A023001 -id:A023001 - cp's OEIS frontend

A125836 Numbers whose base 8 or octal representation is 555555555......5.

0, 5, 45, 365, 2925, 23405, 187245, 1497965, 11983725, 95869805, 766958445, 6135667565, 49085340525, 392682724205, 3141461793645, 25131694349165, 201053554793325, 1608428438346605, 12867427506772845, 102939420054182765

Offset: 1

Zerinvary Lajos, Feb 03 2007

			Octal...............decimal
0........................0
5........................5
55......................45
555....................365
5555..................2925
55555................23405
555555..............187245
5555555............1497965
55555555..........11983725
555555555.........95869805
5555555555.......766958445
etc. ...............etc.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A023001, A024088.

List([1..30], n-> 5*(8^(n-1) -1)/7); # G. C. Greubel, Aug 03 2019

[5*(8^(n-1) -1)/7: n in [1..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(5*(8^n-1)/7, n=0..30);
```

5*(8^(Range[30]-1) -1)/7 (* G. C. Greubel, Aug 03 2019 *)

vector(30, n, 5*(8^(n-1) -1)/7) \\ G. C. Greubel, Aug 03 2019

[5*(8^(n-1) -1)/7 for n in (1..30)] # G. C. Greubel, Aug 03 2019

a(n) = 5*(8^(n-1) -1)/7 = 5*A023001(n-1).
a(n) = 8*a(n-1) + 5, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 5*x^2/( (1-x)*(1-8*x)). - R. J. Mathar, Sep 30 2013
From G. C. Greubel, Aug 03 2019: (Start)
a(n) = 5*A024088(n-1)/7.
E.g.f.: 5*(exp(8*x) - exp(x))/7. (End)

A229393 Number of shapes of balanced 8-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one.

Original entry on oeis.org

1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 64, 1792, 28672, 286720, 1835008, 7340032, 16777216, 16777216, 469762048, 5754585088, 40282095616, 176234168320, 493455671296, 863547424768, 863547424768, 377801998336, 6044831973376, 42313823813632, 169255295254528

Offset: 0

Alois P. Heinz, Sep 21 2013

nonn

a(n) = 1 for n in { A023001 }.

Alois P. Heinz, Table of n, a(n) for n = 0..585

Column k=8 of A221857.

a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 8, 'm'); binomial(8, m) *a(r+1)^m *a(r)^(8-m) fi
    end:
seq(a(n), n=0..73);

A267703 Conjectured list of numbers whose trajectory under the '7x+1' map eventually reaches 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 32, 36, 40, 41, 64, 72, 73, 80, 82, 128, 144, 146, 160, 164, 167, 256, 288, 292, 320, 328, 329, 334, 512, 576, 584, 585, 640, 656, 658, 668, 1024, 1152, 1168, 1170, 1280, 1312, 1316, 1336, 1337, 1965, 2048, 2304, 2336, 2340, 2560

Offset: 1

Michel Lagneau, Jan 19 2016

nonn

This is conjectural in that there is no known proof that the missing numbers 3, 6, 7, ... are really missing. It may be that after a very large number of iterations they will cycle. - N. J. A. Sloane, Jan 23 2016

Note that the computer program does not actually calculate a complete list of "numbers k such that the Collatz-like map T: if x odd, x -> 7*x+1 and if x even, x -> x/2, when started at k, eventually reaches 1".

			5 is in the sequence because the trajectory of 5 is 5 -> 36 -> 18 -> 9 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1.

Dmitry Kamenetsky, Table of n, a(n) for n = 1..164

Cf. A000079, A006577, A023001, A232711, A267969, A267970.

nn:=10000:
for n from 1 to 2340 do:
  m:=n:cyc:={n}:
    for i from 1 to nn do:
     if irem(m,2)=0
      then
       m:=m/2:
      else
      m:=7*m+1:
     fi:
    cyc:=cyc union {m}:
    od:
    n0:=nops(cyc):
    if n0N. J. A. Sloane, Jan 23 2016)

Entry revised by N. J. A. Sloane, Jan 23 2016

a(19)-a(55) from Dmitry Kamenetsky, Jun 24 2024

A299913 a(n) = a(n-1) + 2a(n-2) if n even, or 3a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.

Original entry on oeis.org

0, 1, 1, 7, 9, 55, 73, 439, 585, 3511, 4681, 28087, 37449, 224695, 299593, 1797559, 2396745, 14380471, 19173961, 115043767, 153391689, 920350135, 1227133513, 7362801079, 9817068105, 58902408631, 78536544841, 471219269047, 628292358729, 3769754152375, 5026338869833

Offset: 0

N. J. A. Sloane, Mar 10 2018

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

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

Bisections give A023001, A083068.

Cf. A299914, A299915, A299916.

a:= n-> (<<0|1|0>, <0|0|1>, <8|8|-1>>^n. <<0, 1, 1>>)[1,1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2018

Fold[Append[#1, Inner[Times, 2 Boole[OddQ@ #2] + {1, 2}, {#1[[-1]], #1[[-2]]}, Plus]] &, {0, 1}, Range[2, 30]] (* or *)
CoefficientList[Series[-x (2 x + 1)/((x + 1) (8 x^2 - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Mar 10 2018 *)
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],b+2a,3b+4a]}; NestList[nxt,{1,0,1},30][[;;,2]] (* Harvey P. Dale, Mar 02 2025 *)

concat(0, Vec(x*(1 + 2*x) / ((1 + x)*(1 - 8*x^2)) + O(x^40))) \\ Colin Barker, Mar 11 2018

G.f.: -x*(2*x+1)/((x+1)*(8*x^2-1)). - Alois P. Heinz, Mar 10 2018
From Colin Barker, Mar 11 2018: (Start)
a(n) = (2^(3*n/2) - 1) / 7 for n even.
a(n) = 3*2^((3*(n-1))/2+1)/7 + 1/7 for n odd.
a(n) = -a(n-1) + 8*a(n-2) + 8*a(n-3) for n>2.
(End)

More terms from Altug Alkan, Mar 10 2018

A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 7, 4, 1, 0, 15, 13, 5, 1, 0, 31, 40, 21, 6, 1, 0, 63, 121, 85, 31, 7, 1, 0, 127, 364, 341, 156, 43, 8, 1, 0, 255, 1093, 1365, 781, 259, 57, 9, 1, 0, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 0, 1023, 9841, 21845, 19531, 9331, 2801, 585, 91, 11, 1, 0

Offset: 0

Stefano Spezia, Mar 13 2023

			The array begins:
   0,  0,  0,   0,   0, ...
   1,  1,  1,   1,   1, ...
   3,  4,  5,   6,   7, ...
   7, 13, 21,  31,  43, ...
  15, 40, 85, 156, 259, ...
  ...

Cf. A003992, A361291 (k=2*n+1), A361476 (antidiagonal sums).

Cf. A000225 (k=2), A003462 (k=3), A002450 (k=4), A003463 (k=5), A003464 (k=6), A023000 (k=7), A023001 (k=8), A002452 (k=9), A002275 (k=10), A016123 (k=11).

A[n_,k_]:=(k^n-1)/(k-1); Flatten[Table[A[n-k+2,k],{n,0,10},{k,2,n+2}]]

E.g.f. of column k: exp(x)*(exp((k-1)*x) - 1)/(k - 1).
E.g.f. of column k: 2*exp((k+1)*x/2)*sinh((k-1)*x/2)/(k - 1).
A(n, k) = Sum_{i=0..n-1} k^i.

A033120 Base-2 digits of a(n) are, in order, the first n terms of the periodic sequence with initial period 1,0,1.

Original entry on oeis.org

1, 2, 5, 11, 22, 45, 91, 182, 365, 731, 1462, 2925, 5851, 11702, 23405, 46811, 93622, 187245, 374491, 748982, 1497965, 2995931, 5991862, 11983725, 23967451, 47934902, 95869805, 191739611, 383479222, 766958445, 1533916891

Offset: 1

Clark Kimberling

Minimal number of moves required, under the proviso of a classical tower-of-Hanoi game, to segregate an initial n-disc peg into even and odd numbered discs pegs. - Lekraj Beedassy, Sep 12 2006

B. Averbach & O. Chein, "A Variant Of The Tower Of Brahma" in 'The Journal of Recreational Mathematics', pp. 48-55, vol. 33, no. 1, 2004-5, Baywood, NY.

Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Cf. A023001, A033137 (similar in base 10).

Table[FromDigits[PadRight[{},n,{1,0,1}],2],{n,40}] (* Harvey P. Dale, Aug 26 2016 *)

a(n)=if(n%3==0,5*8^(n/3)-5,if(n%3==1,10*8^((n-1)/3)-3,20*8^((n-2)/3)-6))/7 \\ Ralf Stephan

a(n)=(5*2^n)\7 \\ Tani Akinari, Jul 15 2014

From Ralf Stephan, May 05 2004: (Start)
a(3*n) = (5*8^n - 5)/7, a(3*n+1) = (10*8^n - 3)/7, a(3*n+2) = (20*8^n - 6)/7.
G.f.: (1+x^2)/((1-x)*(1-2*x)*(1+x+x^2)). (End)
a(n) = a(n-6) + 45*2^(n-6). - Lekraj Beedassy, Sep 12 2006
The following recurrence produces this sequence: if(n==1) a(n)=1; else if(n%3==2) a(n)=a(n-1)*2; otherwise a(n)=a(n-1)*2+1. - Piotr Kakol, Jan 24 2011 (in an email message to N. J. A. Sloane).
a(n) = floor( (5/7)*2^n ). - Tani Akinari, Jul 15 2014
From Jorijn Lamberink and Paul van de Veen, Oct 14 2019: (Start)
a(n) = T(n-1) + 1 + T(n-3) + 1 + a(n-3), where T(n) = A000225(n) = 2^n-1 is the number of moves for a classic Tower of Hanoi with n discs.
a(n) = (5/8)*2^n + a(n-3).
a(n) = (5/7)*2^n - 2/3 - (1/21)*cos((2/3)*Pi*n) + (1/7)*sqrt(3)*sin((2/3)*Pi*n). (End)

More terms from Lekraj Beedassy, Sep 12 2006

A096042 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^8-M)/7, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

Original entry on oeis.org

1, 9, 2, 73, 27, 3, 585, 292, 54, 4, 4681, 2925, 730, 90, 5, 37449, 28086, 8775, 1460, 135, 6, 299593, 262143, 98301, 20475, 2555, 189, 7, 2396745, 2396744, 1048572, 262136, 40950, 4088, 252, 8, 19173961, 21570705, 10785348, 3145716, 589806

Offset: 1

Gary W. Adamson, Jun 17 2004

			Triangle begins:
1
9 2
73 27 3
585 292 54 4
4681 2925 730 90 5
37449 28086 8775 1460 135 6

Cf. A007318. First column gives A023001. Row sums give A016133.

P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^8-M)/7 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11); # Alois P. Heinz, Oct 07 2009

P[n_] := P[n] = With[{M = Array[Binomial[#1-1, #2-1]&, {n, n}]}, (MatrixPower[M, 8] - M)/7]; T[n_, k_] := P[n+1][[n+1, k]]; Table[ Table[T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

Edited with more terms by Alois P. Heinz, Oct 07 2009

A103968 Sigma((8^n - 1)/7), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

1, 13, 74, 1092, 4864, 59200, 346112, 4756752, 19436692, 251916288, 1294876800, 20786304000, 79541043200, 1073030266880, 5303003316224, 81061789640832, 324992122224640, 4102172934143680, 20588576135708672, 376372958524932096

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 31 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..402

Cf. A000203, A023001.

DivisorSigma[1,(8^Range[20]-1)/7] (* Harvey P. Dale, Aug 21 2011 *)

a(n) = A000203(A023001(n)). - Amiram Eldar, Feb 24 2020

A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).

Original entry on oeis.org

1, 8, 47, 250, 1281, 6468, 32467, 162590, 813461, 4068328, 20343687, 101722530, 508620841, 2543120588, 12715635707, 63578244070, 317891351421, 1589457019248, 7947285620527, 39736429151210, 198682147853201, 993410743460308, 4967053725690147, 24835268645227950

Offset: 0

N. J. A. Sloane, Dec 11 1999

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016209, A016218, A016256, A023000, A023001.

Cf. A016127.

a:=n->sum((5^(n-j)-2^(n-j))/3,j=0..n): seq(a(n), n=1..20); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=8},Table[c=7*b-10*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

a(n) = (25*5^n - 16*2^n + 3)/12. - Bruno Berselli, Feb 09 2011
a(n) = [(5^0-2^0) + (5^1-2^1) + ... + (5^n-2^n)]/3. - r22lou(AT)cox.net, Nov 14 2005
a(0)=1, a(n) = 5*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 07 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(25*exp(4*x) - 16*exp(x) + 3)/12.
a(n) = 8*a(n-1) - 17*a(n-2) + 10*a(n-3).
a(n) = A016127(n+1) - A003463(n+2). (End)

More terms from Wesley Ivan Hurt, May 05 2014

A327593 Numbers m where an integer b that is a power of two > 2 with 1 < b < m exists such that m is a base-b repdigit.

Original entry on oeis.org

5, 9, 10, 15, 17, 18, 21, 27, 33, 34, 36, 42, 45, 51, 54, 63, 65, 66, 68, 73, 85, 99, 102, 119, 129, 130, 132, 136, 146, 153, 165, 170, 187, 195, 198, 204, 219, 221, 231, 238, 255, 257, 258, 260, 264, 273, 292, 297, 325, 330, 341, 363, 365, 387, 390, 396, 429

Offset: 1

Felix Fröhlich, Sep 18 2019

Let b(n) = A226542(n)-1. This sequence is a supersequence of b.
Conjecture 1: Let c(n) = A001220(n)-1. This sequence is a supersequence of c.
Conjecture 2: This is a supersequence of A240719.
From Bernard Schott, Sep 19 2019: (Start)
There are 3 distinct families of terms in this sequence:
1) Integers of the form: 2^q + 1 = 11_2^q with q >= 2.
First few terms: 5, 9, 17, 33, 65, 129, ...; this is A000051 \ {2, 3}. As 11_b is not a Brazilian representation, five of these terms are not Brazilian, they are 9 and the four known Fermat primes in A019434: 5, 17, 257 and 65537; all the other terms are composite and Brazilian but in a base that is not a power of two as 65 = 11_64 = 55_12.
2) Integers of the form: m * (2^q+1) = (mm)_2^q with q >= 2 and 1 < m < 2^q.
First few terms: 10, 15, 18, 27, 34, 36, ... These numbers are Brazilian with 2 digits in a base that is a power of two >= 4 as 10 = 22_4, 15 = 33_4 or 18 = 22_8.
3) Integers of the form: m * ((2^q)^s - 1)/(2^q - 1) = (mm...m)_ 2^q with q >= 2, s >= 1 and 1 <= m <= 2^q - 1.
First few terms: 21, 42, 63, 73, 85, ... These numbers are Brazilian repdigits with 3 digits or more in a base that is a power of two >= 4 as 42 = 222_4, 73 = 111_8 or 85 = 1111_4. The repunits (4^n-1)/3, (8^n-1)/7, (16^n-1)/15, (32^n-1)/31 respectively in A002450 (when >= 5), A023001 (when >=9), A131865 (when >=17), A132469 (when >=33) are subsequences of this last family.
Remark: there exist numbers that are in this sequence for two reasons as 63 = 77_8 = 333_4. (End)

			18 written in base 8 is 22. 8 is a power of two and 22 is a repdigit, so 18 is a term of the sequence.

Cf. A001220, A226542, A240719.

is(n) = my(b=4, d=0); while(b < n, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1)); b=2*b); 0

cp's OEIS Frontend

A125836 Numbers whose base 8 or octal representation is 555555555......5.

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A229393 Number of shapes of balanced 8-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one.

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A267703 Conjectured list of numbers whose trajectory under the '7x+1' map eventually reaches 1.

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A299913 a(n) = a(n-1) + 2*a(n-2) if n even, or 3*a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.

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A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

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A033120 Base-2 digits of a(n) are, in order, the first n terms of the periodic sequence with initial period 1,0,1.

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A096042 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^8-M)/7, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

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A299913 a(n) = a(n-1) + 2a(n-2) if n even, or 3a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.

A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).