 5 "sort of" gives the n-th row of Pascal's triangle, but now the binomial coefficients with more than one digit overlap. - Daniel Forgues, Aug 12 2012
Numbers k such that sigma(11*k) = 11*k + sigma(k). - Jahangeer Kholdi, Nov 13 2013

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..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 275.
Tanya Khovanova, Recursive Sequences.
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.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (11).

Cf. A096884, A097659, A007318.

[11^n: n in [0..100]]; // Vincenzo Librandi, Apr 24 2011

A001020:=-1/(-1+11*z); # Simon Plouffe in his 1992 dissertation

Table[11^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

makelist(11*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=n^11 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-11*x).
E.g.f.: exp(11*x).
a(n) = 11*a(n-1), n > 0; a(0)=1. - Philippe Deléham, Nov 23 2008

A097659 a(n) = 1001^n.

Original entry on oeis.org

1, 1001, 1002001, 1003003001, 1004006004001, 1005010010005001, 1006015020015006001, 1007021035035021007001, 1008028056070056028008001, 1009036084126126084036009001, 1010045120210252210120045010001, 1011055165330462462330165055011001, 1012066220495792924792495220066012001

Offset: 0

Reinhard Zumkeller, Aug 18 2004

694 is the smallest exponent e such that 1001^e begins with a digit greater than 1: A000030(a(694)) = 2, A000030(a(693)) = 1. - Reinhard Zumkeller, Nov 05 2010

a(n) gives the n-th row of Pascal's triangle (A007318) as long as all the binomial coefficients have at most three digits, otherwise the binomial coefficients with more than three digits overlap. - Daniel Forgues, Aug 12 2012

Rozsa Peter, Playing with Infinity, New York, Dover Publications, 1957.

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (1001).

Cf. A001020, A007318, A096884, A097660.

1001^n \\ Charles R Greathouse IV, Jan 30 2012

my(x='x+O('x^13)); Vec(-1/(1001*x-1)) \\ Elmo R. Oliveira, Jul 06 2025

From Philippe Deléham, Nov 30 2008: (Start)
a(n) = 1001*a(n-1), n > 0; a(0)=1.
G.f.: 1/(1-1001*x). (End)
From Elmo R. Oliveira, Jul 06 2025: (Start)
E.g.f.: exp(1001*x).
a(n) = 91^n * A001020(n). (End)

More terms from Elmo R. Oliveira, Jul 06 2025

A185817 Smallest m such that n is a prefix of 101^m in its decimal representation.

Original entry on oeis.org

0, 70, 111, 140, 162, 181, 196, 209, 221, 1, 10, 19, 27, 34, 41, 48, 54, 60, 65, 70, 75, 80, 84, 88, 93, 97, 100, 104, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 137, 140, 142, 145, 147, 149, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 175, 177, 179, 181

Offset: 1

Reinhard Zumkeller, Feb 07 2011

a(n) = MIN{e: floor(A096884(e)/10^k) = n for some k}.

a(n) <= a(10*n + d), 0 <= d < 10.

			a(1) = 0; a(10) = 1; a(101) = 1;
a(2) = 70, as A000030(101^e) = 1 for e < 70 and A000030(101^70) = 2,
101^70=2006763368395383712973746195325904225117468781576180838692428661200863034768\
95435690179367146267108054577096030854694073677909106884264157001.

Reinhard Zumkeller, Table of n, a(n) for n = 1..7777

Cf. A000030, A096884,

import Data.List (isPrefixOf)
a185817 n = pref101pow 0 1 where
   pref101pow e pow101 = if isPrefixOf (show n) (show pow101)
                            then e
                            else pref101pow (e + 1) (101 * pow101)

A291945 Powers of 1111.

Original entry on oeis.org

1, 1111, 1234321, 1371330631, 1523548331041, 1692662195786551, 1880547699518858161, 2089288494165451416871, 2321199517017816524143681, 2578852663406794158323629591, 2865105309044948309897552475601, 3183131998348937572296180800392711, 3536459650165669642821056869236301921, 3929006671334058973174194181721531434231

Offset: 0

Seiichi Manyama, Mar 09 2018

Index entries for linear recurrences with constant coefficients, signature (1111).

Powers of ((10^k - 1)/9): A000012 (k=1), A001020 (k=2), A225374 (k=3), this sequence (k=4), A291946 (k=5), A109716 (k=6).

Cf. A096884.

1111^Range[0,20] (* Harvey P. Dale, Jul 27 2022 *)

a(n) = 1111^n; \\ Michel Marcus, Mar 11 2018

a(n) = 1111^n.
G.f.: 1/(1 - 1111*x).
From Elmo R. Oliveira, Aug 16 2024: (Start)
E.g.f.: exp(1111*x).
a(n) = 1111*a(n-1) for n > 0.
a(n) = A001020(n)*A096884(n). (End)

A327926 a(n) = 99^n.

Original entry on oeis.org

1, 99, 9801, 970299, 96059601, 9509900499, 941480149401, 93206534790699, 9227446944279201, 913517247483640899, 90438207500880449001, 8953382542587164451099, 886384871716129280658801, 87752102299896798785221299, 8687458127689783079736908601

Offset: 0

Eliora Ben-Gurion, Nov 09 2019

			a(4) = 96059601 = 100^4 - 4*100^3 + 6*100^2 - 4*100 + 1 = 100000000 - 4000000 + 60000 - 400 - 1: 100000000 -> 96000000 -> 96060000 -> 96059600 -> 96059601.
a(8) = 99^8 = 9227446944279201.

Index entries for linear recurrences with constant coefficients, signature (99).

Cf. A096884, A007318.

a(n) = 99^n.
From Elmo R. Oliveira, Aug 11 2024: (Start)
G.f.: 1/(1-99*x).
E.g.f.: exp(99*x).
a(n) = 99*a(n-1) for n > 0. (End)

A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

Original entry on oeis.org

16, 25, 27, 32, 49, 64, 81, 125, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 15625, 16384, 16807, 17161, 19683, 22801, 32761, 32768, 36481, 59049, 65536, 78125, 97969, 117649, 124609, 131072, 139129, 146689, 161051, 177147, 262144

Offset: 1

Bernard Schott, Dec 10 2020

Equivalently: numbers m with only one prime factor such that the LCM of their palindromic divisors is neither 1 nor m: subsequence of A334392.
G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see Simmons p. 98). According to this conjecture, these perfect powers are terms: {2^k, k>=4}, {3^k, k>=3}, {5^k, k>=2}, {7^k, k=2 and k>=4}, {11^k, k>=5}, {101^k, k>= 5}, {131^k, k>=2}, ...
From a(1) = 16 to a(17) = 2187, the data is the same as A056781(10) until A056781(26), then a(18) = 2401 and A056781(27) = 4096.

			5^2 = 25, 2^6 = 64, 3^4 = 81 are terms.
7^2 = 49 is a term, 7^3 = 343 is not a term, and 7^4 = 2401 is a term.
101^2 = 10201 and 11^4 = 14641 are not terms.

Murray S. Klamkin, Problems in applied mathematics: selections from SIAM review, (1990), p. 520.

Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].
Wikipedia, Palindromic number, Perfect Powers.

Intersection of A025475 and A334392.
Cf. A087990, A087999, A334139, A334391.
Subsequences: A000079 \ {1,2,4,8}, A000244 \ {1,3,9}, A000351 \ {1,5}, A000420 \ {1,7,343}, A001020 \ {1,11,121,1331,14641}, A096884 \ {1,101, 10201, 1030301, 104060401}.

q[n_] := Module[{f = FactorInteger[n]}, Length[f] == 1 && f[[1, 2]] > 1 && PalindromeQ[f[[1, 1]]]]; Select[Range[10^5], !PalindromeQ[#] && q[#] &] (* Amiram Eldar, Dec 10 2020 *)

ispal(n) = my(d=digits(n)); Vecrev(d)==d;
isok(k) = my(p); isprimepower(k, &p) && isprime(p) && ispal(p) &&!ispal(k); \\ Michel Marcus, Dec 10 2020

More terms from Amiram Eldar, Dec 10 2020

A096885 Related to diagonals of Pascal's triangle.

Original entry on oeis.org

1, 100, 10001, 1000200, 100030001, 10004000300, 1000500060001, 100060010000400, 10007001500100001, 1000800210020000500, 100090028003500150001, 10010003600560035000600, 1001100450084007000210001, 100120055012001260056000700, 10013006601650210012600280001, 1001400780220033002520084000800

Offset: 0

Paul Barry, Jul 14 2004

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (100,1).

Cf. A096884, A007318, A000045.

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*100^(n-2k).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 100*a(n-1) + a(n-2), n > 1; a(0)=1, a(1)=100.
G.f.: 1/(1-100*x-x^2). (End)
E.g.f.: exp(50*x)*(cosh(sqrt(2501)*x) + 50*sinh(sqrt(2501)*x)/sqrt(2501)). - Stefano Spezia, Aug 05 2024

a(12)-a(15) from Stefano Spezia, Aug 05 2024

cp's OEIS Frontend

A001020 Powers of 11: a(n) = 11^n.

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A097659 a(n) = 1001^n.

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A185817 Smallest m such that n is a prefix of 101^m in its decimal representation.

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A291945 Powers of 1111.

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A327926 a(n) = 99^n.

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A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

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Mathematica

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A096885 Related to diagonals of Pascal's triangle.

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