A003463 -id:A003463 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A353144 Decimal repunits written in base 5.

Original entry on oeis.org

0, 1, 21, 421, 13421, 323421, 12023421, 241023421, 10321023421, 211421023421, 4233421023421, 140223421023421, 3310023421023421, 121201023421023421, 2424021023421023421, 104030421023421023421, 2131113421023421023421, 43122323421023421023421, 1413002023421023421023421

Offset: 0

Seiichi Manyama, Apr 26 2022

Cf. A002275, A291962, A353142, A353143, A353145, A353146, A353147, A353148.

Cf. A003463, A007091.

a(n) = fromdigits(digits((10^n-1)/9, 5));

a(n) = A007091(A002275(n)).

A016234 Expansion of 1/((1-x) * (1-5x) (1-9*x)).

Original entry on oeis.org

1, 15, 166, 1650, 15631, 144585, 1320796, 11984820, 108351661, 977606355, 8810664226, 79357013190, 714518294491, 6432190529325, 57897344158456, 521114244398760, 4690218934452121, 42212924084385495, 379921085131051486, 3419313608037373530, 30773941681625912551

Offset: 0

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15, -59, 45).

Cf. A000012, A003463.
Cf. A016209, A016223, A016247.
Cf. A016307, A018054.

CoefficientList[Series[1/((1-x)(1-5x)(1-9x)),{x,0,30}],x] (* or *) LinearRecurrence[{15,-59,45},{1,15,166},30] (* Harvey P. Dale, Oct 16 2014 *)

Vec(1/((1-x)*(1-5*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (9^(n+2) - 2*5^(n+2) + 1)/32; \\ Joerg Arndt, Aug 13 2013

a(0)=1, a(1)=15, a(n) = 14*a(n-1) - 45*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
a(n) = (9^(n+2) - 2*5^(n+2) + 1)/32. - Yahia Kahloune, Aug 13 2013
a(0)=1, a(1)=15, a(2)=166, a(n) = 15*a(n-1) - 59*a(n-2) + 45*a(n-3). - Harvey P. Dale, Oct 16 2014
O.g.f.: see the name.
E.g.f.: (d^2/dx^2) (exp(x)*((exp(4*x) - 1)^2)/(4^2*2!)) = exp(x)*(1 - 50*exp(4*x) + 81*exp(8*x))/32.
From Seiichi Manyama, May 05 2025: (Start)
a(n) = Sum_{k=0..n} 4^k * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-4)^k * 9^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

A125833 Numbers whose base-5 representation is 333333.......3.

Original entry on oeis.org

0, 3, 18, 93, 468, 2343, 11718, 58593, 292968, 1464843, 7324218, 36621093, 183105468, 915527343, 4577636718, 22888183593, 114440917968, 572204589843, 2861022949218, 14305114746093, 71525573730468, 357627868652343

Offset: 0

Zerinvary Lajos, Feb 03 2007

			Base 5.................decimal
0.........................0
3.........................3
33.......................18
333......................93
3333....................468
33333..................2343
333333................11718
3333333...............58593
33333333.............292968, etc.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A003463.

List([0..30], n-> 3*(5^n -1)/4); G. C. Greubel, Aug 03 2019

[3*(5^n -1)/4: n in [0..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(3*(5^n-1)/4, n=0..30);
```

Table[FromDigits[PadRight[{},n,3],5],{n,0,30}] (* or *) LinearRecurrence[ {6,-5},{0,3},30] (* Harvey P. Dale, Sep 23 2016 *)
3*(5^Range[0,30] -1)/4 (* G. C. Greubel, Aug 03 2019 *)

vector(30, n, n--; 3*(5^n -1)/4) \\ G. C. Greubel, Aug 03 2019

[3*(5^n -1)/4 for n in (0..30)] # G. C. Greubel, Aug 03 2019

a(n) = 3*(5^n - 1)/4.
a(n) = 5*a(n-1) + 3 for n > 0, a(0)=0. - Vincenzo Librandi, Sep 30 2010
From G. C. Greubel, Aug 03 2019: (Start)
a(n) = 3*A003463(n).
G.f.: 3*x/((1-x)*(1-5*x)).
E.g.f.: 3*(exp(5*x) - exp(x))/4. (End)

A157832 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (5^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 5, -6, 1, 125, -155, 31, -1, 15625, -19500, 4030, -156, 1, 9765625, -12203125, 2538250, -101530, 781, -1, 30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1, 476837158203125, -596038818359375

Offset: 0

Roger L. Bagula, Mar 07 2009

Except for n=0, the row sums are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=5)= [1,4,25,120,625,3100,15625,...] DELTA [ -1,0,-5,0,-25,0,-125,0,-625,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins
  1;
  1, -1;
  5, -6, 1;
  125, -155, 31, -1;
  15625, -19500, 4030, -156, 1;
  9765625, -12203125, 2538250, -101530, 781, -1;
  30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1;
  476837158203125, -596038818359375, 124166806640625, -5005123921875, 40040991375, -63573405, 19531, -1;

Cf. A135950, A157783, A109345 (first column), A003463 (first subdiagonal).

A157832 := proc(n,k)
        product( 5^(i-1)-x,i=1..n) ;
        coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

p[x_, n_] = If[n == 0, 1, Product[q^(i - 1) - x, {i, 1, n}]];
q = 5;
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

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

cp's OEIS Frontend

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

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

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

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

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A353144 Decimal repunits written in base 5.

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A016234 Expansion of 1/((1-x) * (1-5*x) * (1-9*x)).

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PARI

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A125833 Numbers whose base-5 representation is 333333.......3.

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A016234 Expansion of 1/((1-x) * (1-5x) (1-9*x)).