A102807 -id:A102807 - cp's OEIS frontend

A075415 Squares of A002280 or numbers (666...6)^2.

0, 36, 4356, 443556, 44435556, 4444355556, 444443555556, 44444435555556, 4444444355555556, 444444443555555556, 44444444435555555556, 4444444444355555555556, 444444444443555555555556, 44444444444435555555555556, 4444444444444355555555555556, 444444444444443555555555555556

Offset: 0

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 66^2 = 4356.
From _Reinhard Zumkeller_, May 31 2010: (Start)
n=1: ..................... 36 = 9 * 4;
n=2: ................... 4356 = 99 * 44;
n=3: ................. 443556 = 999 * 444;
n=4: ............... 44435556 = 9999 * 4444;
n=5: ............. 4444355556 = 99999 * 44444;
n=6: ........... 444443555556 = 999999 * 444444;
n=7: ......... 44444435555556 = 9999999 * 4444444;
n=8: ....... 4444444355555556 = 99999999 * 44444444;
n=9: ..... 444444443555555556 = 999999999 * 444444444. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..500
Gérard Villemin, Variations sur les carrés.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417.
Cf. A059988, A178630, A178631, A178632, A178633, A178634, A178635. [From Reinhard Zumkeller, May 31 2010]
Cf. A002275, A002278, A002279,  A002280, A002283, A002477, A052041, A102807, A309827.

Table[FromDigits[PadRight[{},n,6]]^2,{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{0,36,4356},20] (* Harvey P. Dale, May 20 2021 *)

a(n) = A002280(n)^2 = (6*A002275(n))^2 = 36*A002275(n)^2.
a(n) = (6*(10^n-1)/9)^2 = (4/9)*(10^(2*n) - 2*10^n + 1), which is n-1 4's, followed by a 3, n-1 5's and a 6. - Ignacio Larrosa Cañestro, Feb 26 2005
From Reinhard Zumkeller, May 31 2010: (Start)
a(n) = ((A002278(n-1)*10 + 3)*10^(n-1) + A002279(n-1))*10 + 6 for n>0.
a(n) = A002283(n)*A002278(n). (End)
G.f.: 36*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Arkadiusz Wesolowski, Dec 26 2011
From Elmo R. Oliveira, Jul 27 2025: (Start)
E.g.f.: 4*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
a(n) = 36*A002477(n). (End)

Edited by Alois P. Heinz, Aug 21 2019 (merged with A102794, submitted by Richard C. Schroeppel, Feb 26 2005)

A052041 Squares lacking the digit zero in their decimal expansion.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 169, 196, 225, 256, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 961, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2116, 2916, 3136, 3249, 3364, 3481, 3721, 3844, 3969, 4225, 4356

Offset: 1

Patrick De Geest, Dec 15 1999

This sequence is infinite: see A075415 or A102807 for a constructive proof.

Intersection of A052382 and A000290; A168046(a(n))*A010052(a(n))=1. - Reinhard Zumkeller, Dec 01 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Zerofree [From _Reinhard Zumkeller_, Dec 01 2009]

Cf. A104264, A102265, A102266.

Squares: A052040, A052042, A052043. Cubes: A051750, A051751, A051832, A051833.

Select[Range[66]^2, FreeQ[IntegerDigits[#],0]==True &] (* Jayanta Basu, May 25 2013 *)

a(n) = A052040(n)^2. - R. J. Mathar, Jul 23 2025

A093137 Expansion of (1-7x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 4, 34, 334, 3334, 33334, 333334, 3333334, 33333334, 333333334, 3333333334, 33333333334, 333333333334, 3333333333334, 33333333333334, 333333333333334, 3333333333333334, 33333333333333334, 333333333333333334, 3333333333333333334, 33333333333333333334

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 3*A001045(3n)/3+(-1)^n. Partial sums of A093138. A convex combination of 10^n and 1. In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1,1+k,1+11k,1+111k,... This is the case for k=3.
a(n) is the number of n-length sequences of decimal digits whose sum is divisible by 3. - Geoffrey Critzer, Jan 18 2014
This sequence appears in a family of curious cubic identities based on the Armstrong number 407 = A005188(13). See the formula section. For the analog identities based on 153 = A005188(10) see a comment on A246057 with the van der Poorten et al. reference and A281857. For those based on 370 = A005188(11) see A067275, A002277 and A281858. - Wolfdieter Lang, Feb 08 2017

			a(1)^2 = 16
a(2)^2 = 1156
a(3)^2 = 111556
a(4)^2 = 11115556
a(5)^2 = 1111155556
a(6)^2 = 111111555556
a(7)^2 = 11111115555556
a(8)^2 = 1111111155555556
a(9)^2 = 111111111555555556, etc... (see A102807). - _Philippe Deléham_, Oct 03 2011
Curious cubic identities: 407 = 4^3 + 0^3 + 7^3, 340067 = 34^3 + (00)^3 + 67^3, 334000677 = 334^3 + (000)^3 + 677^3, ... - _Wolfdieter Lang_, Feb 08 2017

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 3334 at p. 168.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A005188, A067275, A246057, A281857, A281859.
Cf. A001045, A002277, A093138, A281858.
Cf. A102807 (squared).

nn=20; r=Solve[{s==4x s+3 x a+3x b+1,a==4x a+3x s+3x b,b==4x b+3x s+3x a},{s,a,b}]; CoefficientList[Series[s/.r,{x,0,nn}],x] (* Geoffrey Critzer, Jan 18 2014 *)
Table[3*10^n/9 + 6/9, {n, 0, 20}] (* or *) NestList[10 # - 6 &, 1, 20] (* Michael De Vlieger, Feb 08 2017 *)
LinearRecurrence[{11,-10},{1,4},20] (* Harvey P. Dale, Oct 07 2017 *)

Vec((1-7*x)/((1-x)*(1-10*x)) + O (x^30)) \\ Michel Marcus, Feb 09 2017

a(n) = 3*10^n/9 + 6/9.
a(n) = 10*a(n-1)-6 with a(0)=1. - Vincenzo Librandi, Aug 02 2010
a(n)^3 + 0(n)^3 + A067275(n+1)^3 = concatenation(a(n), 0(n), A067275(n+1)) = A281859(n), where 0(n) denotes n 0's, n >= 1. - Wolfdieter Lang, Feb 08 2017
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(x)*(exp(9*x) + 2)/3.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1. (End)

A104264 Number of n-digit squares with no zero digits.

Original entry on oeis.org

3, 6, 19, 44, 136, 376, 1061, 2985, 8431, 24009, 67983, 193359, 549697, 1563545, 4446173, 12650545, 35999714, 102439796, 291532841, 829634988, 2360947327, 6719171580, 19122499510, 54423038535, 154888366195

Offset: 1

Reinhard Zumkeller and Ron Knott, Feb 26 2005

Comments from David W. Wilson, Feb 26 2005: (Start)
"There are approximately s(d) = (10^d)^(1/2) - (10^(d-1))^(1/2) d-digit squares. A random d-digit number has the probability p(d) = (9/10)^(d-1) of being zeroless (exponent d-1 as opposed to d because the first digit is not zero). So we expect p(d)s(d) zeroless d-digit squares.
"For d = 1 through 12, we get (truncating): 1, 5, 15, 44, 127, 363, 1034, 2943, 8377, 23841, 67854, 193117, ... The elements grow approximately geometrically with limit ratio (9/10)*10^(1/2) = 2.846+.
"The same naive estimate can easily be generalize to k-th powers, giving the estimate s(d) = (10^d)^(1/k) - (10^(d-1))^(1/k) for d-digit k-th powers. p(d) remains the same. The resulting estimates have ratio (9/10)*10^(1/k).
"We should expect an infinite number of zeroless k-th powers when this ratio is >= 1, which it is for k <= 21. For k >= 22, the ratio is < 1 and we should expect a finite number of zeroless k-th powers." (End)

			a(3) = #{121, 144, 169, 196, 225, 256, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 961} = 19.

Cf. A052041, A104265, A104266, A075415, A102807.

def aupton(terms):
  c, k, kk = [0 for i in range(terms)], 1, 1
  while kk < 10**terms:
    s = str(kk)
    c[len(s)-1], k, kk = c[len(s)-1] + (s.count('0')==0), k+1, kk + 2*k + 1
  return c
print(aupton(14)) # Michael S. Branicky, Mar 06 2021

a(14)-a(18) from Donovan Johnson, Nov 05 2009
a(19)-a(21) from Donovan Johnson, Mar 23 2011
a(22)-a(25) from Donovan Johnson, Jan 29 2013

A104265 Smallest n-digit square with no zero digits.

Original entry on oeis.org

1, 16, 121, 1156, 11236, 111556, 1115136, 11115556, 111112681, 1111155556, 11111478921, 111111555556, 1111118377216, 11111115555556, 111111226346761, 1111111155555556, 11111112515384644, 111111111555555556, 1111111112398242916, 11111111115555555556, 111111111113333185156, 1111111111155555555556

Offset: 1

Reinhard Zumkeller, Feb 26 2005

			a(3) = Min{121, 144, 169, 196, ....} = 121.

Chai Wah Wu, Table of n, a(n) for n = 1..362

Cf. A102807, A104266, A104264, A052041.

f[n_] := Block[{k = Ceiling[ Sqrt[10^n]]}, While[ Union[ IntegerDigits[ k^2]][[1]] == 0, k++ ]; k^2]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, Mar 02 2005 *)
snds[n_]:=Module[{c=Ceiling[Sqrt[FromDigits[Join[PadRight[{},n-1,1], {0}]]]]^2},While[DigitCount[c,10,0]>0,c=(1+Sqrt[c])^2];c]; Array[ snds,22] (* Harvey P. Dale, Jun 12 2020 *)

from sympy import integer_nthroot
def A104265(n):
    m, a = integer_nthroot((10**n-1)//9,2)
    if not a:
        m += 1
    k = m**2
    while '0' in str(k):
        m += 1
        k += 2*m-1
    return k # Chai Wah Wu, Mar 24 2020

From Chai Wah Wu, Mar 24 2020: (Start)
a(n) >= (10^n-1)/9.
a(2n) = (10^n+2)^2/9 = A102807(n). Proof: the smallest 2n-digit number without zero digits is (10^(2n)-1)/9. ((10^n-1)/3)^2 = (10^(2n)-2*10^n+1)/9 < (10^(2n)-1)/9 for n >= 1. Thus a(2n) > ((10^n-1)/3)^2. The next square is ((10^n+2)/3)^2 = (10^(2n)-1)/9 + 4*(10^(n)-1)/9 + 1, i.e. it is n 1's followed by n-1 5's followed by the digit 6, and has no zero digits.
(End)

More terms from Robert G. Wilson v, Mar 02 2005
Two more terms from Jon E. Schoenfield, Mar 29 2015
a(21)-a(22) from Chai Wah Wu, Mar 24 2020

A309907 a(n) is the square of the number consisting of one 1 and n 3's: (133...3)^2.

Original entry on oeis.org

1, 169, 17689, 1776889, 177768889, 17777688889, 1777776888889, 177777768888889, 17777777688888889, 1777777776888888889, 177777777768888888889, 17777777777688888888889, 1777777777776888888888889, 177777777777768888888888889, 17777777777777688888888888889

Offset: 0

Alois P. Heinz, Aug 21 2019

All terms are zeroless (elements of A052382).

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A052382, A097166, A102807.

a:= n-> parse(cat(1, 3$n))^2:
seq(a(n), n=0..18);

{a(n) = ((4*10^n-1)/3)^2} \\ Seiichi Manyama, Aug 23 2019

G.f.: -(40*x^2+58*x+1)/((x-1)*(100*x-1)*(10*x-1)).

a(n) = A097166(n)^2 = ((4*10^n-1)/3)^2. - Seiichi Manyama, Aug 23 2019

A034979 Smallest square starting with a string of n 1's.

Original entry on oeis.org

1, 1156, 111556, 11115556, 111112681, 111111555556, 11111115555556, 1111111155555556, 111111111555555556, 11111111115555555556, 111111111113333185156, 11111111111122500004921, 11111111111115555555555556, 111111111111119590793871025, 11111111111111115805344390916, 11111111111111115805344390916, 111111111111111115889741773581604, 11111111111111111167337156002376484, 11111111111111111115555555555555555556, 1111111111111111111155555555555555555556

Offset: 1

Patrick De Geest, Nov 15 1998

Robert Israel, Table of n, a(n) for n = 1..498

Cf. A034978, A102807.

f:= proc(n) local x,k,s;
  x:= (10^n-1)/9;
  for k from 0 do
    s:= ceil(sqrt(10^k*x))^2;
    if s < (x+1)*10^k then return s fi
  od
end proc:
map(f, [$1..20]); # Robert Israel, May 31 2023

a(n) <= A102807(n). - Robert Israel, May 31 2023

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 23 2001

A093140 Expansion of (1-6x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 5, 45, 445, 4445, 44445, 444445, 4444445, 44444445, 444444445, 4444444445, 44444444445, 444444444445, 4444444444445, 44444444444445, 444444444444445, 4444444444444445, 44444444444444445, 444444444444444445, 4444444444444444445, 44444444444444444445, 444444444444444444445

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 4*A001045(3n)/3+(-1)^n. Partial sums of A093141. A convex combination of 10^n and 1. In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1, 1+k, 1+11k, 1+111k, ... This is the case for k=4.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A001045, A002477, A093141, A102807.

CoefficientList[Series[(1-6x)/((1-x)(1-10x)),{x,0,30}],x] (* or *) LinearRecurrence[{11,-10},{1,5},30] (* or *) Join[{1},Table[FromDigits[PadLeft[{5},n,4]],{n,30}]] (* Harvey P. Dale, Dec 17 2022 *)

G.f.: (1-6*x)/((1-x)*(1-10*x)).
a(n) = 4*10^n/9 + 5/9.
a(n+1) = (A102807(n+1)-A002477(n))/((Sum_{i=1..n} 2*10^i) + 3). [Roger L. Bagula, May 22 2010]
a(n) = 10*a(n-1)-5 with a(0)=1. - Vincenzo Librandi, Aug 02 2010
a(n) = 11*a(n-1)-10*a(n-2). - Wesley Ivan Hurt, May 20 2021
E.g.f.: exp(x)*(4*exp(9*x) + 5)/9. - Elmo R. Oliveira, Aug 17 2024

a(19)-a(22) from Elmo R. Oliveira, Aug 17 2024

A177769 a(n) = 111*n.

Original entry on oeis.org

111, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220, 2331, 2442, 2553, 2664, 2775, 2886, 2997, 3108, 3219, 3330, 3441, 3552, 3663, 3774, 3885, 3996, 4107, 4218, 4329, 4440, 4551, 4662, 4773, 4884, 4995, 5106

Offset: 1

Paul Curtz, May 13 2010

The reference contains also sequences A102807, A109344, A075415, and A109492.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Émile Fourrey, Le nombre 37, Récreations arithmétiques, 1899 and later, Vuibert, Paris, page 11.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A075415, A102807, A109344, A109492.

[111*n: n in [1..50]]; // Vincenzo Librandi, Aug 04 2011

G.f.: 111*x/(x-1)^2.
a(n) = 2*a(n-1) - a(n-2).
a(n) = a(n-1) + 111.
E.g.f.: 111*x*exp(x). - Stefano Spezia, Sep 15 2023

A309827 a(n) is the square of the number consisting of one 1 and n 6's: (166...6)^2.

Original entry on oeis.org

1, 256, 27556, 2775556, 277755556, 27777555556, 2777775555556, 277777755555556, 27777777555555556, 2777777775555555556, 277777777755555555556, 27777777777555555555556, 2777777777775555555555556, 277777777777755555555555556, 27777777777777555555555555556

Offset: 0

Seiichi Manyama, Aug 23 2019

All terms are zeroless (element of A052382).

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

((k*10^n+3-k)/3)^2: A102807 (k=1), A109344 (k=2), A098608 (k=3), A309907 (k=4), this sequence (k=5).

Cf. A052382, A246057.

LinearRecurrence[{111,-1110,1000},{1,256,27556},20] (* Harvey P. Dale, Dec 13 2021 *)

PARI
```
{a(n) = ((5*10^n-2)/3)^2}
```

N=20; x='x+O('x^N); Vec((1+145*x+250*x^2)/((1-x)*(1-10*x)*(1-100*x)))

a(n) = A246057(n)^2 = ((5*10^n-2)/3)^2.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
G.f.: (1+145*x+250*x^2)/((1-x)*(1-10*x)*(1-100*x)).

cp's OEIS Frontend

A075415 Squares of A002280 or numbers (666...6)^2.

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A052041 Squares lacking the digit zero in their decimal expansion.

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A093137 Expansion of (1-7*x)/((1-x)*(1-10*x)).

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A104264 Number of n-digit squares with no zero digits.

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A104265 Smallest n-digit square with no zero digits.

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A309907 a(n) is the square of the number consisting of one 1 and n 3's: (133...3)^2.

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A034979 Smallest square starting with a string of n 1's.

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A093140 Expansion of (1-6*x)/((1-x)*(1-10*x)).

A093137 Expansion of (1-7x)/((1-x)(1-10*x)).

A093140 Expansion of (1-6x)/((1-x)(1-10*x)).