cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A107583 a(n) = 3^n - 3*n.

Original entry on oeis.org

1, 0, 3, 18, 69, 228, 711, 2166, 6537, 19656, 59019, 177114, 531405, 1594284, 4782927, 14348862, 43046673, 129140112, 387420435, 1162261410, 3486784341, 10460353140, 31381059543, 94143178758, 282429536409, 847288609368, 2541865828251, 7625597484906, 22876792454877
Offset: 0

Views

Author

Zak Seidov, May 16 2005

Keywords

Comments

a(n) is the number k such that the number m with n 3's and k 1's has digit product = digit sum = 3^n.

Examples

			Corresponding numbers m are 1, 3, 11133, 111111111111111111333, ...

Links

Crossrefs

Cf. A107584, A107585.

Programs

Formula

From Elmo R. Oliveira, Sep 09 2024: (Start)
G.f.: (1 - 5*x + 10*x^2)/((1 - 3*x)*(1 - x)^2).
E.g.f.: exp(x)*(exp(2*x) - 3*x).
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3) for n > 2. (End)

Extensions

Corrected by Charles R Greathouse IV, Sep 08 2012

A107585 a(n) = 5^n - 5*n.

Original entry on oeis.org

1, 0, 15, 110, 605, 3100, 15595, 78090, 390585, 1953080, 9765575, 48828070, 244140565, 1220703060, 6103515555, 30517578050, 152587890545, 762939453040, 3814697265535, 19073486328030, 95367431640525, 476837158203020, 2384185791015515, 11920928955078010, 59604644775390505, 298023223876953000
Offset: 0

Views

Author

Zak Seidov, May 16 2005

Keywords

Comments

Numbers a(n) = k such that the number m with n 5's and k 1's has digit product = digit sum = 5^n.

Examples

			Corresponding numbers m are 1, 5, 11111111111111155, ...

Links

Crossrefs

Cf. A107583, A107584.

Programs

Formula

a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3), n >= 3. - Vincenzo Librandi, Oct 26 2011
G.f.: (-1-26*x^2+7*x)/((5*x-1)*(x-1)^2). - R. J. Mathar, Oct 26 2011
E.g.f.: exp(x)*(exp(4*x) - 5*x). - Elmo R. Oliveira, Sep 10 2024

Extensions

Corrected by Charles R Greathouse IV, Sep 08 2012

A198396 a(n) = 6^n - 6*n.

Original entry on oeis.org

1, 0, 24, 198, 1272, 7746, 46620, 279894, 1679568, 10077642, 60466116, 362796990, 2176782264, 13060693938, 78364164012, 470184984486, 2821109907360, 16926659444634, 101559956668308, 609359740010382, 3656158440062856, 21936950640377730, 131621703842267004, 789730223053602678
Offset: 0

Views

Author

Vincenzo Librandi, Oct 26 2011

Keywords

Links

Crossrefs

Cf. A005803, A107583, A107584, A107585.

Programs

  • Magma
    [6^n-6*n: n in [0..25]];
  • Mathematica
    CoefficientList[Series[(1 - 8*x + 37*x^2)/((1 - 6*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
Table[6^n-6n,{n,0,30}] (* or *) LinearRecurrence[{8,-13,6},{1,0,24},30] (* Harvey P. Dale, Jul 25 2019 *)
  • PARI
    a(n)=6^n-6*n \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3) for n > 2.
G.f.: (1-8*x+37*x^2)/((1-6*x)*(1-x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(5*x) - 6*x). - Elmo R. Oliveira, Sep 10 2024

A198397 a(n) = 7^n - 7*n.

Original entry on oeis.org

1, 0, 35, 322, 2373, 16772, 117607, 823494, 5764745, 40353544, 282475179, 1977326666, 13841287117, 96889010316, 678223072751, 4747561509838, 33232930569489, 232630513987088, 1628413597910323, 11398895185373010, 79792266297611861, 558545864083283860, 3909821048582987895
Offset: 0

Views

Author

Vincenzo Librandi, Oct 26 2011

Keywords

Links

Crossrefs

Cf. A005803, A107583, A107584, A107585.

Programs

  • Magma
    [7^n-7*n: n in [0..25]];
  • Mathematica
    CoefficientList[Series[(1 - 9*x + 50*x^2)/((1 - 7*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
  • PARI
    a(n)=7^n-7*n \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3) for n > 2.
G.f.: (1 - 9*x + 50*x^2)/((1 - 7*x)*(1 -x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(6*x) - 7*x). - Elmo R. Oliveira, Sep 10 2024

A198398 a(n) = 8^n - 8n.

Original entry on oeis.org

1, 0, 48, 488, 4064, 32728, 262096, 2097096, 16777152, 134217656, 1073741744, 8589934504, 68719476640, 549755813784, 4398046510992, 35184372088712, 281474976710528, 2251799813685112, 18014398509481840, 144115188075855720, 1152921504606846816, 9223372036854775640
Offset: 0

Views

Author

Vincenzo Librandi, Oct 26 2011

Keywords

Links

Crossrefs

Cf. A005803, A107583, A107584, A107585.

Programs

  • Magma
    [8^n-8*n: n in [0..25]];
  • Mathematica
    CoefficientList[Series[(1 - 10*x + 65*x^2)/((1 - 8*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
LinearRecurrence[{10,-17,8},{1,0,48},30] (* Harvey P. Dale, Mar 13 2023 *)
  • PARI
    a(n)=8^n-8*n \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3) for n > 2.
G.f.: (1 - 10*x + 65*x^2)/((1 - 8*x)*(1 -x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(7*x) - 8*x). - Elmo R. Oliveira, Sep 10 2024

A198399 a(n) = 9^n - 9*n.

Original entry on oeis.org

1, 0, 63, 702, 6525, 59004, 531387, 4782906, 43046649, 387420408, 3486784311, 31381059510, 282429536373, 2541865828212, 22876792454835, 205891132094514, 1853020188851697, 16677181699666416, 150094635296998959, 1350851717672991918, 12157665459056928621, 109418989131512359020
Offset: 0

Views

Author

Vincenzo Librandi, Oct 26 2011

Keywords

Links

Crossrefs

Cf. A005803, A107583, A107584, A107585.

Programs

  • Magma
    [9^n-9*n: n in [0..25]];
  • Mathematica
    CoefficientList[Series[(1 - 11*x + 82*x^2)/((1 - 9*x)*(1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
  • PARI
    a(n)=9^n-9*n \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(0)=1, a(1)=0, a(2)=63, a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
G.f.: (1 - 11*x + 82*x^2)/((1 - 9*x)*(1 - x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(8*x) - 9*x). - Elmo R. Oliveira, Sep 09 2024

A198400 a(n) = 10^n - 10*n.

Original entry on oeis.org

1, 0, 80, 970, 9960, 99950, 999940, 9999930, 99999920, 999999910, 9999999900, 99999999890, 999999999880, 9999999999870, 99999999999860, 999999999999850, 9999999999999840, 99999999999999830, 999999999999999820, 9999999999999999810, 99999999999999999800, 999999999999999999790
Offset: 0

Views

Author

Vincenzo Librandi, Oct 26 2011

Keywords

Links

Crossrefs

Cf. A005803, A107583, A107584, A107585.

Programs

  • Magma
    [10^n-10*n: n in [0..25]];
  • Mathematica
    CoefficientList[Series[(1-12*x+101*x^2)/((1-10*x)*(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)
  • PARI
    a(n)=10^n-10*n \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 2.
G.f.: (1-12*x+101*x^2)/((1-10*x)*(1-x)^2). - Vincenzo Librandi, Jul 06 2012
E.g.f.: exp(x)*(exp(9*x) - 10*x). - Elmo R. Oliveira, Aug 29 2024

A221906 a(n) = 4^n + 4*n.

Original entry on oeis.org

1, 8, 24, 76, 272, 1044, 4120, 16412, 65568, 262180, 1048616, 4194348, 16777264, 67108916, 268435512, 1073741884, 4294967360, 17179869252, 68719476808, 274877907020, 1099511627856, 4398046511188, 17592186044504, 70368744177756, 281474976710752, 1125899906842724
Offset: 0

Views

Author

Vincenzo Librandi, Mar 02 2013

Keywords

Links

Crossrefs

Cf. A107584, A370659.

Programs

  • Magma
    [4^n + 4*n: n in [0..30]];
  • Magma
    I:=[1, 8, 24]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+4*Self(n-3): n in [1..30]];
  • Mathematica
    Table[(4^n + 4 n), {n, 0, 30}] (* or *) CoefficientList[Series[(1 + 2 x - 15 x^2)/((1 - x)^2 (1 - 4 x)), {x, 0, 30}], x]

Formula

G.f.: (1 + 2*x - 15*x^2)/((1-x)^2*(1-4*x)).
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3).
E.g.f.: exp(x)*(exp(3*x) + 4*x). - Elmo R. Oliveira, Sep 10 2024

A367629 a(n) = 2^2^(n + 1) - 2^(n + 2).

Original entry on oeis.org

8, 240, 65504, 4294967232, 18446744073709551488, 340282366920938463463374607431768211200, 115792089237316195423570985008687907853269984665640564039457584007913129639424
Offset: 1

Views

Author

J Gregory Moxness, Nov 24 2023

Keywords

Comments

a(n) is the absolute value totals of the characteristic polynomial coefficients of n-qubit normalized Hadamard matrices, excluding the 2nd and next-to-last nonzero entries.
a(n) is also the total of the entries in row 2^(n+1) of Pascal's triangle (A007318), excluding the 2nd and next-to-last entries.

Links

Crossrefs

Cf. A007318, A107584.

Programs

  • Mathematica
    Table[2^2^(n+1)-2^(n+2),{n,10}] (* Harvey P. Dale, Aug 23 2024 *)
  • Python
    def A367629(n): return (1<<(m:=1<Chai Wah Wu, Nov 29 2023

Formula

a(n) = A107584(2^n).

Extensions

Previous Mathematica program replaced by Harvey P. Dale, Aug 23 2024
Showing 1-9 of 9 results.

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