A021093 -id:A021093 - cp's OEIS frontend

A031324 Decimal digits of successive Fibonacci numbers.

0, 1, 1, 2, 3, 5, 8, 1, 3, 2, 1, 3, 4, 5, 5, 8, 9, 1, 4, 4, 2, 3, 3, 3, 7, 7, 6, 1, 0, 9, 8, 7, 1, 5, 9, 7, 2, 5, 8, 4, 4, 1, 8, 1, 6, 7, 6, 5, 1, 0, 9, 4, 6, 1, 7, 7, 1, 1, 2, 8, 6, 5, 7, 4, 6, 3, 6, 8, 7, 5, 0, 2, 5, 1, 2, 1, 3, 9, 3, 1, 9, 6, 4, 1, 8, 3, 1, 7, 8, 1, 1, 5

Offset: 0

Clark Kimberling

Decimal concatenation of Fibonacci numbers in base 10. - Daniel Forgues, Mar 25 2018

			0.011235813213455891442333776109871597...

Robert Israel, Table of n, a(n) for n = 0..10000
Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.

Cf. A000045, A021093, A031325 - A031334.

Cf. A033307, A033308.

F:= [seq(combinat:-fibonacci(n),n=0..50)]:
map(t -> op(ListTools:-Reverse(convert(t,base,10))),F); # Robert Israel, Oct 11 2024

Flatten[IntegerDigits/@Fibonacci[Range[0,30]]] (* Harvey P. Dale, Jan 28 2015 *)

An approximation, where each successive Fibonacci number is shifted right by one place (thus causing an overlap when numbers have more than one digit), is given by 10/89 (A021093). - Daniel Forgues, Mar 25 2018

A021113 Decimal expansion of 1/109.

Original entry on oeis.org

0, 0, 9, 1, 7, 4, 3, 1, 1, 9, 2, 6, 6, 0, 5, 5, 0, 4, 5, 8, 7, 1, 5, 5, 9, 6, 3, 3, 0, 2, 7, 5, 2, 2, 9, 3, 5, 7, 7, 9, 8, 1, 6, 5, 1, 3, 7, 6, 1, 4, 6, 7, 8, 8, 9, 9, 0, 8, 2, 5, 6, 8, 8, 0, 7, 3, 3, 9, 4, 4, 9, 5, 4, 1, 2, 8, 4, 4, 0, 3, 6, 6, 9, 7, 2, 4, 7, 7, 0, 6, 4, 2, 2, 0, 1, 8, 3, 4, 8, 6, 2, 3, 8, 5, 3, 2, 1, 1

Offset: 0

N. J. A. Sloane

From Paul Curtz, Feb 23 2012: (Start)
The sequence of digits is periodic with period length 108. A feature of the period reading from the least significant digit back to the most significant digit is (see the blogspot link and A064737) that it "contains" the single-digit of every Fibonacci subsequence if the digits are added with carry of the previous sum. A064737 starts with the A000045 sequence, and then 5+8 = (1)3, 3+8+1=(1)2. "Every" Fibonacci sequence means (as illustrated in the blog) that one could also start from seeds like 6 and 7, or 7 and 8.
Similar observations are made for the digits of 1/89 in A021093, but following a Fibonacci pattern while reading in the other direction, starting with the most significant digits.
The frequency distribution of the digits 0 to 9 among the 108 digits (which sum to 486) of the period is well-balanced: 10, 11, 11, 11, 11, 11, 11, 11, 11, 10. If one sums over each 2nd, each 3rd, each 6th, each 9th or each 18th digit of the period, one gets 1/2, 1/3, 1/6, 1/9 and 1/18 of 486; again a feature of balance in the digits. There is a half-period in the sense that a(n) + a(n+54) = 9. (End)

			0.00917431192660550458715596330275229357798165137614...

"Xochipilli", Le Webinet des curiosités:les meilleures recettes de Kaprekar (in French)
Guillaume Yoda, Dictionnaire des nombres - 109 (in French)
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A000045, A018481, A021093, A064737.

RealDigits[1/109, 10, 100][[1]] (* Alonso del Arte, Feb 11 2012 *)

1/109. \\ Charles R Greathouse IV, Feb 13 2012

Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/10^(k+1). - Amiram Eldar, Feb 05 2022

A165155 a(n) = 100*a(n-1) + 11^(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 1, 111, 11221, 1123431, 112357741, 11235935151, 1123595286661, 112359548153271, 11235955029685981, 1123595505326545791, 112359550558592003701, 11235955056144512040711, 1123595505617589632447821, 112359550561793485956926031, 11235955056179728345526186341

Offset: 0

Mark Dols, Sep 05 2009

Generalization of A000225. - Mark Dols, Jan 28 2010

			From _Mark Dols_, Jan 28 2010: (Start)
As triangle:
  ........... 1
  ......... 1 1 1
  ....... 1 1 2 2 1
  ..... 1 1 2 3 4 3 1
  ... 1 1 2 3 5 7 7 4 1
  . 1 1 2 3 5 9 3 5 1 5 1
  1 1 2 3 5 9 5 2 8 6 6 6 1
(Mirrored version of A162741) (End)

G. C. Greubel, Table of n, a(n) for n = 0..495
Index entries for linear recurrences with constant coefficients, signature (111,-1100).

Cf. A000225, A021093, A094704, A162741, A164913.

[(1/89)*(100^n-11^n): n in [0..40]] // Vincenzo Librandi, Dec 05 2010

RecurrenceTable[{a[0]==0,a[n]==100a[n-1]+11^(n-1)},a,{n,40}] (* Harvey P. Dale, Feb 20 2016 *)

[(10^(2*n) - 11^n)/89 for n in range(41)] # G. C. Greubel, Feb 09 2023

G.f.: x/((1-11*x)*(1-100*x)). - R. J. Mathar, Nov 02 2016

E.g.f.: (1/89)*(exp(100*x) - exp(11*x)). - G. C. Greubel, Feb 09 2023

a(0) prepended by Bruno Berselli, Oct 02 2015

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)

A227093 Decimal expansion of 1/9899.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 1, 3, 2, 1, 3, 4, 5, 5, 9, 0, 4, 6, 3, 6, 8, 3, 2, 0, 0, 3, 2, 3, 2, 6, 4, 9, 7, 6, 2, 6, 0, 2, 2, 8, 3, 0, 5, 8, 8, 9, 4, 8, 3, 7, 8, 6, 2, 4, 1, 0, 3, 4, 4, 4, 7, 9, 2, 4, 0, 3, 2, 7, 3, 0, 5, 7, 8, 8, 4, 6, 3, 4, 8, 1, 1, 5, 9, 7, 1, 3, 1, 0, 2, 3, 3, 3, 5, 6, 9, 0, 4, 7

Offset: 0

Roland Kneer, Jul 01 2013

Group the terms 2 by 2 to get the first 11 Fibonacci numbers (A000045): 00 01 01 02 03 05 08 13 21 34 55 (89, 144, 233, ...).

			0.00010102030508132134559046368320032326497626022830588...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A000045, A021093, A034948.

Maple
```
Digits := 140; evalf(1/9899);
```

First[RealDigits[1/9899, 10, 100, -1]] (* Paolo Xausa, Jun 16 2024 *)

Equals Sum_{i>=0} Fibonacci(i)/100^(i+1).

A021182 Decimal expansion of 1/178.

Original entry on oeis.org

0, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Cf. A021093.

Digits:=100: evalf(1/178); # Wesley Ivan Hurt, Jan 24 2017

Join[{0,0},RealDigits[1/178,10,120][[1]]] (* or *) PadRight[{0},120,{5,0,5,6,1,7,9,7,7,5,2,8,0,8,9,8,8,7,6,4,0,4,4,9,4,3,8,2,0,2,2,4,7,1,9,1,0,1,1,2,3,5,9,5}] (* Harvey P. Dale, Sep 25 2017 *)

1/178. \\ Charles R Greathouse IV, Dec 05 2011

A021360 Decimal expansion of 1/356.

Original entry on oeis.org

0, 0, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6

Offset: 0

N. J. A. Sloane

			0.002808988764044943820224719101123595505617977528...

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Cf. A021093.

First[RealDigits[1/356, 10, 100, -1]] (* Paolo Xausa, Aug 12 2025 *)

1/356. \\ Charles R Greathouse IV, Dec 05 2011

A021449 Decimal expansion of 1/445.

Original entry on oeis.org

0, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1

Offset: 0

N. J. A. Sloane

Expansion in any base b >= 3 of 2/(b(b-1)(b+1)) = 2/(b^3-b). E.g., 1/12 in base 3, 1/30 in base 4, 1/60 in base 5, etc. - Franklin T. Adams-Watters, Nov 07 2006

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Cf. A027480, A021093.

Join[{0,0},RealDigits[1/445,10,120][[1]]] (* Harvey P. Dale, Aug 25 2015 *)

1/445. \\ Charles R Greathouse IV, Dec 05 2011

From Chai Wah Wu, May 08 2025: (Start)
a(n) = a(n-1) - a(n-22) + a(n-23) for n > 23.
G.f.: x^2*(-2*x^21 - 6*x^20 + 5*x^19 - x^18 - 5*x^17 + 5*x^16 + 4*x^14 - 4*x^13 - 2*x^12 - x^11 - x^10 - x^8 + x^7 + 8*x^6 - 8*x^5 + 6*x^4 - 3*x^3 - 2*x^2 - 2)/(x^23 - x^22 + x - 1). (End)

A021716 Decimal expansion of 1/712.

Original entry on oeis.org

0, 0, 1, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8

Offset: 0

N. J. A. Sloane

			0.0014044943820224719101123595505617977528089887640449...

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Cf. A021093.

First[RealDigits[1/712, 10, 100, -1]] (* Paolo Xausa, Aug 27 2025 *)

1/712. \\ Charles R Greathouse IV, Dec 05 2011

From Chai Wah Wu, Apr 22 2024: (Start)
a(n) = a(n-1) - a(n-22) + a(n-23) for n > 25.
G.f.: x^2*(-5*x^23 - 2*x^22 - x^21 - x^19 + x^18 + 8*x^17 - 8*x^16 + 6*x^15 - 3*x^14 - 2*x^13 - 2*x^11 + 2*x^10 + 6*x^9 - 5*x^8 + x^7 + 5*x^6 - 5*x^5 - 4*x^3 + 4*x^2 - 3*x - 1)/(x^23 - x^22 + x - 1). (End)

A177940 Decimal expansion of 190/89.

Original entry on oeis.org

2, 1, 3, 4, 8, 3, 1, 4, 6, 0, 6, 7, 4, 1, 5, 7, 3, 0, 3, 3, 7, 0, 7, 8, 6, 5, 1, 6, 8, 5, 3, 9, 3, 2, 5, 8, 4, 2, 6, 9, 6, 6, 2, 9, 2, 1, 3, 4, 8, 3, 1, 4, 6, 0, 6, 7, 4, 1, 5, 7, 3, 0, 3, 3, 7, 0, 7, 8, 6, 5, 1, 6, 8, 5, 3, 9, 3, 2, 5, 8, 4, 2, 6, 9, 6, 6, 2, 9, 2, 1, 3, 4, 8, 3, 1, 4, 6, 0, 6, 7, 4, 1, 5, 7, 3

Offset: 1

Mark Dols, May 15 2010

			The sum of the shifted Lucas numbers begins:
2
0.1
0.03
0.004
0.0007
0.00011
0.000018
.......29
---------- +
2.134831...

Cf. A000032, A021093.

RealDigits[190/89,10,120][[1]] (* Harvey P. Dale, Jul 18 2018 *)

190/89. \\ Charles R Greathouse IV, Apr 25 2016

equals sum_{n=0..infinity} A000032(n)/10^n.

Keyword:cons added, formula and offset adjusted - R. J. Mathar, May 19 2010

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A031324 Decimal digits of successive Fibonacci numbers.

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A021113 Decimal expansion of 1/109.

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A165155 a(n) = 100*a(n-1) + 11^(n-1) for n>0, a(0)=0.

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

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A227093 Decimal expansion of 1/9899.

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A021182 Decimal expansion of 1/178.

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A021360 Decimal expansion of 1/356.

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