A023001 -id:A023001 - cp's OEIS frontend

A016218 Expansion of 1/((1-x)(1-4x)(1-5x)).

1, 10, 71, 440, 2541, 14070, 75811, 400900, 2091881, 10808930, 55442751, 282806160, 1436400421, 7271480590, 36715316891, 185008240220, 930767824161, 4676745613050, 23475354034231, 117743274047080, 590182385739101, 2956775990710310, 14807336201610771

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-29,20).

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

Vec(1/((1-x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

From Vincenzo Librandi, Feb 10 2011: (Start)
a(n) = a(n-1) + 5^(n+1) - 4^(n+1), n >= 1.
a(n) = 9*a(n-1) - 20*a(n-2) + 1, n >= 2. (End)
a(n) = 1/12 - 4^(n+2)/3 + 5^(n+2)/4. - R. J. Mathar, Mar 15 2011

A016256 Expansion of 1/((1-x)(1-8x)(1-9x)).

Original entry on oeis.org

1, 18, 235, 2700, 28981, 298278, 2984095, 29253600, 282456361, 2695498938, 25486623955, 239196683700, 2231306698141, 20710052641998, 191416812647815, 1762962024789000, 16188343910770321, 148268580698287458, 1355005110295423675, 12359749064745505500

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (18,-89,72)

Cf. A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125.

a:=n->sum(9^(n-j)-8^(n-j),j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 04 2007

Table[(-8^(n + 2) + 7*9^(n + 1) + 1)/56, {n, 40}] (* and *) CoefficientList[Series[1/((1 - z) (1 - 8*z) (1 - 9*z)), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)

Vec(1/((1-x)*(1-8*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

G.f.: 1/((1-x)*(1-8*x)*(1-9*x)).
a(n) = 17*a(n-1) - 72*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
a(n) = 9^(n+2)/8 - 8^(n+2)/7 + 1/56. - R. J. Mathar, Mar 14 2011
a(n) = 18*a(n-1) - 89*a(n-2) + 72*a(n-3). - Wesley Ivan Hurt, Apr 20 2023

A083713 a(n) = (8^n - 1)*3/7.

Original entry on oeis.org

0, 3, 27, 219, 1755, 14043, 112347, 898779, 7190235, 57521883, 460175067, 3681400539, 29451204315, 235609634523, 1884877076187, 15079016609499, 120632132875995, 965057063007963, 7720456504063707, 61763652032509659

Offset: 0

Klaus Brockhaus, Jun 14 2003

Fixed points of the mapping defined by A067585. In binary these numbers show a regular pattern: 0, 11, 11011, 11011011, 11011011011, etc.
From Reinhard Zumkeller, Feb 22 2010: (Start)
a(n) = A173593(6*n-5) for n > 0:
terms of A173593 beginning and ending with digits '11' in binary representation;
for n > 0: a(n) = A033129(3*n-1); a(n) - a(n-1) = A103333(n). (End)

			From _Zerinvary Lajos_, Jan 14 2007: (Start)
Octal..........decimal:
0....................0
3....................3
33..................27
333................219
3333..............1755
33333............14043
333333..........112347
3333333.........898779
33333333.......7190235
333333333.....57521883
3333333333...460175067
etc. (End)

Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A067585, A023001.

(3/7)(8^Range[0,20]-1) (* or *) LinearRecurrence[{9,-8},{0,3},30] (* or *) NestList[8#+3&,0,30] (* Harvey P. Dale, Jun 06 2013 *)

a(n)=(8^n-1)*3/7 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*A023001(n).
Recursion: a(0) = 0, a(n+1) = (((a(n)*2)*2+1)*2+1).
a(n) = 8*a(n-1) + 3 (with a(0)=0). - Vincenzo Librandi, Aug 08 2010
a(0)=0, a(1)=3, a(n) = 9*a(n-1) - 8*a(n-2). - Harvey P. Dale, Jun 06 2013
From Stefano Spezia, Feb 23 2025: (Start)
G.f.: 3*x/((1 - x)*(1 - 8*x)).
E.g.f.: 3*exp(x)*(exp(7*x) - 1)/7. (End)

A218750 a(n) = (47^n - 1)/46.

Original entry on oeis.org

0, 1, 48, 2257, 106080, 4985761, 234330768, 11013546097, 517636666560, 24328923328321, 1143459396431088, 53742591632261137, 2525901806716273440, 118717384915664851681, 5579717091036248029008, 262246703278703657363377, 12325595054099071896078720

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 47 (A009991).

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 (48,-47)

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. A009991.

[n le 2 select n-1 else 48*Self(n-1) - 47*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

Table[(47^n - 1)/46, {n, 0, 19}] (* Alonso del Arte, Nov 04 2012 *)
LinearRecurrence[{48, -47}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218750(n):=(47^n-1)/46$ makelist(A218750(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218750(n)=47^n\46
```

a(n) = floor(47^n/46).
G.f.: x/(47*x^2-48*x+1) = x/((1-x)*(1-47*x)). [Colin Barker, Nov 06 2012]
a(0)=0, a(n) = 47*a(n-1) + 1. - Vincenzo Librandi, Nov 08 2012
a(n) = 48*a(n-1) - 47*a(n-2). - Wesley Ivan Hurt, Jan 25 2022
E.g.f.: exp(24*x)*sinh(23*x)/23. - Elmo R. Oliveira, Aug 27 2024

A033118 Base 8 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.

Original entry on oeis.org

1, 8, 65, 520, 4161, 33288, 266305, 2130440, 17043521, 136348168, 1090785345, 8726282760, 69810262081, 558482096648, 4467856773185, 35742854185480, 285942833483841, 2287542667870728, 18300341342965825, 146402730743726600

Offset: 1

Clark Kimberling

Partial sums of A015565. - Mircea Merca, Dec 28 2010

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

Pairwise sums are (8^n - 1)/7 (A023001).

[Round((8*8^n-8)/63): n in [1..30]]; // Vincenzo Librandi, Jun 25 2011

seq(1/7*floor(8^(n+1)/9),n=1..22); # Mircea Merca, Dec 27 2010

Table[FromDigits[PadRight[{},n,{1,0}],8],{n,20}] (* or *) LinearRecurrence[ {8,1,-8},{1,8,65},20] (* Harvey P. Dale, Jan 20 2021 *)

a(n) = 8*a(n-1) + a(n-2) - 8*a(n-3).
a(n) = 2^(3*n+3)/63 - 1/14 - (-1)^n/18. - R. J. Mathar, Jan 08 2011
From Paul Barry, Apr 04 2008: (Start)
G.f. x/((1-x^2)*(1-8*x));
a(n) = (1/3)*Sum_{k=0..n} A001045(3k). (End)
a(n) = floor(8^(n+1)/9)/7 = floor((8*8^n-1)/63) = round((8*8^n-8)/63) = round((16*8^n-9)/63) = ceiling((8*8^n-8)/63). a(n) = a(n-2) + 8^(n-1), n > 2. - Mircea Merca, Dec 28 2010

A218726 a(n) = (23^n - 1)/22.

Original entry on oeis.org

0, 1, 24, 553, 12720, 292561, 6728904, 154764793, 3559590240, 81870575521, 1883023236984, 43309534450633, 996119292364560, 22910743724384881, 526947105660852264, 12119783430199602073, 278755018894590847680, 6411365434575589496641, 147461404995238558422744

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 23, q-integers for q=23: diagonal k=1 in triangle A022187.

Partial sums are in A014909. Also, the sequence is related to A014941 by A014941(n) = n*a(n) - Sum{a(i), i=0..n-1} 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 (24,-23).

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. A014909, A014941, A022187.

[n le 2 select n-1 else 24*Self(n-1)-23*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{24, -23}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(23^Range[0,20]-1)/22 (* Harvey P. Dale, Nov 09 2012 *)

A218726(n):=(23^n-1)/22$
makelist(A218726(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218726(n)=23^n\22
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-23*x)).
a(n) = floor(23^n/22).
a(n) = 24*a(n-1) - 23*a(n-2). (End)
E.g.f.: exp(12*x)*sinh(11*x)/11. - Elmo R. Oliveira, Aug 27 2024

A218732 a(n) = (29^n - 1)/28.

Original entry on oeis.org

0, 1, 30, 871, 25260, 732541, 21243690, 616067011, 17865943320, 518112356281, 15025258332150, 435732491632351, 12636242257338180, 366451025462807221, 10627079738421409410, 308185312414220872891, 8937374060012405313840, 259183847740359754101361

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 29 (A009973).

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 (30,-29).

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. A009973.

[n le 2 select n-1 else 30*Self(n-1)-29*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{30, -29}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218732(n):=(29^n-1)/28$
makelist(A218732(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=29^n\28
```

a(n) = floor(29^n/28).
G.f.: x/((1-x)*(1-29*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = 30*a(n-1) - 29*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(15*x)*sinh(14*x)/14. - Elmo R. Oliveira, Aug 27 2024

A218733 a(n) = (30^n - 1)/29.

Original entry on oeis.org

0, 1, 31, 931, 27931, 837931, 25137931, 754137931, 22624137931, 678724137931, 20361724137931, 610851724137931, 18325551724137931, 549766551724137931, 16492996551724137931, 494789896551724137931, 14843696896551724137931, 445310906896551724137931, 13359327206896551724137931

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 30 (A009974).

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 (31,-30).

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. A009974.

[n le 2 select n-1 else 31*Self(n-1) - 30*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{31, -30}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(30^Range[0,20]-1)/29 (* Harvey P. Dale, Nov 22 2022 *)

A218733(n):=floor((30^n-1)/29)$ makelist(A218733(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218733(n)=30^n\29
```

a(n) = floor(30^n/29).
From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-30*x)).
a(n) = 31*a(n-1) - 30*a(n-2). (End)
E.g.f.: exp(x)*(exp(29*x) - 1)/29. - Elmo R. Oliveira, Aug 29 2024

A218740 a(n) = (37^n - 1)/36.

Original entry on oeis.org

0, 1, 38, 1407, 52060, 1926221, 71270178, 2636996587, 97568873720, 3610048327641, 133571788122718, 4942156160540567, 182859777940000980, 6765811783780036261, 250335035999861341658, 9262396331994869641347, 342708664283810176729840, 12680220578500976539004081

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 37 (A009981).

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 (38,-37).

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. A009981.

[n le 2 select n-1 else 38*Self(n-1)-37*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{38, -37}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218740(n):=(37^n-1)/36$
makelist(A218740(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218740(n)=37^n\36
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 37*x)).
a(n) = 38*a(n-1) - 37*a(n-2).
a(n) = floor(37^n/36). (End)
E.g.f.: exp(x)*(exp(36*x) - 1)/36. - Stefano Spezia, Mar 28 2023

A218744 a(n) = (41^n - 1)/40.

Original entry on oeis.org

0, 1, 42, 1723, 70644, 2896405, 118752606, 4868856847, 199623130728, 8184548359849, 335566482753810, 13758225792906211, 564087257509154652, 23127577557875340733, 948230679872888970054, 38877457874788447772215, 1593975772866326358660816, 65353006687519380705093457

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 41 (A009985).

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 (42,-41).

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. A009985.

[n le 2 select n-1 else 42*Self(n-1)-41*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{42, -41}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218744(n):=(41^n-1)/40$
makelist(A218744(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218744(n)=41^n\40
```

a(n) = floor(41^n/40).
G.f.: x/((1-x)*(1-41*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = 42*a(n-1) - 41*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(21*x)*sinh(20*x)/20. - Elmo R. Oliveira, Aug 27 2024

cp's OEIS Frontend

A016218 Expansion of 1/((1-x)*(1-4*x)*(1-5*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A016256 Expansion of 1/((1-x)*(1-8*x)*(1-9*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A083713 a(n) = (8^n - 1)*3/7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A218750 a(n) = (47^n - 1)/46.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A033118 Base 8 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A218726 a(n) = (23^n - 1)/22.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A218732 a(n) = (29^n - 1)/28.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

A016218 Expansion of 1/((1-x)(1-4x)(1-5x)).

A016256 Expansion of 1/((1-x)(1-8x)(1-9x)).