A094028 -id:A094028 - cp's OEIS frontend

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

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

A261544 a(n) = Sum_{k=0..n} 1000^k.

Original entry on oeis.org

1, 1001, 1001001, 1001001001, 1001001001001, 1001001001001001, 1001001001001001001, 1001001001001001001001, 1001001001001001001001001, 1001001001001001001001001001, 1001001001001001001001001001001, 1001001001001001001001001001001001

Offset: 0

Ilya Gutkovskiy, Aug 24 2015

A sequence of palindromic numbers.

			From _Bruno Berselli_, Aug 25 2015: (Start)
a(n)   is the binary representation of    A023001
-------------------------------------------------
1  ...........................................  1
1001  ........................................  9
1001001 .....................................  73
1001001001  ................................  585
1001001001001  ............................  4681
1001001001001001  ........................  37449
1001001001001001001  ....................  299593
1001001001001001001001  ................  2396745
1001001001001001001001001  ............  19173961, etc.
(End)

Colin Barker, Table of n, a(n) for n = 0..333 (corrected by Michel Marcus, Jan 19 2019)
John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 2019.
Eric Weisstein's World of Mathematics, Palindromic Number.
Index entries for linear recurrences with constant coefficients, signature (1001,-1000).

Cf. A000533, A002113, A023001.
Subsequence of A033146.
Sums of 100^k: A094028; sums of 10^k: A000042.
Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

[(1000^(n+1)-1)/999: n in [0..30]]; // Vincenzo Librandi, Aug 24 2015

Table[(1000^(n + 1) - 1)/999, {n, 0, 15}]
LinearRecurrence[{1001, -1000}, {1, 1001}, 20] (* Vincenzo Librandi, Aug 24 2015 *)

Vec(1 / ((x-1)*(1000*x-1)) + O(x^20)) \\ Colin Barker, Aug 24 2015

a(n) = (1000^(n + 1) - 1)/999.
a(n) = 1001*a(n-1) - 1000*a(n-2). - Colin Barker, Aug 24 2015
G.f.: 1 / ((x-1)*(1000*x-1)). - Colin Barker, Aug 24 2015
E.g.f.: (1/999)*(1000000*exp(1000*x) - exp(x)). - G. C. Greubel, Aug 29 2015

A344822 Numbers m with decimal expansion (d_1, ..., d_k) such that d_i = m * i mod 10 for i = 1..k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 505, 50505, 246802, 482604, 628406, 864208, 5050505, 505050505, 12345678901, 24680246802, 36925814703, 48260482604, 50505050505, 62840628406, 74185296307, 86420864208, 98765432109, 5050505050505, 505050505050505, 2468024680246802

Offset: 1

Rémy Sigrist, May 29 2021

This sequence is infinite as it contains 5 * A094028(k) for any k > 0.

Also contains terms with patterns 2(46802)^k, 4(82604)^k, 6(28406)^k, 8(64208)^k, 1(2345678901)^k, 3(6925814703)^k, 7(4185296307)^k, 9(8765432109)^k for k >= 0, where ^ denotes repeated concatenation; all terms have first and last digits the same. - Michael S. Branicky, May 29 2021

			- 4 * 1 = 4 mod 10,
- 4 * 2 = 8 mod 10,
- 4 * 3 = 2 mod 10,
- 4 * 4 = 6 mod 10,
- 4 * 5 = 0 mod 10,
- 4 * 6 = 4 mod 10,
so 482604 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..426
Rémy Sigrist, PARI program for A344822

Cf. A094028, A344748, A344823.

is(n) = { my (d=digits(n)); for (k=1, #d, if (d[k] != (n*k)%10, return (0))); return (1) }

PARI
```
See Links section.
```

def ok(m):
  d = str(m)
  return all(d[i-1] == str((m*i)%10) for i in range(1, len(d)+1))
print(list(filter(ok, range(10**6)))) # Michael S. Branicky, May 29 2021

def auptod(maxdigits):
  alst = [0]
  for k in range(1, maxdigits+1):
    for d1 in range(1, 10):
      d = [(d1*i)%10 for i in range(1, k+1)]
      if d1 == d[-1]: alst.append(int("".join(map(str, d))))
  return alst
print(auptod(16)) # Michael S. Branicky, May 29 2021

A098610 a(n) = 10^n + (-1)^n.

Original entry on oeis.org

2, 9, 101, 999, 10001, 99999, 1000001, 9999999, 100000001, 999999999, 10000000001, 99999999999, 1000000000001, 9999999999999, 100000000000001, 999999999999999, 10000000000000001, 99999999999999999, 1000000000000000001, 9999999999999999999, 100000000000000000001

Offset: 0

Henry Bottomley, Sep 17 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,10).

Cf. A015585, A094028, A098609, A098611.

[10^n+(-1)^n: n in [0..20]]; // Vincenzo Librandi, Sep 23 2016

Total/@Partition[Riffle[10^Range[0,20],{1,-1}],2] (* or *) Table[10^n+(-1)^n,{n,0,20}] (* Harvey P. Dale, Aug 20 2012 *)

a(n) = A098611(n) + 2*(-1)^n.
a(n) = A098609(n)/A098611(n).
a(n) = A098609(n)/(11*A015585(n)).
a(n) = 9*A094028(n+1)/A015585(n).
From Chai Wah Wu, Sep 22 2016: (Start)
a(n) = 9*a(n-1) + 10*a(n-2) for n > 1.
G.f.: (9*x - 2)/((x + 1)*(10*x - 1)). (End)
E.g.f.: exp(-x)*(exp(11*x) + 1). - Elmo R. Oliveira, Aug 17 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

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

cp's OEIS Frontend

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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A261544 a(n) = Sum_{k=0..n} 1000^k.

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A344822 Numbers m with decimal expansion (d_1, ..., d_k) such that d_i = m * i mod 10 for i = 1..k.

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A098610 a(n) = 10^n + (-1)^n.

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

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