A001018 -id:A001018 - cp's OEIS frontend

A164675 a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 12.

1, 12, 8, 96, 64, 768, 512, 6144, 4096, 49152, 32768, 393216, 262144, 3145728, 2097152, 25165824, 16777216, 201326592, 134217728, 1610612736, 1073741824, 12884901888, 8589934592, 103079215104, 68719476736, 824633720832

Offset: 1

Klaus Brockhaus, Aug 20 2009

Interleaving of A001018 and 12*A001018.

Binomial transform is A164539.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A001018 (powers of 8), A164539.

[ n le 2 select 11*n-10 else 8*Self(n-2): n in [1..26] ];

Riffle[#, 12*#] & [8^Range[0, 14]] (* or *)
LinearRecurrence[{0, 8}, {1, 12}, 30] (* Paolo Xausa, Apr 22 2024 *)

a(n) = (5+(-1)^n)*2^(1/4*(6*n-11+3*(-1)^n)).

G.f.: x*(1+12*x)/(1-8*x^2).

A240841 a(n) = floor(8^n/(1+2sin(6Pi/13)/(2*sin(Pi/13)))^n).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 14, 21, 34, 52, 82, 127, 198, 308, 478, 744, 1156, 1796, 2792, 4339, 6742, 10477, 16282, 25302, 39318, 61100, 94947, 147545, 229281, 356295, 553672, 860388, 1337014, 2077676, 3228640, 5017200, 7796562, 12115600, 18827241, 29256909, 45464268

Offset: 0

Kival Ngaokrajang, Apr 13 2014

a(n) is the perimeter (rounded down) of a tridecaflake after n iterations, let a(0) = 1. The total number of sides is 13*A001018(n). The total number of holes is A091030(n), n >= 1.

Kival Ngaokrajang, Illustration of tridecaflake for n = 0..3
Wikipedia, n-flake

Cf. A001018, A091030, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake).

A240841:=n->floor(8^n/(1+2*sin(6*Pi/13)/(2*sin(Pi/13)))^n); seq(A240841(n), n=0..50); # Wesley Ivan Hurt, Apr 13 2014

Table[Floor[8^n/(1 + 2*Sin[6*Pi/13]/(2*Sin[Pi/13]))^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 13 2014 *)

{a(n)=floor(8^n/(1+2*sin(6*Pi/13)/(2*sin(Pi/13)))^n)}

A248217 a(n) = 8^n - 2^n.

Original entry on oeis.org

0, 6, 60, 504, 4080, 32736, 262080, 2097024, 16776960, 134217216, 1073740800, 8589932544, 68719472640, 549755805696, 4398046494720, 35184372056064, 281474976645120, 2251799813554176, 18014398509219840, 144115188075331584, 1152921504605798400

Offset: 0

Vincenzo Librandi, Oct 04 2014

If 2^(n+1) is the length of the even leg of a primitive Pythagorean triangle (PPT) then it constrains the odd leg to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n+2^n, an area of 8^n-2^n and an inradius of 2^n-1. For n > 0, a(n) is the area of such triangles. - Frank M Jackson, Sep 07 2018

Maximum anomalous cancellation multiplicity of (2n+1)-digit integers: number of (2n+1)-digit integers which can be anomalously canceled with a fixed (2n+1)-digit integer. The maximum is obtained at 88...88911...11 containing n 8's and n 1's (see Example below). Anomalous cancellation is a "canceling" of digits of a and b in the numerator and denominator of a fraction a/b which results in a fraction equal to the original, and no 0 or digits that appear different times in a and b are canceled. For example, 49/98 = 4/8, 138/184 = 3/4, 1985/5955 = 185/555, 88911/43956 = 8811/4356, but 120/340 is not because canceling the 0's is not an anomalous cancellation. - Xiaohan Zhang, Nov 21 2019

			For n=1, there are 6 numbers with 3 digits that can be anomalously canceled with 891: 297, 396, 495, 594, 693, 792. For n=2 there are 60 numbers with 88911: 12987, 13986, 14985, 15984, 16983, 17982, 21978, 22977, 23976, 24975, 25974, 26973, 27972, 28971, 31968, 32967, 33966, 34965, 35964, 36963, 37962, 38961, 41958, 42957, 43956, 44955, 45954, 46953, 47952, 48951, 51948, 52947, 53946, 54945, 55944, 56943, 57942, 58941, 61938, 62937, 63936, 64935, 65934, 66933, 67932, 68931, 71928, 72927, 73926, 74925, 75924, 76923, 77922, 78921, 82917, 83916, 84915, 85914, 86913, 87912. For n=3 504 numbers with 8889111, and no other (2n+1)-digit number has greater multiplicity. There seems to be a pattern of integer partitions in these examples, because the sum of the digits of numbers above are all multiples of 9. - _Xiaohan Zhang_, Nov 21 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Anomalous Cancellation
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. similar sequences listed in A248216.

Cf. A000079, A024036.

Magma
```
[8^n-2^n: n in [0..25]];
```

Table[8^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 2 x) (1 - 8 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-16},{0,6},30] (* Harvey P. Dale, Mar 29 2015 *)

a(n) = 8^n-2^n; \\ Altug Alkan, Sep 07 2018

def A248217(n): return 6*binomial(pow(2,n) +1, 3)
print([A248217(n) for n in range(41)]) # G. C. Greubel, Dec 26 2024

G.f.: 6*x/((1-2*x)*(1-8*x)).
a(n) = 10*a(n-1) - 16*a(n-2).
a(n) = 2^n*(4^n-1) = A000079(n) * A024036(n) = A001018(n) - A000079(n).
E.g.f.: exp(2*x)*(-1 + exp(6*x)). - Stefano Spezia, Sep 07 2018
a(n) = 6*A016131(n-1). - R. J. Mathar, Mar 10 2022

A013713 a(n) = 8^(2n+1).

Original entry on oeis.org

8, 512, 32768, 2097152, 134217728, 8589934592, 549755813888, 35184372088832, 2251799813685248, 144115188075855872, 9223372036854775808, 590295810358705651712, 37778931862957161709568, 2417851639229258349412352

Offset: 0

N. J. A. Sloane

Additive digital root of a(n) = 8. - Miquel Cerda, Jul 03 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..160
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (64).

Cf. A001018, A013734, A013735, A089357.

[8^(2*n+1): n in [0..20]]; // Vincenzo Librandi, May 25 2011

8^(2*Range[0,20]+1) (* or *) NestList[64#&,8,20] (* Harvey P. Dale, Feb 27 2013 *)

x='x+O('x^99); Vec(8/(1-64*x)) \\ Altug Alkan, Jul 09 2016

a(n)=8<<(3*n) \\ Charles R Greathouse IV, Jul 11 2016

a(n) = 64a(n-1), n>0. O.g.f.: 8/(1-64x). - R. J. Mathar, May 07 2008
a(n) = 8*A089357(n) = 2*A013734(n) = A013735(n)/2 . - Philippe Deléham, Nov 24 2008
From Ilya Gutkovskiy, Jul 03 2016: (Start)
E.g.f.: 8*exp(64*x).
a(n) = A001018(A005408(n)). (End)

A014832 a(1)=1; for n>1, a(n) = 9*a(n-1) + n.

Original entry on oeis.org

1, 11, 102, 922, 8303, 74733, 672604, 6053444, 54481005, 490329055, 4412961506, 39716653566, 357449882107, 3217048938977, 28953440450808, 260580964057288, 2345228676515609, 21107058088640499, 189963522797764510, 1709671705179880610, 15387045346618925511, 138483408119570329621

Offset: 1

N. J. A. Sloane

			For n=5, a(5) = 1*15 + 8*20 + 8^2*15 + 8^3*6 + 8^4*1 = 8303. [_Bruno Berselli_, Nov 13 2015]

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 18.
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. A001018, A104712.

a:=n->sum((9^(n-j)-1)/8,j=0..n): seq(a(n), n=1..18); # Zerinvary Lajos, Jan 15 2007
a:= n-> (Matrix([[1,0,1],[1,1,1],[0,0,9]])^n)[2,3]: seq(a(n), n=1..18); # Alois P. Heinz, Aug 06 2008

RecurrenceTable[{a[1]==1,a[n]==9a[n-1]+n},a,{n,20}] (* or *) LinearRecurrence[ {11,-19,9},{1,11,102},20] (* Harvey P. Dale, May 01 2012 *)

a(n) = (9^(n+1) - 8*n - 9)/64. - Rolf Pleisch, Oct 22 2010
From Harvey P. Dale, May 01 2012: (Start)
a(1)=1, a(2)=11, a(3)=102; for n>3, a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
G.f.: -x/((x-1)^2*(9*x-1)). (End)
a(n) = Sum_{i=0..n-1} 8^i*binomial(n+1,n-1-i). - Bruno Berselli, Nov 13 2015
E.g.f.: exp(x)*(9*exp(8*x) - 8*x - 9)/64. - Elmo R. Oliveira, Mar 29 2025

A036294 a(n) = n*8^n.

Original entry on oeis.org

0, 8, 128, 1536, 16384, 163840, 1572864, 14680064, 134217728, 1207959552, 10737418240, 94489280512, 824633720832, 7146825580544, 61572651155456, 527765581332480, 4503599627370496, 38280596832649216, 324259173170675712, 2738188573441261568, 23058430092136939520

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018, A053539.

[n*8^n: n in [0..20]]; // Vincenzo Librandi, Aug 09 2017

Table[n 7^n, {n, 0, 30}] (* or *) LinearRecurrence[{16,-64},{0,8},31] (* Vincenzo Librandi, Aug 09 2017 *)

From Vincenzo Librandi, Aug 09 2017: (Start)
G.f.: 8*x/(8*x-1)^2.
a(n) = 16*a(n-1) - 64*a(n-2) for n > 1. (End)
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(8/7).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(9/8). (End)
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 8*x*exp(8*x).
a(n) = n*A001018(n) = 8*A053539(n). (End)

A060195 a(n) = 8^(n-1)*(2^n - 1).

Original entry on oeis.org

1, 24, 448, 7680, 126976, 2064384, 33292288, 534773760, 8573157376, 137304735744, 2197949513728, 35175782154240, 562881233944576, 9006649498927104, 144110790029344768, 2305807824841605120, 36893206672442392576, 590293558558891966464, 9444714951340780945408

Offset: 1

Manish Kumar Gupta (M.Gupta(AT)math.canterbury.ac.nz), Mar 21 2001

Harry J. Smith, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (24,-128).

Cf. A000225, A001018.

[8^(n-1)*(2^n-1): n in [1..40]]; // G. C. Greubel, Aug 01 2024

Table[8^(n-1) (2^n-1),{n,20}] (* or *) LinearRecurrence[{24,-128},{1,24},20] (* Harvey P. Dale, Oct 20 2014 *)

{ a(n) = 8^(n - 1)*(2^n - 1) } \\ Harry J. Smith, Jul 02 2009

[8^(n-1)*(2^n-1) for n in range(1,41)] # G. C. Greubel, Aug 01 2024

a(1)=1, a(2)=24, a(n) = 24*a(n-1) - 128*a(n-2). - Harvey P. Dale, Oct 20 2014
From G. C. Greubel, Aug 01 2024: (Start)
a(n) = A000225(n)*A001018(n-1).
G.f.: x/((1 - 8*x)*(1 - 16*x)).
E.g.f.: (1/4)*exp(12*x)*sinh(4*x). (End)

More terms from Jason Earls and Larry Reeves (larryr(AT)acm.org), Mar 21 2001

A139541 There are 4n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4n)!/(n!*8^n) different ways.

Original entry on oeis.org

1, 3, 315, 155925, 212837625, 618718975875, 3287253918823875, 28845653137679503125, 388983632561608099640625, 7637693625347175036443671875, 209402646126143497974176151796875, 7752714167528210725497923667975703125, 377130780679409810741846496828678078515625

Offset: 0

Reinhard Zumkeller, Apr 25 2008

nonn

From Karol A. Penson, Oct 05 2009: (Start)
Integral representation as n-th moment of a positive function on a positive semi-axis (solution of the Stieltjes moment problem), in Maple notation:
a(n)=int(x^n*((1/4)*sqrt(2)*(Pi^(3/2)*2^(1/4)*hypergeom([], [1/2, 3/4], -(1/32)*x)*sqrt(x)-2*Pi*hypergeom([], [3/4, 5/4], -(1/32)*x)*GAMMA(3/4)*x^(3/4)+sqrt(Pi)*GAMMA(3/4)^2*2^(1/4)*hypergeom([], [5/4, 3/2],-(1/32)*x)*x)/(Pi^(3/2)*GAMMA(3/4)*x^(5/4))), x=0..infinity), n=0,1... .
This solution may not be unique. (End)

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Appendix: Problem 203.1, p164.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Tournament
Index entries for sequences related to tornaments.

Cf. A008299, A000142, A100733, A001018.

a(n)={(4*n)!/(n!*8^n)} \\ Andrew Howroyd, Jan 07 2020

a(n) = (4*n)!/(n!*8^n).
a(n) = A001147(n)*A001147(2*n).
a(n) = A008977(n)*(A049606(n)/A001316(n))^3. - Reinhard Zumkeller, Apr 28 2008

Terms a(11) and beyond from Andrew Howroyd, Jan 07 2020

A164683 a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 8.

Original entry on oeis.org

1, 8, 8, 64, 64, 512, 512, 4096, 4096, 32768, 32768, 262144, 262144, 2097152, 2097152, 16777216, 16777216, 134217728, 134217728, 1073741824, 1073741824, 8589934592, 8589934592, 68719476736, 68719476736, 549755813888

Offset: 1

Klaus Brockhaus, Aug 22 2009

Interleaving of A001018 and A001018 without initial term 1.

Binomial transform is A083100. Second binomial transform is A164607.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (0, 8).

Cf. A001018 (powers of 8), A083100, A164607.

[ n le 2 select 7*n-6 else 8*Self(n-2): n in [1..26] ];

With[{c=8^Range[0,15]},Riffle[c,c]]//Rest (* Harvey P. Dale, Aug 08 2017 *)

a(n) = 2^(1/4*(6*n-3+3*(-1)^n)).

G.f.: x*(1+8*x)/(1-8*x^2).

A164737 a(n) = 8*a(n-2) for n > 2; a(1) = 5, a(2) = 12.

Original entry on oeis.org

5, 12, 40, 96, 320, 768, 2560, 6144, 20480, 49152, 163840, 393216, 1310720, 3145728, 10485760, 25165824, 83886080, 201326592, 671088640, 1610612736, 5368709120, 12884901888, 42949672960, 103079215104, 343597383680, 824633720832

Offset: 1

Klaus Brockhaus, Aug 24 2009

nonn

Interleaving of 5*A001018 and 12*A001018.

Binomial transform is A096980 without initial terms 1. Second binomial transform is A164593. Third binomial transform is A101386.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A001018 (powers of 8), A067412, A096980, A101386, A164593.

[ n le 2 select 7*n-2 else 8*Self(n-2): n in [1..26] ];

seq(coeff(series( x*(5+12*x)/(1-8*x^2) , x, n+1), x, n), n=1..30); # G. C. Greubel, Apr 16 2020

LinearRecurrence[{0,8}, {5,12}, 30] (* G. C. Greubel, Apr 16 2020 *)

[(13 -7*(-1)^n)*2^((6*n -11 +3*(-1)^n)/4) for n in (1..30)] # G. C. Greubel, Apr 16 2020

a(n) = (13 - 7*(-1)^n)*2^(1/4*(6*n - 11 + 3*(-1)^n)).

G.f.: x*(5 + 12*x)/(1 - 8*x^2).

cp's OEIS Frontend

A164675 a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 12.

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A240841 a(n) = floor(8^n/(1+2*sin(6*Pi/13)/(2*sin(Pi/13)))^n).

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A248217 a(n) = 8^n - 2^n.

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A013713 a(n) = 8^(2n+1).

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A014832 a(1)=1; for n>1, a(n) = 9*a(n-1) + n.

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A036294 a(n) = n*8^n.

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A060195 a(n) = 8^(n-1)*(2^n - 1).

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A240841 a(n) = floor(8^n/(1+2sin(6Pi/13)/(2*sin(Pi/13)))^n).

A139541 There are 4n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4n)!/(n!*8^n) different ways.