A010727 -id:A010727 - cp's OEIS frontend

A021040 Decimal expansion of 1/36.

0, 2, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

N. J. A. Sloane

			0.0277777777777777777777777777777777777777777777777777777777777777777777777777...

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

Cf. A010727.

Join[{0},RealDigits[1/36,10,120][[1]]] (* or *) PadRight[{0,2},120,{7}] (* Harvey P. Dale, Jul 14 2017 *)

PARI
```
1./36
```

A321358 a(n) = (2*4^n + 7)/3.

Original entry on oeis.org

3, 5, 13, 45, 173, 685, 2733, 10925, 43693, 174765, 699053, 2796205, 11184813, 44739245, 178956973, 715827885, 2863311533, 11453246125, 45812984493, 183251937965, 733007751853, 2932031007405, 11728124029613, 46912496118445, 187649984473773, 750599937895085, 3002399751580333

Offset: 0

Paul Curtz, Nov 07 2018

Difference table:
3,  5, 13,  45,  173,  685,  2733, ...   (this sequence)
2,  8, 32, 128,  512, 2048,  8192, ...    A004171
6, 24, 96, 384, 1536, 6144, 24576, ...    A002023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A010701, A010727, A020988, A083594, A002023, A004171, A155980, A171382, A193579.

a[n_]:= (2*4^n + 7)/3; Array[a, 20, 0] (* or *)
CoefficientList[Series[1/3 (7 E^x + 2 E^(4 x)), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 10 2018 *)

a(n) = (2*4^n + 7)/3; \\ Michel Marcus, Nov 08 2018

Vec((3 - 10*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 10 2018

O.g.f.: (3 - 10*x) / ((1 - x)*(1 - 4*x)). - Colin Barker, Nov 10 2018
E.g.f.: (1/3)*(7*exp(x) + 2*exp(4*x)). - Stefano Spezia, Nov 10 2018
a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 3, a(1) = 5.
a(n) = 4*a(n-1) - 7, a(0) = 3.
a(n) = (2/3)*(4^n-1)/3 + 3.
a(n) = A171382(2*n) = A155980(2*n+2).
a(n) = A193579(n)/3.
a(n) = A007583(n) + 2 = A001045(2*n+1) + 2.

More terms from Michel Marcus, Nov 08 2018

A321421 a(n) = 10*(4^n - 1)/3 + 1.

Original entry on oeis.org

1, 11, 51, 211, 851, 3411, 13651, 54611, 218451, 873811, 3495251, 13981011, 55924051, 223696211, 894784851, 3579139411, 14316557651, 57266230611, 229064922451, 916259689811, 3665038759251, 14660155037011, 58640620148051, 234562480592211, 938249922368851

Offset: 0

Paul Curtz, Nov 09 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000302, A002450, A010727.

List([0..25],n->10*(4^n-1)/3+1); # Muniru A Asiru, Nov 10 2018

seq(coeff(series((1+6*x)/((1-x)*(1-4*x)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Nov 10 2018

a[n_]:=10*(4^n - 1)/3 + 1 ; Array[a, 20, 0] (* or *)
CoefficientList[Series[-((7 E^x)/3) + (10 E^(4 x))/3 , {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 10 2018 *)
LinearRecurrence[{5,-4},{1,11},30] (* Harvey P. Dale, Aug 22 2020 *)

Vec((1 + 6*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 10 2018

a(n) = 4*a(n-1) + 7, a(0) = 1 for n > 0.
a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 1, a(1) = 11, n > 1.
a(n) = a(n-1) + 10*4^(n-1), a(0) = 1, n > 0.
a(n) = A086462(n) + 1 for n > 0. - Michel Marcus, Nov 09 2018
G.f.: (1 + 6*x) / ((1 - x)*(1 - 4*x)). - Colin Barker, Nov 10 2018
E.g.f.: (-7*exp(x) + 10*exp(4*x))/3. - Stefano Spezia, Nov 10 2018
a(n) = 10*A002450(n) + 1. - Omar E. Pol, Nov 10 2018

More terms from Colin Barker, Nov 10 2018

A044567 Numbers n such that string 6,6 occurs in the base 7 representation of n but not of n+1.

Original entry on oeis.org

48, 97, 146, 195, 244, 293, 342, 391, 440, 489, 538, 587, 636, 685, 734, 783, 832, 881, 930, 979, 1028, 1077, 1126, 1175, 1224, 1273, 1322, 1371, 1420, 1469, 1518, 1567, 1616, 1665, 1714, 1763, 1812, 1861, 1910, 1959, 2008, 2057, 2106, 2155, 2204, 2253, 2302

Offset: 1

Clark Kimberling

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Non Recursions

Cf. A157362, A010727 [From Vincenzo Librandi, Mar 12 2009]

SequencePosition[Table[If[SequenceCount[IntegerDigits[n,7],{6,6}]>0,1,0],{n,2500}],{1,0}][[All,1]] (* Harvey P. Dale, Aug 20 2021 *)

Incorrect Mathematica program deleted by Harvey P. Dale, Aug 20 2021

A253671 a(n) = floor(A000111(n)/A000111(n-1)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40

Offset: 1

Paul Curtz, Jan 08 2015, with the help of Jean-François Alcover

1, 2, 3, 4, ... first appear at n = 1, 3, 5, 7, 8, 10, 11, 13, ... . a(500) = 318.
Numbers appearing only once: interleave 4+7*n, 6+7*n, 9+7*n = 4, 6, 9, 11, 13, 16, ... .
This is a nondecreasing sequence.
The ratio a(n)/n asymptotically tends to 7/11 = 0.6363... - Jean-François Alcover, Jul 21 2015

			Floor of 1/1, 1/1, 2/1, 5/2, 16/5, 61/16, ... .
1=1*1+0, 1=1*1+0, 2=2*1+0, 5=2*2+1, 16=3*5+1, 61=3*16+13, 272=4*61+28, ... .

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A000111, A010727, A017005, A017029, A017053.

max = 500; ee = Table[2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, max}]; A000111 = Table[Differences[ee, n] // First // Abs, {n, 0, max}]; Table[Quotient[A000111[[n + 1]], A000111[[n]]], {n, 1, max}] (* Jean-François Alcover, Jan 08 2015 *)

Vec(x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1)/((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Jan 22 2015

# requires python 3.2 or higher
from itertools import accumulate
A253671_list, blist, l1, l2 = [1], [1], 1, 1
for n in range(10**2):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    l2, l1 = l1, sum(blist)
    A253671_list.append(l1//l2) # Chai Wah Wu, Jan 29 2015

a(n+2) = a(n+1) + (0, 1, 0, followed by a sequence of period 11: repeat 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1).
a(n+12) = a(n+1) + (6, 7, 6, followed by 7's = A010727).
a(n) = a(n-1) + a(n-11) - a(n-12) for n>15. - Colin Barker, Jan 22 2015
G.f.: x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1) / ((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). - Colin Barker, Jan 22 2015

A255910 Decimal expansion of 16/9.

Original entry on oeis.org

1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Derek Orr, Mar 10 2015

Cutting the unit square [0,1] x [0,1] into two equal areas with a parabolic curve y = A*x^2 requires A to be 16/9. If you extend this to an arbitrary square [0,s] x [0,s], A = (16/9)*s.

Except for the first terms, identical to A186684, A021040 and A010727.

			1.7777777777777777777777777777...

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

Cf. A010727, A021040, A122553, A186684.

RealDigits[16/9, 10, 100][[1]] (* Vincenzo Librandi, Mar 24 2015 *)

x=16/9; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d,", "))

From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: x + 7*x^2/(1 - x).
E.g.f.: 7*(exp(x) - 1) - 6*x.
a(n) = 7 - 6*0^(n-1).
a(n) = 7, n > 1. (End)

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A021040 Decimal expansion of 1/36.

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A321358 a(n) = (2*4^n + 7)/3.

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A321421 a(n) = 10*(4^n - 1)/3 + 1.

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A044567 Numbers n such that string 6,6 occurs in the base 7 representation of n but not of n+1.

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A253671 a(n) = floor(A000111(n)/A000111(n-1)).

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A255910 Decimal expansion of 16/9.

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