A001020 -id:A001020 - cp's OEIS frontend

A013617 Triangle of coefficients in expansion of (1+10x)^n.

1, 1, 10, 1, 20, 100, 1, 30, 300, 1000, 1, 40, 600, 4000, 10000, 1, 50, 1000, 10000, 50000, 100000, 1, 60, 1500, 20000, 150000, 600000, 1000000, 1, 70, 2100, 35000, 350000, 2100000, 7000000, 10000000, 1, 80, 2800, 56000, 700000, 5600000, 28000000, 80000000, 100000000

Offset: 0

N. J. A. Sloane

T(n,k) equals the number of n-length words on {0,1,...,10} having n-k zeros. - Milan Janjic, Jul 24 2015

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+10*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 24 2015

G.f.: 1 / (1 - x(1+10y)).

T(n,k) = 10^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*9^(n-i). Row sums are 11^n = A001020. - Mircea Merca, Apr 28 2012

A016823 a(n) = (4n+1)^11.

Original entry on oeis.org

1, 48828125, 31381059609, 1792160394037, 34271896307633, 350277500542221, 2384185791015625, 12200509765705829, 50542106513726817, 177917621779460413, 550329031716248441, 1532278301220703125, 3909821048582988049, 9269035929372191597, 20635899893042801193

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A001020, A013669, A016813.

[(4*n+1)^11 : n in [0..20]]; // Wesley Ivan Hurt, Oct 10 2014

A016823:=n->(4*n+1)^11: seq(A016823(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014

Table[(4 n + 1)^11, {n, 0, 20}] (* Wesley Ivan Hurt, Oct 10 2014 *)
CoefficientList[Series[(1 + 48828113 x + 30795122175 x^2 + 1418810334759 x^3 + 14826379326378 x^4 + 50417667664170 x^5 + 64020606756990 x^6 + 31088834650350 x^7 + 5356480404741 x^8 + 261595441397 x^9 + 1975200979 x^10 + 177147 x^11)/(x - 1)^12, {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 10 2014 *)

From Wesley Ivan Hurt, Oct 10 2014 : (Start)
G.f.: (1 + 48828113*x + 30795122175*x^2 + 1418810334759*x^3 + 14826379326378*x^4 + 50417667664170*x^5 + 64020606756990*x^6 + 31088834650350*x^7 + 5356480404741*x^8 + 261595441397*x^9 + 1975200979*x^10 + 177147*x^11) / (x - 1)^12.
Recurrence: a(n) = 12*a(n-1)-66*a(n-2)+220*a(n-3)-495*a(n-4)+792*a(n-5)-924*a(n-6)+792*a(n-7)-495*a(n-8)+220*a(n-9)-66*a(n-10)+12*a(n-11)-a(n-12).
a(n) = A016813(n)^11 = A001020(A016813(n)). (End)
Sum_{n>=0} 1/a(n) = 50521*Pi^11/29727129600 + 2047*zeta(11)/4096. - Amiram Eldar, Apr 21 2023

A056002 a(n) = (10^2)*11^(n-2); a(0)=1, a(1)=9.

Original entry on oeis.org

1, 9, 100, 1100, 12100, 133100, 1464100, 16105100, 177156100, 1948717100, 21435888100, 235794769100, 2593742460100, 28531167061100, 313842837672100, 3452271214393100, 37974983358324100, 417724816941565100

Offset: 0

Barry E. Williams, Jun 18 2000

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7,8,9,10,11} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7,8,9,10,11} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 10*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A001020.

Join[{1,9},100*11^Range[0,20]] (* or *) Join[{1,9},NestList[11#&,100,20]] (* Harvey P. Dale, May 24 2012 *)

a(n)=11a(n-1)+[(-1)^n]*C(2, 2-n). G.f.(x)=(1-x)^2/(1-11x).

a(n) = Sum_{k, 0<=k<=n} A201780(n,k)*9^k. - Philippe Deléham, Dec 05 2011

More terms from James Sellers, Jul 04 2000

A091946 a(n) = floor(11^n/10^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 23, 25, 28, 30, 34, 37, 41, 45, 49, 54, 60, 66, 72, 80, 88, 97, 106, 117, 129, 142, 156, 171, 189, 207, 228, 251, 276, 304, 334, 368, 405, 445, 490, 539, 593, 652, 717, 789

Offset: 0

Reinhard Zumkeller, Feb 16 2004

			a(2) = floor(1.1^2) = floor(1.21) = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A001020, A011557.

Cf. A002379, A094969-A094500, A091947.

[Floor(11^n / 10^n): n in [0..70]]; // Vincenzo Librandi, Sep 08 2011

Mathematica
```
Table[ Floor[(11/10)^n], {n, 0, 70}]
```

a(n) = floor(1.1^n) = floor(A001020(n)/A011557(n)).

More terms from Robert G. Wilson v, May 26 2004

A092846 a(n) = 100...001^n, where there are just enough zeros for the result to display the terms in the n-th row of Pascal's triangle.

Original entry on oeis.org

1, 11, 121, 1331, 14641, 10510100501, 1061520150601, 107213535210701, 10828567056280801, 1009036084126126084036009001, 1010045120210252210120045010001, 1011055165330462462330165055011001, 1012066220495792924792495220066012001

Offset: 0

Jorge Coveiro, Apr 15 2004

			a(0)=11^0
a(1)=11^1
a(2)=11^2
a(3)=11^3
a(4)=11^4
a(5)=101^5
a(6)=101^6
a(7)=101^7
a(8)=101^8
a(9)=1001^9
a(10)=1001^10
a(11)=1001^11

Eric M. Schmidt, Table of n, a(n) for n = 0..50

Cf. A007318, A001020.

def A092846(n) : return (10^binomial(n, n//2).ndigits()+1)^n # Eric M. Schmidt, Apr 04 2014

a(n) = (10^k + 1)^n, where k is the number of digits in A001405(n). - Eric M. Schmidt, Apr 04 2014

A139743 a(n) = 11^n - 5^n.

Original entry on oeis.org

0, 6, 96, 1206, 14016, 157926, 1755936, 19409046, 213968256, 2355994566, 25927658976, 285262842486, 3138184236096, 34521491440806, 379743730067616, 4177217651837526, 45949577275681536, 505446265559840646, 5559913498794965856, 61159071374928218166, 672749899565128368576

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 (16,-55).

Cf. A000351 (5^n), A001020 (11^n), A016165.

[11^n-5^n: n in [0..30]]; // Vincenzo Librandi, Jun 02 2011

a(n) = 16*a(n-1) - 55*a(n-2). - Vincenzo Librandi, Jun 02 2011
a(n) = 6*A016165(n-1) for n >= 1. - Philippe Deléham, Mar 23 2023
From Elmo R. Oliveira, Apr 01 2025: (Start)
G.f.: 6*x/((1-5*x)*(1-11*x)).
E.g.f.: 2*exp(8*x)*sinh(3*x). (End)

A159850 Numerator of Hermite(n, 17/22).

Original entry on oeis.org

1, 17, 47, -7429, -160415, 4464217, 269993839, -1892147821, -489536076223, -4658915114335, 987008017069999, 28053710866880683, -2150502256703365727, -118026514721378720791, 4759029349325350323695, 480777330814562061542723, -9102061914203466628786559

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 17/11, 47/121, -7429/1331, -160415/14641, ...

Robert Israel, Table of n, a(n) for n = 0..435
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Cf. A001020 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(17/11)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 09 2018

f:= gfun:-rectoproc({a(n) = 17*a(n-1)+242*(1-n)*a(n-2), a(0)=1,a(1)=17},a(n),remember):
map(f, [$0..40]); # Robert Israel, Dec 07 2017

Numerator[Table[HermiteH[n,17/22],{n,0,30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
Table[11^n*HermiteH[n, 17/22], {n,0,30}] (* G. C. Greubel, Jul 09 2018 *)

a(n)=numerator(polhermite(n, 17/22)) \\ Charles R Greathouse IV, Jan 29 2016

D-finite with recurrence a(n) = 17*a(n-1) + 242*(1-n)*a(n-2). - Robert Israel, Dec 07 2017
E.g.f.: exp(17*x - 121*x^2). - Simon Plouffe, Jun 23 2018
From G. C. Greubel, Jun 02 2018: (Start)
a(n) = 11^n * Hermite(n, 17/22).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(17/11)^(n-2*k)/(k!*(n-2*k)!)). (End)

A161999 For n even a(n) = a(n-1) + 10*a(n-2), for n odd a(n) = a(n-3) + 10 a(n-2); with a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 1, 10, 20, 101, 301, 1030, 4040, 10601, 51001, 110050, 620060, 1151501, 7352101, 12135070, 85656080, 128702801, 985263601, 1372684090, 11225320100, 14712104501, 126965305501, 158346365110, 1427999420120

Offset: 1

Mark Dols, Jun 24 2009, Jun 28 2009, Jul 13 2009

			As pairs:
0, 1
1, 10
20, 101
301, 1030
4040, 10601
51001, 110050
620060, 1151501
7352101, 12135070
85656080, 128702801

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 20, 0, -99).

Combination of A081192 and A016190. Triangle A007318 (even /uneven rows). Partly same function as A015446. A001020 (as sum of pairs of 2n).A001019 (as difference of pairs of 2n)

Cf. A162849.

nxt[{n_,a_,b_,c_}]:={n+1,b,c,If[OddQ[n],c+10b,a+10b]}; NestList[nxt,{2,0,1,1},30][[All,2]] (* or *) LinearRecurrence[{0,20,0,-99},{0,1,1,10},30] (* Harvey P. Dale, May 03 2018 *)

a(n)=20*a(n-2)-99*a(n-4). G.f.: -x^2*(-1-x+10*x^2)/((3*x-1)*(3*x+1)*(11*x^2-1)). [From R. J. Mathar, Jul 13 2009]

Edited by N. J. A. Sloane, Jun 30 2009

NAME adapted to offset. - R. J. Mathar, Jun 19 2021

A180689 Smallest power of 11 that begins with n.

Original entry on oeis.org

1, 214358881, 3138428376721, 4177248169415651, 505447028499293771, 61159090448414546291, 7400249944258160101211, 81402749386839761113321, 9849732675807611094711841, 108347059433883722041830251, 11, 121, 1331, 14641

Offset: 1

Daniel Mondot, Sep 17 2010

Cf. A001020, A018869, A180690.

a(14) from Michel Marcus, Aug 30 2013

A216156 Period of powers of 11 mod 10^n.

Original entry on oeis.org

10, 50, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000

Offset: 1

V. Raman, Sep 02 2012

Essentially the same as A093143. - R. J. Mathar, Sep 06 2012

Also period of squares mod 10^n. - Mohammed Yaseen, Apr 18 2023

			a(2) = 50 because 11^50 = 11739085287969531650666649599035831993898213898723001 = 1 mod 1000.

Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A001020, A216099.

Join[{10}, NestList[10*# &, 50, 20]] (* Paolo Xausa, Feb 20 2024 *)

a(n) = 5*10^(n-1), n >= 2.

cp's OEIS Frontend

A013617 Triangle of coefficients in expansion of (1+10x)^n.

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Maple

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A016823 a(n) = (4n+1)^11.

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Magma

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A056002 a(n) = (10^2)*11^(n-2); a(0)=1, a(1)=9.

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A091946 a(n) = floor(11^n/10^n).

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Magma

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A092846 a(n) = 100...001^n, where there are just enough zeros for the result to display the terms in the n-th row of Pascal's triangle.

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Sage

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A139743 a(n) = 11^n - 5^n.

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Magma

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A159850 Numerator of Hermite(n, 17/22).

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A161999 For n even a(n) = a(n-1) + 10*a(n-2), for n odd a(n) = a(n-3) + 10 a(n-2); with a(1) = 0, a(2) = 1.

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