A073551 -id:A073551 - cp's OEIS frontend

A002280 a(n) = 6*(10^n - 1)/9.

0, 6, 66, 666, 6666, 66666, 666666, 6666666, 66666666, 666666666, 6666666666, 66666666666, 666666666666, 6666666666666, 66666666666666, 666666666666666, 6666666666666666, 66666666666666666, 666666666666666666, 6666666666666666666, 66666666666666666666, 666666666666666666666

Offset: 0

N. J. A. Sloane

a(n-1) = number of Fibonacci numbers F(k), k <= 10^n, which end in 0. a(1)=6 because there are 6 Fibonacci numbers up to 10^2 which end in 0. - Shyam Sunder Gupta and Benoit Cloitre, Aug 15 2002

a(n) is the total number of holes in a certain triangle fractal (start with 10 triangles, 6 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Feb 21 2015

Ivan Panchenko, Table of n, a(n) for n = 0..200
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Different from A072912. Cf. A073548, etc.
Cf. A002275, A002276, A002277, A002278, A002279, A002281, A002282, A002283, A075415, A178631, A178633.
Cf. A010785, A017233, A073551.

LinearRecurrence[{11,-10},{0,6},20] (* Harvey P. Dale, Dec 20 2012 *)

a(n)=6*(10^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 6*A002275(n).
From Jaume Oliver Lafont, Feb 03 2009: (Start)
G.f.: 6*x/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=6. (End)
a(n) = A178633(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 6*10^(n-1) with a(0)=0. - Vincenzo Librandi, Jul 22 2010
E.g.f.: 2*exp(x)*(exp(9*x) - 1)/3. - Stefano Spezia, Sep 13 2023
From Elmo R. Oliveira, Jul 21 2025: (Start)
a(n) = A073551(n+1)/2 for n >= 1.
a(n) = A010785(A017233(n-1)) for n >= 1. (End)

A073548 Number of Fibonacci numbers F(k), k <= 10^n, which end in 2.

Original entry on oeis.org

1, 6, 66, 666, 6666, 66666, 666666, 6666666, 66666666, 666666666, 6666666666, 66666666666, 666666666666, 6666666666666, 66666666666666, 666666666666666, 6666666666666666, 66666666666666666, 666666666666666666, 6666666666666666666, 66666666666666666666, 666666666666666666666, 6666666666666666666666, 66666666666666666666666

Offset: 1

Shyam Sunder Gupta, Aug 15 2002

			a(2) = 6 because there are 6 Fibonacci numbers up to 10^2 which end in 2.

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

Cf. A000045, A002280, A073549, A073550, A073551, etc.

LinearRecurrence[{11,-10},{1,6,66},30] (* Harvey P. Dale, May 02 2016 *)

If n>1 then a(n) = (10^n - 10)/15. - Robert Gerbicz, Sep 06 2002
From Paul Barry, Mar 24 2004: (Start)
G.f.: (1-5*x+10*x^2)/((1-x)*(1-10*x)).
a(n) = 2*(10^n - 1)/3 + 0^n (offset 0). (End)
From Elmo R. Oliveira, Jul 21 2025: (Start)
E.g.f.: (9 + 15*x - 10*exp(x) + exp(10*x))/15.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 3.
a(n) = A073551(n)/2. (End)

More terms from Robert Gerbicz, Sep 06 2002

A351473 Numbers m such that the largest digit in the decimal expansion of 1/m is 7.

Original entry on oeis.org

27, 36, 37, 44, 132, 135, 148, 234, 270, 288, 292, 297, 308, 315, 360, 364, 369, 370, 404, 407, 440, 468, 576, 616, 636, 657, 707, 728, 756, 808, 864, 1287, 1295, 1313, 1314, 1320, 1332, 1350, 1365, 1375, 1386, 1404, 1408, 1476, 1480, 1485, 1507, 1512, 1752, 1804, 1896

Offset: 1

Bernard Schott and Robert G. Wilson v, Feb 17 2022

If k is a term, 10*k is also a term.
First few primitive terms are 27, 36, 37, 44, 132, 135, 148, 234, 288, ...
The unique prime up to 2.6*10^8 is 37 (see comments in A333237 and example).
Subsequence: {132, 1332, 13332, ...} = A073551 \ {2, 12}.

			As 1/37 = 0.027027027..., 37 is a term.
As 1/148 = 0.00675675675675..., 148 is a term.

Index entries for sequences related to decimal expansion of 1/n.

Similar with largest digit k: A333402 (k=1), A341383 (k=2), A350814 (k=3), A351470 (k=4), A351471 (k=5), A351472 (k=6), this sequence (k=7), A351474 (k=8), A333237 (k=9).

Cf. A073551, A333236.

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 7 &]

from itertools import count, islice
from sympy import multiplicity, n_order
def A351473_gen(startvalue=1): # generator of terms >= startvalue
    for a in count(max(startvalue,1)):
        m2, m5 = (~a&a-1).bit_length(), multiplicity(5,a)
        k, m = 10**max(m2,m5), 10**n_order(10,a//(1<A351473_list = list(islice(A351473_gen(),20)) # Chai Wah Wu, May 02 2023

A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 11, 2, 0, 0, 111, 22, 3, 0, 0, 1111, 222, 33, 4, 0, 0, 11111, 2222, 333, 44, 5, 0, 0, 111111, 22222, 3333, 444, 55, 6, 0, 0, 1111111, 222222, 33333, 4444, 555, 66, 7, 0, 0, 11111111, 2222222, 333333, 44444, 5555, 666, 77, 8, 0

Offset: 0

Stefano Spezia, Sep 14 2023

			The array begins:
  0,     0,     0,     0,     0,     0, ...
  0,     1,     2,     3,     4,     5, ...
  0,    11,    22,    33,    44,    55, ...
  0,   111,   222,   333,   444,   555, ...
  0,  1111,  2222,  3333,  4444,  5555, ...
  0, 11111, 22222, 33333, 44444, 55555, ...
  ...

Cf. A000004 (n=0 or k=0), A001477 (n=1), A002275 (k=1), A002276 (k=2), A002277 (k=3), A002278 (k=4), A002279 (k=5), A002280 (k=6), A002281 (k=7), A002282 (k=8), A002283 (k=9), A008593 (n=2), A053422 (main diagonal), A105279 (k=10), A132583, A177769 (n=3), A365645 (antidiagonal sums), A365646.

A[n_,k_]:=k(10^n-1)/9; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

O.g.f.: x*y/((1 - x)*(1 - 10*x)*(1 - y)^2).
E.g.f.: y*exp(x+y)*(exp(9*x) - 1)/9.
A(n, 11) = A132583(n-1) for n > 0.
A(n, 12) = A073551(n+1) for n > 0.

A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

Original entry on oeis.org

78, 8778, 887778, 88877778, 8888777778, 888887777778, 88888877777778, 8888888777777778, 888888887777777778, 88888888877777777778, 8888888888777777777778, 888888888887777777777778, 88888888888877777777777778, 8888888888888777777777777778

Offset: 1

David Radcliffe, Aug 18 2025

This is one of four infinite families of triangular numbers consisting of two different digits. The other three families are A319170, A037156 (n>1), and A309597 (n>2).

Paolo Xausa, Table of n, a(n) for n = 1..495
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A000217, A037156, A062691, A073551, A256340, A309597, A319170.

Subsequence of: A119216, A119230, A119236.

A383942[n_] := (8*10^(2*n) - 10^(n+1) + 2)/9; Array[A383942, 15] (* or *)
LinearRecurrence[{111, -1110, 1000}, {78, 8778, 887778}, 15] (* Paolo Xausa, Aug 27 2025 *)

def A383942(n): return (8*10**(2*n)-10**(n+1)+2)//9

a(n) = A000217(A073551(n+1)).

G.f.: 6*x*(13 + 20*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Stefano Spezia, Aug 19 2025

A073554 Number of Fibonacci numbers F(k), k <= 10^n, which end in 7.

Original entry on oeis.org

0, 14, 134, 1334, 13334, 133334, 1333334, 13333334, 133333334, 1333333334, 13333333334, 133333333334, 1333333333334, 13333333333334, 133333333333334, 1333333333333334, 13333333333333334, 133333333333333334, 1333333333333333334, 13333333333333333334, 133333333333333333334, 1333333333333333333334, 13333333333333333333334

Offset: 1

Shyam Sunder Gupta, Aug 15 2002

			a(2) = 14 because there are 14 Fibonacci numbers up to 10^2 which end in 7.

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

Cf. A073548 (end in 2), A073549 (6), A073550 (1), A073551 (3), (A073552 (4)), A073553 (5), this sequence (7), A073555 (8), A073556 (9).

Cf. A000045, A067275.

Join[{0},Table[10 FromDigits[PadRight[{1},n,3]]+4,{n,30}]] (* Harvey P. Dale, Mar 29 2023 *)

If n>1 then a(n) = (2*10^n + 10)/15. - Robert Gerbicz, Sep 06 2002
a(n) = A073550(n) for n >= 3. - Georg Fischer, Oct 13 2022
From Elmo R. Oliveira, Jul 22 2025: (Start)
G.f.: 2*x^2*(7 - 10*x)/((1-x)*(1-10*x)).
E.g.f.: 2*(-6 - 15*x + 5*exp(x) + exp(10*x))/15.
a(n) = 2*A067275(n) for n >= 2.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 3. (End)

More terms from Robert Gerbicz, Sep 06 2002

cp's OEIS Frontend

A002280 a(n) = 6*(10^n - 1)/9.

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A073548 Number of Fibonacci numbers F(k), k <= 10^n, which end in 2.

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A351473 Numbers m such that the largest digit in the decimal expansion of 1/m is 7.

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A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

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A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

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A073554 Number of Fibonacci numbers F(k), k <= 10^n, which end in 7.

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Mathematica

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Extensions