A023001 -id:A023001 - cp's OEIS frontend

A218746 a(n) = (43^n - 1)/42.

0, 1, 44, 1893, 81400, 3500201, 150508644, 6471871693, 278290482800, 11966490760401, 514559102697244, 22126041415981493, 951419780887204200, 40911050578149780601, 1759175174860440565844, 75644532518998944331293, 3252714898316954606245600, 139866740627629048068560801

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 43 (A009987).

0 followed by the binomial transform of A170762. - R. J. Mathar, Jul 18 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums
Index entries related to q-numbers
Index entries for linear recurrences with constant coefficients, signature (44,-43)

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009987, A170762.

[n le 2 select n-1 else 44*Self(n-1) - 43*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{44, -43}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[43^Range[0,20]]] (* Harvey P. Dale, Jan 27 2015 *)

A218746(n):=(43^n-1)/42$
makelist(A218746(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218746(n)=43^n\42
```

G.f.: x/((1-x)*(1-43*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = 44*a(n-1) - 43*a(n-2). - Vincenzo Librandi, Nov 07 2012
a(n) = floor(43^n/42).  - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(22*x)*sinh(21*x)/21. - Elmo R. Oliveira, Aug 27 2024

A353147 Decimal repunits written in base 8.

Original entry on oeis.org

0, 1, 13, 157, 2127, 25547, 331007, 4172107, 52305307, 647665707, 10216432707, 122621414707, 1473657200707, 20126330410707, 241540165130707, 3120702223570707, 37450626705270707, 473627744665470707, 6125757360430070707, 75533532545361070707

Offset: 0

Seiichi Manyama, Apr 26 2022

Cf. A002275, A291962, A353142, A353143, A353144, A353145, A353146, A353148.

Cf. A007094, A023001.

a(n) = fromdigits(digits((10^n-1)/9, 8));

def a(n): return 0 if n == 0 else int(oct(int("1"*n))[2:])
print([a(n) for n in range(13)]) # Michael S. Branicky, Apr 26 2022

a(n) = A007094(A002275(n)).

A125835 Numbers whose base-8 or octal representation is 22222222.......2.

Original entry on oeis.org

0, 2, 18, 146, 1170, 9362, 74898, 599186, 4793490, 38347922, 306783378, 2454267026, 19634136210, 157073089682, 1256584717458, 10052677739666, 80421421917330, 643371375338642, 5146971002709138, 41175768021673106, 329406144173384850, 2635249153387078802, 21081993227096630418

Offset: 1

Zerinvary Lajos, Feb 03 2007

			Octal.............decimal
0.......................0
2.......................2
22.....................18
222...................146
2222.................1170
22222................9362
222222..............74898
2222222............599186
22222222..........4793490
222222222........38347922
2222222222......306783378
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.

List([1..30], n-> 2*(8^(n-1) -1)/7); # G. C. Greubel, Aug 03 2019

[2*(8^(n-1) -1)/7: n in [1..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(2*(8^n-1)/7, n=0..30);
```

nxt2[n_]:=Module[{idn=IntegerDigits[n,8]}, FromDigits[PadLeft[idn,Length[idn]+1,2],8]]; Join[{0},NestList[nxt2,2,30]]  (* Harvey P. Dale, Mar 09 2011 *)
Module[{nn=30,c},c=PadRight[{},nn,2];Table[FromDigits[Take[c,n],8],{n,0,nn}]] (* Harvey P. Dale, Sep 05 2015 *)
2*(8^(Range[30]-1) -1)/7 (* G. C. Greubel, Aug 03 2019 *)

a(n)=2*(1<<(3*n-3)\7) \\ Charles R Greathouse IV, Mar 09 2011

vector(30, n, 2*(8^(n-1) -1)/7) \\ G. C. Greubel, Aug 03 2019

[2*(8^(n-1) -1)/7 for n in (1..30)] # G. C. Greubel, Aug 03 2019

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

Offset corrected by N. J. A. Sloane, Oct 02 2010

Terms a(21) onward added by G. C. Greubel, Aug 03 2019

A218728 a(n) = (25^n - 1)/24.

Original entry on oeis.org

0, 1, 26, 651, 16276, 406901, 10172526, 254313151, 6357828776, 158945719401, 3973642985026, 99341074625651, 2483526865641276, 62088171641031901, 1552204291025797526, 38805107275644938151, 970127681891123453776, 24253192047278086344401, 606329801181952158610026

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 25 (A009969); q-integers for q=25.

Partial sums are in A014914. Also, the sequence is related to A014943 by A014943(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Nov 07 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (26,-25).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009969, A014914, A014943.

[n le 2 select n-1 else 26*Self(n-1)-25*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{26, -25}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(25^Range[0,20]-1)/24 (* Harvey P. Dale, Aug 23 2020 *)

A218728(n):=(25^n-1)/24$
makelist(A218728(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218728(n)=25^n\24
```

a(n) = floor(25^n/24).
From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-25*x)).
a(n) = 26*a(n-1) - 25*a(n-2). (End)
E.g.f.: exp(13*x)*sinh(12*x)/12. - Elmo R. Oliveira, Aug 27 2024
a(n) = 25*a(n-1) + 1. - Jerzy R Borysowicz, Sep 05 2025

A218743 a(n) = (40^n - 1)/39.

Original entry on oeis.org

0, 1, 41, 1641, 65641, 2625641, 105025641, 4201025641, 168041025641, 6721641025641, 268865641025641, 10754625641025641, 430185025641025641, 17207401025641025641, 688296041025641025641, 27531841641025641025641, 1101273665641025641025641, 44050946625641025641025641

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 40 (A009983).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (41,-40).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009983.

[n le 2 select n-1 else 41*Self(n-1) - 40*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{41, -40}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218743(n):=floor(40^n/39)$ makelist(A218743(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
a(n)=40^n\39
```

a(n) = floor(40^n/39).
From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-40*x)).
a(n) = 41*a(n-1) - 40*a(n-2). (End)
E.g.f.: exp(x)*(exp(39*x) - 1)/39. - Elmo R. Oliveira, Aug 29 2024

A269025 a(n) = Sum_{k = 0..n} 60^k.

Original entry on oeis.org

1, 61, 3661, 219661, 13179661, 790779661, 47446779661, 2846806779661, 170808406779661, 10248504406779661, 614910264406779661, 36894615864406779661, 2213676951864406779661, 132820617111864406779661, 7969237026711864406779661, 478154221602711864406779661

Offset: 0

Ilya Gutkovskiy, Feb 18 2016

Partial sums of powers of 60 (A159991).
Converges in a 10-adic sense to ...762711864406779661.
More generally, the ordinary generating function for the Sum_{k = 0..n} m^k is 1/((1 - m*x)*(1 - x)). Also, Sum_{k = 0..n} m^k = (m^(n + 1) - 1)/(m - 1).

Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (61,-60).

Cf. A159991.

Cf. similar sequences of the form (k^n-1)/(k-1): 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), A016125 (k=12), A091030 (k=13), A135519 (k=14), A135518 (k=15), A131865 (k=16), A091045 (k=17), A218721 (k=18), A218722 (k=19), A064108 (k=20), A218724-A218734 (k=21..31), A132469 (k=32), A218736-A218753 (k=33..50), this sequence (k=60), A133853 (k=64), A094028 (k=100), A218723 (k=256), A261544 (k=1000).

Table[Sum[60^k, {k, 0, n}], {n, 0, 15}]
Table[(60^(n + 1) - 1)/59, {n, 0, 15}]
LinearRecurrence[{61, -60}, {1, 61}, 15]

a(n)=60^n + 60^n\59 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: 1/((1 - 60*x)*(1 - x)).
a(n) = (60^(n + 1) - 1)/59 = 60^n + floor(60^n/59).
a(n+1) = 60*a(n) + 1, a(0)=1.
a(n) = Sum_{k = 0..n} A159991(k).
Sum_{n>=0} 1/a(n) = 1.016671221665660580331...
E.g.f.: exp(x)*(60*exp(59*x) - 1)/59. - Stefano Spezia, Mar 23 2023

A098436 Triangle of 3rd central factorial numbers T(n,k).

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 73, 36, 1, 1, 585, 1045, 100, 1, 1, 4681, 28800, 7445, 225, 1, 1, 37449, 782281, 505280, 35570, 441, 1, 1, 299593, 21159036, 33120201, 4951530, 130826, 784, 1, 1, 2396745, 571593565, 2140851900, 652061451, 33209946, 399738, 1296, 1

Offset: 0

Ralf Stephan, Sep 08 2004

			  1;
  1,   1;
  1,   9,    1;
  1,  73,   36,   1;
  1, 585, 1045, 100, 1;
  ...

M. Domaratzki, A Generalization of the Genocchi Numbers with Applications to Enumeration of Finite Automata, Technical Report No. 2001-449, September 2001, 11 pages.
John Riordan, Letter, Apr 28 1976.

First column is A023001, first diagonal is A000537.
Row sums are in A098437.
Replace in recurrence (k+1)^3 with k: A008277; with k^2: A008957 (note offsets).

A098436 := proc(n,k)
    option remember;
    if k=0 or k = n then
        1;
    else
        (k+1)^3*procname(n-1,k)+procname(n-1,k-1) ;
    end if;
end proc:
seq(seq( A098436(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jan 13 2025

T[n_, n_] = 1;
T[n_ /; n>=0, k_] /; 0<=k<=n := T[n, k] = (k+1)^3 T[n-1, k]+T[n-1, k-1];
T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 08 2022 *)

Recurrence: T(n, k) = (k+1)^3*T(n-1, k) + T(n-1, k-1), T(0, 0)=1.

A110205 Triangle, read by rows, where T(n,k) equals the sum of cubes of numbers < 2^n having exactly k ones in their binary expansion.

Original entry on oeis.org

1, 9, 27, 73, 368, 343, 585, 3825, 6615, 3375, 4681, 36394, 88536, 86614, 29791, 37449, 332883, 1024002, 1449198, 970677, 250047, 299593, 2979420, 10970133, 20078192, 19714083, 9974580, 2048383, 2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375

Offset: 1

Paul D. Hanna, Jul 16 2005

Compare to triangle A110200 (sum of squares).

			Row 4 is formed by sums of cubes of numbers < 2^4:
  T(4,1) = 1^3 + 2^3 + 4^3 + 8^3 = 585;
  T(4,2) = 3^3 + 5^3 + 6^3 + 9^3 + 10^3 + 12^3 = 3825;
  T(4,3) = 7^3 + 11^3 + 13^3 + 14^3 = 6615;
  T(4,4) = 15^3 = 3375.
Triangle begins:
        1;
        9,       27;
       73,      368,       343;
      585,     3825,      6615,      3375;
     4681,    36394,     88536,     86614,     29791;
    37449,   332883,   1024002,   1449198,    970677,    250047;
   299593,  2979420,  10970133,  20078192,  19714083,   9974580,  2048383;
  2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375; ...
Row g.f.s are:
  row 1: (1 + 2*x + 1*x^2)/(1+x)^2;
  row 2: (9 + 36*x + 27*x^2)/(1+x);
  row 3: (73 + 368*x + 343*x^2);
  row 4: (585 + 3240*x + 3375*x^2)*(1+x).
G.f. for row n is:
  ((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).

Paul D. Hanna, Rows n = 1..45, flattened.

Cf. A110206 (row sums), A110207 (central terms), A023001 (column 1).

b:= func< n,k | Binomial(n-3, k) >;
A110205:= func< n,k | (8^n-1)/7*(b(n,k-1) -b(n,k-2)) +(2^n-1)^2*((2^n+1)*b(n,k-2) +(2^n-1)*b(n,k-3)) >;
[A110205(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2024

b[n_, k_]= Binomial[n-3, k];
T[n_, k_]:= (8^n-1)/7*(b[n,k-1] -b[n,k-2]) + (2^n-1)^2*((2^n+1)*b[n,k-2] + (2^n-1)*b[n,k-3]);
A110205[n_, k_]:= If[n<3, T[n,k]/2, T[n,k]];
Table[A110205[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2024 *)

T(n,k)=(8^n-1)/7*binomial(n-3,k-1)+((2^n-1)*(4^n-1)-(8^n-1)/7)*binomial(n-3,k-2) +(2^n-1)^3*binomial(n-3,k-3)

/* Sum of cubes of numbers<2^n with k 1-bits: */
T(n,k)=local(B=vector(n+1));if(n

def b(n,k): return binomial(n-3, k)
def A110205(n,k): return (8^n-1)/7*(b(n,k-1) - b(n,k-2)) + (2^n-1)^2*((2^n+1)*b(n,k-2) + (2^n-1)*b(n,k-3))
flatten([[A110205(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 03 2024

T(n, k) = (8^n-1)/7*C(n-3, k-1) + ((2^n-1)*(4^n-1)-(8^n-1)/7)*C(n-3, k-2) + (2^n-1)^3*C(n-3, k-3).

G.f. for row n: ((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).

A125837 Numbers whose base 8 or octal representation is 6666666......6.

Original entry on oeis.org

0, 6, 54, 438, 3510, 28086, 224694, 1797558, 14380470, 115043766, 920350134, 7362801078, 58902408630, 471219269046, 3769754152374, 30158033218998, 241264265751990, 1930114126015926, 15440913008127414, 123527304065019318, 988218432520154550, 7905747460161236406

Offset: 1

Zerinvary Lajos, Feb 03 2007

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

Cf. A001018, A023001, A024088, A083068.

List([1..30], n-> 6*(8^(n-1)-1)/7); # G. C. Greubel, Aug 03 2019

[6*(8^(n-1)-1)/7: n in [1..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(6*(8^n-1)/7, n=0..30);
```

FromDigits[#,8]&/@Table[Table[6,{i}],{i,0,30}]  (* Harvey P. Dale, Mar 19 2011 *)
6*(8^(Range[30]-1) -1)/7 (* G. C. Greubel, Aug 03 2019 *)

vector(30, n, 6*(8^(n-1)-1)/7) \\ G. C. Greubel, Aug 03 2019

[6*(8^(n-1)-1)/7 for n in (1..30)] # G. C. Greubel, Aug 03 2019

a(n) = 6*(8^(n-1) -1)/7 = 6*A023001(n-1).
a(n) = 8*a(n-1) + 6 for n>1, a(1)=0. - Vincenzo Librandi, Oct 03 2010
G.f.: 6*x^2/( (1-x)*(1-8*x) ). - R. J. Mathar, Oct 07 2016
E.g.f.: 6*(exp(8*x) - exp(x))/7. - G. C. Greubel, Aug 03 2019
a(n) = -1 + A083068(n-1). - Alois P. Heinz, May 20 2023

A218725 a(n) = (22^n - 1)/21.

Original entry on oeis.org

0, 1, 23, 507, 11155, 245411, 5399043, 118778947, 2613136835, 57489010371, 1264758228163, 27824681019587, 612142982430915, 13467145613480131, 296277203496562883, 6518098476924383427, 143398166492336435395, 3154759662831401578691, 69404712582290834731203

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 22; q-integers for q=22: Diagonal k=1 in the triangle A022186.

Partial sums are in A014907. Also, the sequence is related to A014940 by A014940(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. [Bruno Berselli, Nov 06 2012]

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (23,-22).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A014907, A014940, A022186.

[n le 2 select n-1 else 23*Self(n-1) - 22*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{23, -22}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218725(n):=(22^n-1)/21$ makelist(A218725(n),n,0,30); /* Martin Ettl, Nov 06 2012 */

PARI
```
A218725(n)=22^n\21
```

a(n) = floor(22^n/21).
G.f.: x/((1-x)*(1-22*x)). [Bruno Berselli, Nov 06 2012]
a(n) = 23*a(n-1) - 22*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(x)*(exp(21*x) - 1)/21. - Elmo R. Oliveira, Aug 29 2024

cp's OEIS Frontend

A218746 a(n) = (43^n - 1)/42.

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A353147 Decimal repunits written in base 8.

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A125835 Numbers whose base-8 or octal representation is 22222222.......2.

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A218728 a(n) = (25^n - 1)/24.

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A218743 a(n) = (40^n - 1)/39.

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A269025 a(n) = Sum_{k = 0..n} 60^k.

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A098436 Triangle of 3rd central factorial numbers T(n,k).

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