A020806 -id:A020806 - cp's OEIS frontend

A133390 Period 18: repeat 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8.

1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8, 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8, 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8, 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8, 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8

Offset: 0

Paul Curtz, Nov 23 2007

Disordered, three times 1, 2, 4, 5, 7, 8.

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

PadRight[{},18*5,{1,4,7,2,2,5,4,1,1,8,5,2,7,7,4,5,8,8}] (* Harvey P. Dale, Nov 06 2011 *)

for(i=1,9,print1("1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8, ")) \\ Charles R Greathouse IV, Jun 02 2011

Sum of digits mod 9 of 1, 4, 7, 11, 20, A130625. Digits of 1/7, A020806 or A029898, 1, 1, 2, 4, 8.

A255873 The first nonzero digit of n/7.

Original entry on oeis.org

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

Offset: 1

Kit Scriven, Mar 08 2015

			The leading (most significant) digit of 22/7 in A068028 is 3, so a(22)=3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A020806 (1/7), A068028 (22/7), A216606 (360/7).

A255873 := proc(n)
    local nshf ;
    nshf := n/7 ;
    while nshf < 1 do
        nshf := 10*nshf;
    end do;
    while nshf >= 10 do
        nshf := nshf/10;
    end do;
    floor(nshf) ;
end proc: # R. J. Mathar, May 28 2016

f[n_] := RealDigits[n/7, 10, 9][[1, 1]]; Array[f, 105] (* Robert G. Wilson v, Mar 08 2015 *)

a(n) = {my(x = n/7.0); if (x < 1, x *= 10); while (x >= 10, x /= 10); floor(x);} \\ Michel Marcus, Mar 12 2015

A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 7, 42, 301, 2100, 14707, 102942, 720601, 5044200, 35309407, 247165842, 1730160901, 12111126300, 84777884107, 593445188742, 4154116321201, 29078814248400, 203551699738807, 1424861898171642, 9974033287201501, 69818233010410500, 488727631072873507, 3421093417510114542

Offset: 0

Ilya Gutkovskiy, Apr 13 2016

In general, the ordinary generating function for the recurrence b(n) = k^n - b(n-1), where n>0 and b(0)=0, is k*x/((1 + x)*(1 - k*x)). This recurrence gives the closed form b(n) = k*(k^n - (-1)^n)/(k + 1).

			a(2) = 7^2 - a(2-1) = 49 - 7 = 42.
a(4) = 7^4 - a(4-1) = 2401 - 301 = 2100.

Index entries for linear recurrences with constant coefficients, signature (6,7).

Cf. A000420, A015552.

Cf. similar sequences with the recurrence b(n) = k^n - b(n-1): A125122 (k=1), A078008 (k=2), A054878 (k=3), A109499 (k=4), A109500 (k=5), A109501 (k=6), this sequence (k=7), A093134 (k=8), A001099 (k=n).

LinearRecurrence[{6, 7}, {0, 7}, 30]
Table[7 (7^n - (-1)^n)/8, {n, 0, 30}]

vector(50, n, n--; 7*(7^n-(-1)^n)/8) \\ Altug Alkan, Apr 13 2016

for n in range(0,10**2):print((int)((7*(7**n-(-1)**n))/8))
# Soumil Mandal, Apr 14 2016

O.g.f.: 7*x/(1 - 6*x - 7*x^2).
E.g.f.: (7/8)*(exp(7*x) - exp(-x)).
a(n) = 6*a(n-1) + 7*a(n-2).
a(n) = 7*(7^n - (-1)^n)/8.
a(n) = 7*A015552(n).
Sum_{n>0} 1/(a(n) + a(n-1)) = 1/6 = A020793.
Limit_{n->oo} a(n-1)/a(n) = 1/7 = A020806.

A355068 Square array read by upwards antidiagonals: T(n,k) = k-th digit after the decimal point in decimal expansion of 1/n, for n >= 1 and k >= 1.

Original entry on oeis.org

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

Offset: 1

Chittaranjan Pardeshi, Jun 17 2022

First row is all zeros since n=1 has all zeros after the decimal point.

			Array begins:
      k=1  2  3  4  5  6  7  8
  n=1:  0, 0, 0, 0, 0, 0, 0, 0,
  n=2:  5, 0, 0, 0, 0, 0, 0, 0,
  n=3:  3, 3, 3, 3, 3, 3, 3, 3,
  n=4:  2, 5, 0, 0, 0, 0, 0, 0,
  n=5:  2, 0, 0, 0, 0, 0, 0, 0,
  n=6:  1, 6, 6, 6, 6, 6, 6, 6,
  n=7:  1, 4, 2, 8, 5, 7, 1, 4,
  n=8:  1, 2, 5, 0, 0, 0, 0, 0,
Row n=7 is 1/7 = .142857142857..., whose digits after the decimal point are 1,4,2,8,5,7,1,4,2,8,5,7, ...

Cf. A020806, A021017, A021018.
Cf. A048962, A257927.
Cf. A061480 (diagonal).
Cf. A355202 (binary).

T(n,k) = my(r=lift(Mod(10,n)^(k-1))); floor(10*r/n)%10;

def T(n,k): return (10*pow(10,k-1,n)//n)%10

1/n = Sum_{k>=1} T(n, k)*10^-k, for n > 1.

A021039 Decimal expansion of 1/35.

Original entry on oeis.org

0, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8

Offset: 0

N. J. A. Sloane

Same decimal period as A020806. - R. J. Mathar, Feb 13 2012

			0.0285714285714285..

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

Cf. A020806.

Join[{0},RealDigits[1/35,10,120][[1]]] (* or *) PadRight[{0},120,{4,2,8,5,7,1}] (* Harvey P. Dale, Oct 15 2013 *)

A021101 Decimal expansion of 1/97.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

From Daniel Forgues, Oct 28 2011: (Start)
Generalization:
1/7 = Sum_{i>=0} 3^i/10^(i+1) (A020806)
1/97 = Sum_{i>=0} 3^i/100^(i+1) (this sequence);
1/997 = Sum_{i>=0} 3^i/1000^(i+1) (A022001);
1/9997 = Sum_{i>=0} 3^i/1000^(i+1); ... (End)

			0.0103092783505154639175257731...

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 77.

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,-1,1).

Cf. A020806, A022001.

[0] cat Reverse(Intseq(Floor(10^100/97))); // Vincenzo Librandi, Jan 04 2025

First@ RealDigits[1/97, 10, 100, -1] (* Vincenzo Librandi, Jan 04 2025 *)

localprec(98); 1/97. \\ Michel Marcus, Dec 31 2024

A021116 Decimal expansion of 1/112.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Same decimal period as A020806. - R. J. Mathar, Feb 13 2012

			0.0089285714285714...

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

Cf. A020806.

Join[{0,0},RealDigits[1/112,10,120][[1]]] (* or *) PadRight[{0,0,8,9},120,{5,7,1,4,2,8}] (* Harvey P. Dale, Apr 01 2014 *)

1/112. \\ Charles R Greathouse IV, Feb 13 2012

A021228 Decimal expansion of 1/224.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Same decimal period as A020806. - R. J. Mathar, Feb 13 2012

			0.0044642857142857..

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

Cf. A020806.

Join[{0,0},RealDigits[1/224 ,10,120][[1]]]  (* Harvey P. Dale, Apr 23 2011 *)

A021319 Decimal expansion of 1/315.

Original entry on oeis.org

0, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3

Offset: 0

N. J. A. Sloane

If the initial 0 is ignored, a(n) is periodic with period 6: [0, 3, 1, 7, 4, 6]. - Wesley Ivan Hurt, Oct 10 2014

			1/315 = 0.0031746031746031746031746031...

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

Cf. A020806, A068028.

Digits:=100: evalf(1/315); # Wesley Ivan Hurt, Oct 10 2014

RealDigits[1/315, 10, 100, -1][[1]] (* Wesley Ivan Hurt, Oct 10 2014 *)
Join[{0},LinearRecurrence[{1, 0, -1, 1},{0, 3, 1, 7},98]] (* Ray Chandler, Aug 26 2015 *)

a(n) = a(n-1)-a(n-3)+a(n-4) for n>0, with a(0)=0; a(n) = A068028(n+1)-1 for n>0; a(n+1) = A020806(n)-1. - Wesley Ivan Hurt, Oct 10 2014

G.f.: x^2*(-6*x^2 + 2*x - 3)/(x^4 - x^3 + x - 1). - Chai Wah Wu, Sep 04 2025

A021452 Decimal expansion of 1/448.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Same decimal period as A020806. - R. J. Mathar, Feb 13 2012

			0.00223214285714285714285714285714285714285714285714...

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

Cf. A020806.

First[RealDigits[1/448, 10, 100, -1]] (* Paolo Xausa, Aug 15 2025 *)

cp's OEIS Frontend

A133390 Period 18: repeat 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8.

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A255873 The first nonzero digit of n/7.

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A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

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A355068 Square array read by upwards antidiagonals: T(n,k) = k-th digit after the decimal point in decimal expansion of 1/n, for n >= 1 and k >= 1.

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A021039 Decimal expansion of 1/35.

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A021101 Decimal expansion of 1/97.

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Magma

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A021116 Decimal expansion of 1/112.

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