A135518 -id:A135518 - cp's OEIS frontend

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

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

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

A218737 a(n) = (34^n - 1)/33.

Original entry on oeis.org

0, 1, 35, 1191, 40495, 1376831, 46812255, 1591616671, 54114966815, 1839908871711, 62556901638175, 2126934655697951, 72315778293730335, 2458736461986831391, 83597039707552267295, 2842299350056777088031, 96638177901930420993055, 3285698048665634313763871

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 34 (A009978).

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 (35,-34).

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.

[n le 2 select n-1 else 35*Self(n-1)-34*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{35, -34}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218737(n):=(34^n-1)/33$
makelist(A218737(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218737(n)=34^n\33
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 34*x)).
a(n) = 35*a(n-1) - 34*a(n-2).
a(n) = floor(34^n/33). (End)
E.g.f.: exp(x)*(exp(33*x) - 1)/33. - Stefano Spezia, Mar 26 2023

A218738 a(n) = (35^n - 1)/34.

Original entry on oeis.org

0, 1, 36, 1261, 44136, 1544761, 54066636, 1892332261, 66231629136, 2318107019761, 81133745691636, 2839681099207261, 99388838472254136, 3478609346528894761, 121751327128511316636, 4261296449497896082261, 149145375732426362879136, 5220088150634922700769761

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 35 (A009979).

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 (36,-35).

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. A009979.

[n le 2 select n-1 else 36*Self(n-1)-35*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{36, -35}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218738(n):=(35^n-1)/34$
makelist(A218738(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218738(n)=35^n\34
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 35*x)).
a(n) = 36*a(n-1) - 35*a(n-2).
a(n) = floor(35^n/34). (End)
E.g.f.: exp(x)*(exp(34*x) - 1)/34. - Stefano Spezia, Mar 28 2023

A218745 a(n) = (42^n - 1)/41.

Original entry on oeis.org

0, 1, 43, 1807, 75895, 3187591, 133878823, 5622910567, 236162243815, 9918814240231, 416590198089703, 17496788319767527, 734865109430236135, 30864334596069917671, 1296302053034936542183, 54444686227467334771687, 2286676821553628060410855, 96040426505252378537255911

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 42 (A009986).

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 (43,-42).

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. A009986.

[n le 2 select n-1 else 43*Self(n-1) - 42*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{43, -42}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(42^Range[0,20]-1)/41 (* Harvey P. Dale, May 08 2024 *)

A218745(n):=(42^n-1)/41$
makelist(A218745(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218745(n)=42^n\41
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-42*x)).
a(n) = 43*a(n-1) - 42*a(n-2).
a(n) = floor(42^n/41). (End)
E.g.f.: exp(x)*(exp(41*x) - 1)/41. - Elmo R. Oliveira, Aug 29 2024

A218727 a(n) = (24^n - 1)/23.

Original entry on oeis.org

0, 1, 25, 601, 14425, 346201, 8308825, 199411801, 4785883225, 114861197401, 2756668737625, 66160049703001, 1587841192872025, 38108188628928601, 914596527094286425, 21950316650262874201, 526807599606308980825, 12643382390551415539801, 303441177373233972955225

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 24 (A009968); q-integers for q=24: diagonal k=1 in triangle A022188.

Partial sums are in A014913. Also, the sequence is related to A014942 by A014942(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 (25,-24).

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. A009968, A014942, A022188.

[n le 2 select n-1 else 25*Self(n-1)-24*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{25, -24}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218727(n):=(24^n-1)/23$
makelist(A218727(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218727(n)=24^n\23
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-24*x)).
a(n) = floor(24^n/23).
a(n) = 25*a(n-1) - 24*a(n-2). (End)
E.g.f.: exp(x)*(exp(23*x) - 1)/23. - Elmo R. Oliveira, Aug 29 2024

A218729 a(n) = (26^n - 1)/25.

Original entry on oeis.org

0, 1, 27, 703, 18279, 475255, 12356631, 321272407, 8353082583, 217180147159, 5646683826135, 146813779479511, 3817158266467287, 99246114928149463, 2580398988131886039, 67090373691429037015, 1744349715977154962391, 45353092615406029022167, 1179180408000556754576343

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 26 (A009970); q-integers for q=26.

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 (27,-26).

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. A009970.

[n le 2 select n-1 else 27*Self(n-1)-26*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{27, -26}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218729(n):=(26^n-1)/25$
makelist(A218729(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218729(n)=26^n\25
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-26*x)).
a(n) = floor(26^n/25).
a(n) = 27*a(n-1) - 26*a(n-2). (End)
E.g.f.: exp(x)*(exp(25*x) - 1)/25. - Elmo R. Oliveira, Aug 29 2024

A218730 a(n) = (27^n - 1)/26.

Original entry on oeis.org

0, 1, 28, 757, 20440, 551881, 14900788, 402321277, 10862674480, 293292210961, 7918889695948, 213810021790597, 5772870588346120, 155867505885345241, 4208422658904321508, 113627411790416680717, 3067940118341250379360, 82834383195213760242721, 2236528346270771526553468

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 27 (A009971); q-integers for q=27.

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 (28,-27).

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. A009971.

[n le 2 select n-1 else 28*Self(n-1)-27*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{28, -27}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218730(n):=(27^n-1)/26$
makelist(A218730(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=27^n\26
```

G.f.: x/((1-x)*(1-27*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = floor(27^n/26). - Vincenzo Librandi, Nov 07 2012
a(n) = 28*a(n-1) - 27*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(14*x)*sinh(13*x)/13. - Elmo R. Oliveira, Aug 27 2024

cp's OEIS Frontend

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

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A218737 a(n) = (34^n - 1)/33.

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A218738 a(n) = (35^n - 1)/34.

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A218745 a(n) = (42^n - 1)/41.

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