A218736 -id:A218736 - cp's OEIS frontend

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

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

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

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

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

cp's OEIS Frontend

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

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

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