A133853 -id:A133853 - cp's OEIS frontend

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

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

A138635 a(n) =3a(n-3)-3a(n-6)+2*a(n-9).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 2, 6, 5, 5, 11, 10, 11, 21, 21, 22, 42, 43, 43, 85, 86, 85, 171, 171, 170, 342, 341, 341, 683, 682, 683, 1365, 1365, 1366, 2730, 2731, 2731, 5461, 5462, 5461, 10923, 10923, 10922, 21846, 21845, 21845, 43691, 43690, 43691, 87381

Offset: 0

Paul Curtz, May 14 2008

As the recurrence shows, these are three interleaved sequences which obey recurrences b(n)=3*b(n-1)-3*b(n-2)+2*b(n-3), indicating that the b(n) equal their third differences.
These three sequences are A024495, A024494 (or A131708) and A024493 (or A130781).
Their starting "vectors" b(0,1,2) are 0,0,1 and 0,1,2 and 1,1,1, respectively, therefore linearly independent, such that other sequences with the same recursion as b(n) can be written as linear combinations of these.

Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,2).

Cf. A024493, A024494, A024495, A131708, A130781.

a(18*n) = 21*A133853(n).

G.f.: -x^2*(1+x^2-2*x^3+x^4-x^5+x^6)/((2*x^3-1)*(x^6-x^3+1)). - R. J. Mathar, May 17 2009

Edited by R. J. Mathar, May 17 2009

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

A218731 a(n) = (28^n - 1)/27.

Original entry on oeis.org

0, 1, 29, 813, 22765, 637421, 17847789, 499738093, 13992666605, 391794664941, 10970250618349, 307167017313773, 8600676484785645, 240818941573998061, 6742930364071945709, 188802050194014479853, 5286457405432405435885, 148020807352107352204781, 4144582605859005861733869

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 28 (A009972).

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (29,-28).

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. A009972.

[n le 2 select n-1 else 29*Self(n-1)-28*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{29, -28}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218731(n):=(28^n-1)/27$
makelist(A218731(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218731(n)=28^n\27
```

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

A218739 a(n) = (36^n - 1)/35.

Original entry on oeis.org

0, 1, 37, 1333, 47989, 1727605, 62193781, 2238976117, 80603140213, 2901713047669, 104461669716085, 3760620109779061, 135382323952046197, 4873763662273663093, 175455491841851871349, 6316397706306667368565, 227390317427040025268341, 8186051427373440909660277

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 36 (A009980).

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 (37,-36).

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. A009980.

[n le 2 select n-1 else 37*Self(n-1)-36*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{37, -36}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[36^Range[0,20]]] (* Harvey P. Dale, Jun 03 2015 *)

A218739(n):=(36^n-1)/35$
makelist(A218739(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218739(n)=36^n\35
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 36*x)).
a(n) = 37*a(n-1) - 36*a(n-2).
a(n) = floor(36^n/35). (End)
E.g.f.: exp(x)*(exp(35*x) - 1)/35. - Stefano Spezia, Mar 28 2023

A218741 a(n) = (38^n - 1)/37.

Original entry on oeis.org

0, 1, 39, 1483, 56355, 2141491, 81376659, 3092313043, 117507895635, 4465300034131, 169681401296979, 6447893249285203, 245019943472837715, 9310757851967833171, 353808798374777660499, 13444734338241551098963, 510899904853178941760595, 19414196384420799786902611

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 38 (A009982).

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 (39,-38).

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. A009982.

[n le 2 select n-1 else 39*Self(n-1) - 38*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{39, -38}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218741(n):=(38^n-1)/37$
makelist(A218741(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218741(n)=38^n\37
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-38*x)).
a(n) = 39*a(n-1) - 38*a(n-2).
a(n) = floor(38^n/37). (End)
E.g.f.: exp(x)*(exp(37*x) - 1)/37. - Elmo R. Oliveira, Aug 29 2024

A218742 a(n) = (39^n - 1)/38.

Original entry on oeis.org

0, 1, 40, 1561, 60880, 2374321, 92598520, 3611342281, 140842348960, 5492851609441, 214221212768200, 8354627297959801, 325830464620432240, 12707388120196857361, 495588136687677437080, 19327937330819420046121, 753789555901957381798720, 29397792680176337890150081

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 39 (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 (40,-39).

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 40*Self(n-1) - 39*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{40, -39}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(39^Range[0,20]-1)/38 (* Harvey P. Dale, Mar 05 2023 *)

A218742(n):=(39^n-1)/38$
makelist(A218742(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=39^n\38
```

a(n) = floor(39^n/38).
From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-39*x)).
a(n) = 40*a(n-1) - 39*a(n-2). (End)
E.g.f.: exp(20*x)*sinh(19*x)/19. - Elmo R. Oliveira, Aug 29 2024

A218747 a(n) = (44^n - 1)/43.

Original entry on oeis.org

0, 1, 45, 1981, 87165, 3835261, 168751485, 7425065341, 326702875005, 14374926500221, 632496766009725, 27829857704427901, 1224513738994827645, 53878604515772416381, 2370658598693986320765, 104308978342535398113661, 4589595047071557517001085, 201942182071148530748047741

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 44 (A009988).

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 (45,-44).

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. A009988.

[n le 2 select n-1 else 45*Self(n-1) - 44*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{45, -44}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[44^Range[0,20]]] (* Harvey P. Dale, Dec 28 2015 *)

A218747(n):=(44^n-1)/43$
makelist(A218747(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218747(n)=44^n\43
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-44*x)).
a(n) = 45*a(n-1) - 44*a(n-2).
a(n) = floor(44^n/43). (End)
E.g.f.: exp(x)*(exp(43*x) - 1)/43. - Elmo R. Oliveira, Aug 29 2024

cp's OEIS Frontend

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

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A138635 a(n) =3*a(n-3)-3*a(n-6)+2*a(n-9).

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

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

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

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Magma

Mathematica

Maxima

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Formula

A218731 a(n) = (28^n - 1)/27.

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Programs

Magma

Mathematica

Maxima

PARI

Formula

A218739 a(n) = (36^n - 1)/35.

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A138635 a(n) =3a(n-3)-3a(n-6)+2*a(n-9).