A001018 -id:A001018 - cp's OEIS frontend

A138973 a(n) = 8^n mod 7^n.

0, 1, 15, 169, 1695, 15961, 26846, 450066, 5247614, 13156907, 226316077, 680627620, 13354327932, 65310761853, 328708074010, 1951441519231, 15611532153848, 158125187800385, 101848932467045, 7328445851378156, 35829776440278962, 286638211522231696

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1183

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), this sequence (k=8), A139733 (k=9).

Cf. A000420, A001018.

a[n_]:=PowerMod[8,n,7^n];Array[a,22,0] (* James C. McMahon, Jun 23 2025 *)

a(n) = lift(Mod(8, 7^n)^n); \\ Michel Marcus, Feb 20 2018

[power_mod(8,n,7^n) for n in range(0,22)] # Zerinvary Lajos, Nov 28 2009

A164640 a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

Original entry on oeis.org

1, 6, 8, 48, 64, 384, 512, 3072, 4096, 24576, 32768, 196608, 262144, 1572864, 2097152, 12582912, 16777216, 100663296, 134217728, 805306368, 1073741824, 6442450944, 8589934592, 51539607552, 68719476736, 412316860416

Offset: 1

Klaus Brockhaus, Aug 20 2009

Interleaving of A001018 and A083233 without initial term 1.

Binomial transform is A164544. Third binomial transform is A038761.

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), A083233 ((3*8^n+(0)^n)/4), A164544, A038761.

[ n le 2 select 5*n-4 else 8*Self(n-2): n in [1..26] ];

LinearRecurrence[{0,8},{1,6},40] (* Harvey P. Dale, Nov 06 2013 *)

a(n)=(7-(-1)^n)*2^(1/4*(6*n-15+3*(-1)^n)) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (7-(-1)^n)*2^(1/4*(6*n -15+3*(-1)^n)).

G.f.: x*(1+6*x)/(1-8*x^2).

G.f. corrected by Klaus Brockhaus, Sep 18 2009

A222935 T(n,k)=Sum of neighbor maps: number of nXk binary arrays indicating the locations of corresponding elements equal to the sum mod 4 of their horizontal and antidiagonal neighbors in a random 0..3 nXk array.

Original entry on oeis.org

2, 2, 4, 8, 16, 8, 16, 48, 64, 16, 24, 256, 512, 232, 32, 64, 1024, 4000, 4096, 1024, 64, 128, 4096, 32760, 63488, 32768, 4096, 128, 232, 15820, 262104, 1047664, 1048352, 262144, 16224, 256, 512, 65536, 2097152, 16776992, 33553920, 16777216, 2097152

Offset: 1

R. H. Hardin Mar 09 2013

Table starts
....2.......2.........8.........16..........24..........64.........128
....4......16........48........256........1024........4096.......15820
....8......64.......512.......4000.......32760......262104.....2097152
...16.....232......4096......63488.....1047664....16776992...268435456
...32....1024.....32768....1048352....33553920..1073741824.34358569712
...64....4096....262144...16777216..1073741824.68719413208
..128...16224...2097152..268435456.34359738368
..256...65536..16777216.4294962688
..512..262144.134217728
.1024.1047680
.2048

			Some solutions for n=3 k=4
..0..0..1..1....0..1..1..1....1..1..1..0....0..1..0..0....1..1..1..1
..0..0..1..1....0..0..1..0....0..1..0..0....0..1..1..1....1..1..0..1
..0..0..1..0....1..1..0..0....0..1..0..1....0..1..1..0....0..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..70

Column 1 is A000079
Column 2 is A222381
Column 3 is A001018 except for a(2)
Row 1 is A220172

A239013 Exponents m such that the decimal expansion of 8^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27, 43, 144, 342, 633, 653, 2642, 6966, 16124, 84595, 225177, 4069057, 4890280, 6298187, 39573326, 99250579, 242281125, 1007075831, 4705063695, 5439666500, 5741331846, 6168193506, 9297912451, 34411164318, 36390662612, 265816303567

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030704.
Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.
Not just three times A031142; although {99250579, 6168193506, 9297912451, 34411164318, 36390662612} are possible candidates.

Cf. A001018, A030704, A020665, A031142, A239008, A239009, A239010, A239011, A239012, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[8, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 200000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(29)-a(35) from Bert Dobbelaere, Jan 21 2019

a(36)-a(38) from Chai Wah Wu, Jan 18 2020

A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.

Original entry on oeis.org

1, 8, -7, 64, -84, 21, 512, -896, 224, 196, -35, 4096, -8960, 2240, 3920, -350, -980, 35, 32768, -86016, 21504, 56448, -3360, -18816, 336, -5488, 1470, 1176, -21, 262144, -802816, 200704, 702464, -31360, -263424, 3136, -153664, 27440, 21952, -196, 38416, -1372, -3430, 7

Offset: 1

Gregory Gerard Wojnar, Jun 16 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_8)* eM_1^t_1 * eM_2^t_2 * ...*eM_8^t_8) summed over all length 8 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3+...+8*t_8=k, where SM_k are the averaged k-th power sum symmetric polynomials in 8 data (i.e., SM_k = S_k/8 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(8,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_8) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give: 1,8,85,932,10291,114878,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins
     1;
     8,    -7;
    64,   -84,   21;
   512,  -896,  224,  196,  -35;
  4096, -8960, 2240, 3920, -350, -980, 35;
  ...

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, pp. 22-24, arXiv:1706.08381 [math.GM], 2017.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7). See Girard-Waring A210258. T(n,1)=8^(n-1)=A001018(n).

Java
```
// See Wojnar link.
```

A367248 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 7.

Original entry on oeis.org

0, 5, 111, 1601, 19095, 204545, 2045511, 19508081, 179752215, 1613908385, 14202967911, 123028446161, 1052237271735, 8907026785025, 74758478722311, 623053865857841, 5162154289325655, 42558224511290465, 349394287423788711, 2858263098464575121, 23311522539676521975

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366964.

Index entries for linear recurrences with constant coefficients, signature (21,-146,336).

Cf. A366964.
Cf. A000400, A000420, A001018.
Cf. A367243, A367244, A367245, A367246, A367247, A367249, A367250.

LinearRecurrence[{21,-146,336},{0,5,111},21]

a(n) = 23*8^(n-1) - 41*7^(n-1) + 3*6^n.
a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3) for n > 3.
O.g.f.: x^2*(5 + 6*x)/((1 - 6*x)*(1 - 7*x)*(1 - 8*x)).
E.g.f.: (161*exp(8*x) - 328*exp(7*x) + 168*exp(6*x) - 1)/56.

A367249 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 8.

Original entry on oeis.org

0, 3, 79, 1323, 18175, 223323, 2555119, 27828363, 292407775, 2990349243, 29943991759, 294872615403, 2864776362175, 27525734996763, 262061152909999, 2475899571994443, 23240879960425375, 216963121865909883, 2015960236625789839, 18656492902684557483, 172056837889322101375

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366965.

Index entries for linear recurrences with constant coefficients, signature (24,-191,504).

Cf. A366965.
Cf. A000420, A001018, A001019.
Cf. A367243, A367244, A367245, A367246, A367247, A367248, A367250.

LinearRecurrence[{24,-191,504},{0,3,79},21]

a(n) = 17*9^(n-1) - 31*8^(n-1) + 2*7^n.
a(n) = 24*a(n-1) - 191*a(n-2) + 504*a(n-3) for n > 3.
O.g.f.: x^2*(3 + 7*x)/((1 - 7*x)*(1 - 8*x)*(1 - 9*x)).
E.g.f.: (136*exp(9*x) - 279*exp(8*x) + 144*exp(7*x) - 1)/72.

A367250 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 9.

Original entry on oeis.org

0, 1, 35, 703, 11231, 158311, 2062655, 25466743, 302423471, 3487593511, 39314599775, 435241463383, 4748453693711, 51186327429511, 546278900354495, 5781325731101623, 60750456603203951, 634502309615150311, 6592506388026870815, 68188442304165981463, 702543059232886986191

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366966.

Index entries for linear recurrences with constant coefficients, signature (27,-242,720).

Cf. A366966.
Cf. A001018, A001019, A011557.
Cf. A367243, A367244, A367245, A367246, A367247, A367248, A367249.

LinearRecurrence[{27,-242,720},{0,1,35},21]

a(n) = 9*10^(n-1) - 17*9^(n-1) + 8^n.
a(n) = 27*a(n-1) - 242*a(n-2) + 720*a(n-3) for n > 3.
O.g.f.: x^2*(1 + 8*x)/((1 - 8*x)*(1 - 9*x)*(1 - 10*x)).
E.g.f.: (81*exp(10*x) - 170*exp(9*x) + 90*exp(8*x) - 1)/90.

A060760 a(n) = 8^(n^2).

Original entry on oeis.org

1, 8, 4096, 134217728, 281474976710656, 37778931862957161709568, 324518553658426726783156020576256, 178405961588244985132285746181186892047843328

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 23 2001

Number of n X n matrices over GF(8).

Harry J. Smith, Table of n, a(n) for n = 0..33

Cf. A001018, A000290.

8^Range[0,10]^2 (* Harvey P. Dale, Jun 07 2025 *)

a(n)={8^(n^2)} \\ Harry J. Smith, Jul 11 2009

a(n) = A001018(A000290(n)). - Michel Marcus, Jul 03 2018

More terms from Olaf Voß, Feb 11 2008

A126985 Expansion of 1/(1+8xc(x)), c(x) the g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -8, 56, -400, 2840, -20208, 143664, -1021728, 7265240, -51665200, 367392656, -2612584928, 18578329456, -132112749920, 939467783520, -6680662171200, 47506922377560, -337827035002800, 2402325467002320, -17083203745473120, 121480558396908240, -863861754435010080

Offset: 0

Philippe Deléham, Mar 21 2007

sign

Hankel transform is (-8)^n.

Catalan transform of (-1)^n*A001018(n). - R. J. Mathar, Nov 11 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A001018, A039599.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(10-8*Sqrt(1-4*x)) )); // G. C. Greubel, May 28 2019

c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+8*x*c),x=0,25): seq(coeff(ser,x,n),n=0..21); # Emeric Deutsch, Mar 24 2007

CoefficientList[Series[2/(10-8*Sqrt[1-4*x]), {x,0,30}], x] (* G. C. Greubel, May 28 2019 *)

my(x='x+O('x^30)); Vec(2/(10-8*sqrt(1-4*x))) \\ G. C. Greubel, May 28 2019

(2/(10-8*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 28 2019

a(n) = Sum_{k=0..n} A039599(n,k)*(-9)^k.
G.f.: 2/(10 - 8*sqrt(1-4*x)). - G. C. Greubel, May 28 2019
D-finite with recurrence 9*n*a(n) +2*(14*n+27)*a(n-1) +128*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

More terms from Emeric Deutsch, Mar 24 2007

cp's OEIS Frontend

A138973 a(n) = 8^n mod 7^n.

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A164640 a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

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A222935 T(n,k)=Sum of neighbor maps: number of nXk binary arrays indicating the locations of corresponding elements equal to the sum mod 4 of their horizontal and antidiagonal neighbors in a random 0..3 nXk array.

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A239013 Exponents m such that the decimal expansion of 8^m exhibits its first zero from the right later than any previous exponent.

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A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.

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A367248 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 7.

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A367249 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 8.

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A367250 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 9.

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Mathematica

A126985 Expansion of 1/(1+8xc(x)), c(x) the g.f. of Catalan numbers A000108.