A218723 -id:A218723 - 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

A133752 a(n) = 256^n.

Original entry on oeis.org

1, 256, 65536, 16777216, 4294967296, 1099511627776, 281474976710656, 72057594037927936, 18446744073709551616, 4722366482869645213696, 1208925819614629174706176, 309485009821345068724781056

Offset: 0

Marcel Hetkowski Fabeny (marcelfabeny(AT)yahoo.com.br), Jan 01 2008

Number of different possible values that a file of n bytes can have; each byte has 8 bits and each bit can be 0 or 1.
The sequence shows how many different files can exist even with a very small amount of data (just a few bytes). With just 5 bytes of data, there are 1099511627776 different possible files.
Partial sums are in A218723. - M. F. Hasler, Nov 05 2012

			a(1) = 256^1 = 256 --> there are 256 possible 1-byte files;
a(2) = 256^2 = 65536 --> there are 65536 possible 2-byte files;
a(3) = 256^3 = 16777216 --> there are 16777216 possible 3-byte files;
a(4) = 256^4 = 4294967296 --> there are 4294967296 possible 4-byte files;
a(5) = 256^5 = 1099511627776 --> there are 1099511627776 possible 5-byte files.

Index entries for linear recurrences with constant coefficients, signature (256).

Subsequence of A000079.

Cf. A218723.

lst={};Do[AppendTo[lst,256^n],{n,0,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 02 2009 *)
256^Range[0,20] (* Harvey P. Dale, Aug 21 2014 *)

def A133752(n): return 1<<(n<<3) # Chai Wah Wu, Nov 10 2022

From Elmo R. Oliveira, Aug 11 2024: (Start)
G.f.: 1/(1-256*x).
E.g.f.: exp(256*x).
a(n) = 256*a(n-1), n > 0. (End)

Edited by Jon E. Schoenfield, Jan 24 2015

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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A133752 a(n) = 256^n.

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

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Magma

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Maxima

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Formula

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