A001020 -id:A001020 - cp's OEIS frontend

A013716 a(n) = 11^(2*n + 1).

11, 1331, 161051, 19487171, 2357947691, 285311670611, 34522712143931, 4177248169415651, 505447028499293771, 61159090448414546291, 7400249944258160101211, 895430243255237372246531

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 (121).

Bisection of A001020 (11^n).

Cf. A013708-A013715, A013717-A013729.

[11^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(11^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

11^(2*Range[0,20]+1) (* or *) NestList[121#&,11,20] (* Harvey P. Dale, Jun 21 2024 *)

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

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 121*a(n-1), a(0)=11.
G.f.: 11/(1-121*x). (End)

A073211 Sum of two powers of 11.

Original entry on oeis.org

2, 12, 22, 122, 132, 242, 1332, 1342, 1452, 2662, 14642, 14652, 14762, 15972, 29282, 161052, 161062, 161172, 162382, 175692, 322102, 1771562, 1771572, 1771682, 1772892, 1786202, 1932612, 3543122, 19487172, 19487182, 19487292, 19488502, 19501812, 19648222, 21258732, 38974342

Offset: 0

Jeremy Gardiner, Jul 20 2002

			T(2,0) = 11^2 + 11^0 = 122.
Table T(n,m) begins:
      2;
     12,    22;
    122,   132,   242;
   1332,  1342,  1452,  2662;
  14642, 14652, 14762, 15972, 29282;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A001020 (powers of 11).
Equals twice A073219.
Sums of two powers of n: A073423 (0), A007395 (1), A173786 (2), A055235 (3), A055236 (4), A055237 (5), A055257 (6), A055258 (7), A055259 (8), A055260 (9), A052216 (10), A194887 (12), A072390 (13), A055261 (16), A073213 (17), A073214 (19), A073215 (23).

t = 11^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)

from math import isqrt
def A073211(n): return 11**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+11**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

T(n,m) = 11^n + 11^m, n = 0, 1, 2, 3, ..., m = 0, 1, 2, 3, ... n.

Bivariate g.f.: (2 - 12*x)/((1 - x)*(1 - 11*x)*(1 - 11*x*y)). - J. Douglas Morrison, Jul 26 2021

A009988 Powers of 44.

Original entry on oeis.org

1, 44, 1936, 85184, 3748096, 164916224, 7256313856, 319277809664, 14048223625216, 618121839509504, 27197360938418176, 1196683881290399744, 52654090776777588736, 2316779994178213904384, 101938319743841411792896, 4485286068729022118887424, 197352587024076973231046656

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 44), L(1, 44), P(1, 44), T(1, 44). Essentially same as Pisot sequences E(44, 1936), L(44, 1936), P(44, 1936), T(44, 1936). 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 44-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 (44).

Cf. A000079, A000302, A001020, A008776, A009966.

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

44^Range[0,20] (* Harvey P. Dale, May 22 2017 *)

G.f.: 1/(1-44*x). - Philippe Deléham, Nov 24 2008
a(n) = 44^n; a(n) = 44*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(44*x).
a(n) = A000079(n)*A009966(n) = A000302(n)*A001020(n). (End)

A139745 a(n) = 11^n - 7^n.

Original entry on oeis.org

0, 4, 72, 988, 12240, 144244, 1653912, 18663628, 208594080, 2317594084, 25654949352, 283334343868, 3124587089520, 34425823133524, 379071610510392, 4172500607905708, 45916496933002560, 505214397985306564, 5558288899894321032, 61147691553229173148, 672670202666262397200

Offset: 0

N. J. A. Sloane, May 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (18,-77).

Cf. A000420, A001020, A016183.

[11^n-7^n: n in [0..30]]; // Vincenzo Librandi, Jun 02 2011

Table[11^n-7^n,{n,0,30}] (* or *) LinearRecurrence[{18,-77},{0,4},30] (* Harvey P. Dale, Jun 17 2014 *)

a(n) = 18*a(n-1) - 77*a(n-2). - Vincenzo Librandi, Jun 02 2011
From Stefano Spezia, Mar 09 2025: (Start)
G.f.: 4*x/((1 - 11*x)*(1 - 7*x)).
E.g.f.: exp(7*x)*(exp(4*x) - 1). (End)
a(n) = 4*A016183(n-1) for n >= 1. - Elmo R. Oliveira, Apr 01 2025

A220653 a(n) = n^11 + 11*n + 11^n.

Original entry on oeis.org

1, 23, 2191, 178511, 4208989, 48989231, 364568683, 1996813991, 8804293561, 33739007399, 125937424711, 570623341343, 3881436747541, 36314872538111, 383799398753059, 4185897925275191, 45967322049616753, 505481300395601591, 5559981581902310911, 61159206938673444719

Offset: 0

Jonathan Vos Post, Dec 17 2012

This is to A220425 as 11 is to 2, to A220509 as 11 is to 3, to A220511 as 11 is to 5, and to A220528 as 11 is to 7.

The subsequence of primes begins: 23, 4185897925275191, see A220787 for the associated n.

			a(1) = 1^11 + 11*1 + 11^1 = 23.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A008455, A008593, A001020, A220425, A220509, A220511, A220528, A220787.

Table[n^11 + 11*n + 11^n, {n, 0, 30}] (* T. D. Noe, Dec 17 2012 *)

A220653(n):=n^11+11*n+11^n$ makelist(A220653(n),n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n) = A008455(n) + A008593(n) + A001020(n).

G.f.: (131*x^12 +21186*x^11 +1682460*x^10 +24070936*x^9 +104942001*x^8 +163196604*x^7 +91422264*x^6 +14484216*x^5 -518211*x^4 -131726*x^3 -1860*x^2 -1) / ((x -1)^12*(11*x -1)). - Colin Barker, May 09 2013

a(19) from Stefano Spezia, May 03 2025

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

A130652 a(n) = 11^n - 2.

Original entry on oeis.org

9, 119, 1329, 14639, 161049, 1771559, 19487169, 214358879, 2357947689, 25937424599, 285311670609, 3138428376719, 34522712143929, 379749833583239, 4177248169415649, 45949729863572159, 505447028499293769, 5559917313492231479, 61159090448414546289, 672749994932560009199

Offset: 1

Alexander Adamchuk, Jun 20 2007

There are only two known primes in a(n): a(4) = 14639 and a(6) = 1771559 (see A128472 = smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists). 3 divides a(2k-1). 7 divides a(3k-1). 13 divides a(12k-5). 17 divides a(16k-14).
Final digit of a(n) is 9.
Final two digits of a(n) are periodic with period 10: a(n) mod 100 = {09, 19, 29, 39, 49, 59, 69, 79, 89, 99}.
Final three digits of a(n) are periodic with period 50: a(n) mod 1000 = {009, 119, 329, 639, 049, 559, 169, 879, 689, 599, 609, 719, 929, 239, 649, 159, 769, 479, 289, 199, 209, 319, 529, 839, 249, 759, 369, 079, 889, 799, 809, 919, 129, 439, 849, 359, 969, 679, 489, 399, 409, 519, 729, 039, 449, 959, 569, 279, 089, 999}.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A001020, A024127, A034524. Cf. A104096 = Largest prime <= 11^n. Cf. A084714 = smallest prime of the form (2n-1)^k - 2, or 0 if no such number exists. Cf. A128472 = smallest prime of the form (2n-1)^k - 2 for k>(2n-1), or 0 if no such number exists. Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.

[11^n - 2: n in [1..50]]; // Vincenzo Librandi, Jun 08 2011

LinearRecurrence[{12, -11},{9, 119},17] (* Ray Chandler, Aug 26 2015 *)

a(n) = 11*a(n-1) + 20; a(1)=9. - Vincenzo Librandi, Jun 08 2011
From Elmo R. Oliveira, Jun 16 2025: (Start)
G.f.: x*(11*x+9)/((11*x-1)*(x-1)).
E.g.f.: 1 + exp(x)*(exp(10*x) - 2).
a(n) = 12*a(n-1) - 11*a(n-2) for n > 2. (End)

A139740 a(n) = 11^n - 2^n.

Original entry on oeis.org

0, 9, 117, 1323, 14625, 161019, 1771497, 19487043, 214358625, 2357947179, 25937423577, 285311668563, 3138428372625, 34522712135739, 379749833566857, 4177248169382883, 45949729863506625, 505447028499162699, 5559917313491969337, 61159090448414022003, 672749994932558960625

Offset: 0

N. J. A. Sloane, May 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (13,-22).

Cf. A000079, A001020, A016135.

[11^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jun 02 2011

Table[11^n - 2^n, {n, 0, 20}] (* or *)
LinearRecurrence[{13, -22}, {0, 9}, 21] (* Paolo Xausa, Mar 16 2025 *)

a(n)=11^n-2^n \\ Charles R Greathouse IV, Mar 26 2012

[11**n-2**n for n in range(20)] # Gennady Eremin, Mar 05 2022

[11^n - 2^n for n in range(0,20)] # Zerinvary Lajos, Jun 05 2009

a(n) = 13*a(n-1) - 22*a(n-2). - Vincenzo Librandi, Jun 02 2011
a(n) = A001020(n) - A000079(n). - Michel Marcus, Feb 28 2022
a(n) = 9*A016135(n-1), n > 0. - Bernard Schott, Mar 09 2022
E.g.f.: exp(2*x)*(exp(9*x) - 1). - Stefano Spezia, Mar 09 2025
G.f.: 9*x/((1-2*x)*(1-11*x)). - Elmo R. Oliveira, Mar 15 2025

A139742 a(n) = 11^n - 4^n.

Original entry on oeis.org

0, 7, 105, 1267, 14385, 160027, 1767465, 19470787, 214293345, 2357685547, 25936376025, 285307476307, 3138411599505, 34522645035067, 379749565147785, 4177247095673827, 45949725568604865, 505447011319424587, 5559917244772754745, 61159090173536639347, 672749993833048381425

Offset: 0

N. J. A. Sloane, May 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (15,-44).

Cf. A000302, A001020, A016158.

[11^n-4^n: n in [0..30]]; // Vincenzo Librandi, Jun 02 2011

Table[11^n-4^n,{n,0,30}] (* or *) LinearRecurrence[{15,-44},{0,7},30] (* Harvey P. Dale, Dec 10 2013 *)

a(n) = 15*a(n-1) - 44*a(n-2). - Vincenzo Librandi, Jun 02 2011
From Elmo R. Oliveira, Apr 01 2025: (Start)
G.f.: 7*x/((1-4*x)*(1-11*x)).
E.g.f.: 2*exp(15*x/2)*sinh(7*x/2).
a(n) = 7*A016158(n-1) for n >= 1. (End)

A164899 Binomial matrix (1,10^n) read by antidiagonals.

Original entry on oeis.org

1, 1, 10, 1, 11, 100, 1, 12, 110, 1000, 1, 13, 121, 1100, 10000, 1, 14, 133, 1210, 11000, 100000, 1, 15, 146, 1331, 12100, 110000, 1000000, 1, 16, 160, 1464, 13310, 121000, 1100000, 10000000, 1, 17, 175, 1610, 14641, 133100, 1210000, 11000000, 100000000

Offset: 1

Mark Dols, Aug 30 2009

			Matrix array, A(n, k), begins:
  1, 10, 100, 1000, ...
  1, 11, 110, 1100, ...
  1, 12, 121, 1210, ...
  1, 13, 133, 1331, ...
  1, 14, 146, 1464, ...
  1, 15, 160, 1610, ...
Antidiagonal triangle, T(n, k), begins as:
  1;
  1, 10;
  1, 11, 100;
  1, 12, 110, 1000;
  1, 13, 121, 1100, 10000;
  1, 14, 133, 1210, 11000, 100000;
  1, 15, 146, 1331, 12100, 110000, 1000000;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Cf. A001020, A007318, A021093, A164844.

Cf. A094704 (antidiagonal row sums).

function T(n,k) // T = A164899
  if k eq n then return 10^(n-1);
  elif k eq 1 then return 1;
  else return T(n-1,k) + T(n-2,k-1);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Feb 10 2023

T[n_, k_]:= T[n,k]= If[k==n, 10^(n-1), If[k==1, 1, T[n-1,k] +T[n-2, k-1]]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Feb 10 2023 *)

def T(n,k): # T = A164899
    if (k==n): return 10^(n-1)
    elif (k==1): return 1
    else: return T(n-1,k) + T(n-2,k-1)
flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Feb 10 2023

Sum_{k=1..n} T(n, k) = A094704(n).
As a triangle T(n,k) read by rows, T(n,1) = 1, T(n,n) = 10^(n-1), and T(n,k) = T(n-1, k) + T(n-2, k-1) otherwise. - Joerg Arndt, Dec 10 2016
From G. C. Greubel, Feb 10 2023: (Start)
A(n, k) = A(n-1, k) + A(n-1, k-1), with A(n, 1) = 1 and A(1, k) = 10^(k-1) (array).
T(n, k) = A(n-k+1, k) (antidiagonal triangle). (End)

cp's OEIS Frontend

A013716 a(n) = 11^(2*n + 1).

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A073211 Sum of two powers of 11.

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A009988 Powers of 44.

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A139745 a(n) = 11^n - 7^n.

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A220653 a(n) = n^11 + 11*n + 11^n.

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A097659 a(n) = 1001^n.

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A130652 a(n) = 11^n - 2.

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