A107583 -id:A107583 - cp's OEIS frontend

A107584 a(n) = 4^n - 4*n.

1, 0, 8, 52, 240, 1004, 4072, 16356, 65504, 262108, 1048536, 4194260, 16777168, 67108812, 268435400, 1073741764, 4294967232, 17179869116, 68719476664, 274877906868, 1099511627696, 4398046511020, 17592186044328, 70368744177572, 281474976710560, 1125899906842524

Offset: 0

Zak Seidov, May 16 2005

Numbers a(n) = k such that number m with n 4's and k 1's has digit product = digit sum = 4^n.

			Corresponding numbers m are 1, 4, 1111111144, ...

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A107583, A107585.

[(4^n - 4*n): n in [0..25]]; // Vincenzo Librandi, Dec 16 2010

Table[4^m-4*m, {m, 0, 20}]
LinearRecurrence[{6,-9,4},{1,0,8},30] (* Harvey P. Dale, Oct 21 2011 *)

a(n)=4^n-4*n \\ Charles R Greathouse IV, Sep 08 2012

def A107584(n): return (1<<(n<<1))-(n<<2) # Chai Wah Wu, Nov 29 2023

From Harvey P. Dale, Oct 21 2011: (Start)
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=0, and a(2)=8.
G.f.: (-17*x^2+6*x-1)/((x-1)^2*(4*x-1)). (End)
E.g.f.: exp(x)*(exp(3*x) - 4*x). - Elmo R. Oliveira, Sep 10 2024

More terms from Vincenzo Librandi, Dec 16 2010

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

Zak Seidov, May 16 2005

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A107583, A107584.

[(5^n - 5*n): n in [0..25]]; // Vincenzo Librandi, Dec 16 2010

Table[5^m-5*m, {m, 0, 10}]
LinearRecurrence[{7,-11,5},{1,0,15},30] (* Harvey P. Dale, Oct 21 2015 *)

a(n)=5^n-5*n \\ Charles R Greathouse IV, Oct 26 2011

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

Corrected by Charles R Greathouse IV, Sep 08 2012

A221905 a(n) = 3^n + 3*n.

Original entry on oeis.org

1, 6, 15, 36, 93, 258, 747, 2208, 6585, 19710, 59079, 177180, 531477, 1594362, 4783011, 14348952, 43046769, 129140214, 387420543, 1162261524, 3486784461, 10460353266, 31381059675, 94143178896, 282429536553, 847288609518, 2541865828407, 7625597485068, 22876792455045

Offset: 0

Vincenzo Librandi, Mar 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A107583, A176805, A370658.

Magma
```
[3^n+3*n: n in [0..30]];
```

I:=[1, 6, 15]; [n le 3 select I[n] else 5*Self(n-1)-7*Self(n-2)+3*Self(n-3): n in [1..30]];

Table[(3^n + 3 n), {n, 0, 30}] (* or *) CoefficientList[Series[(1 + x - 8 x^2)/((1 - x)^2 (1 -3 x)), {x, 0, 30}], x]

G.f.: (1 + x - 8*x^2)/((1-x)^2*(1-3*x)).
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
a(n) = A176805(n) - 1.
E.g.f.: exp(x)*(exp(2*x) + 3*x). - Elmo R. Oliveira, Sep 10 2024

A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 05 2017

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).
Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.
Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).
The partial sum of this sequence is A184985.
a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.
a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.
Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

			For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
V. Manturov, Knot Theory, CRC Press, 2004.

I. Altintas, An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput. 194 (2007) 168-178.
J. A. Baldwin and A. S. Levine, A combinatorial spanning tree model for knot Floer homology, Advances in Mathematics, Vol. 231 (2012), 1886-1939.
A. Banerjee, Knot theory [Foil knot family].
D. Denton and P. Doyle, Shadow movies not arising from knots, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.

CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* Wesley Ivan Hurt, Nov 05 2017 *)
f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PadRight[{0,1,2},120,{1}] (* Harvey P. Dale, Feb 20 2023 *)

makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);

PARI
```
a(n) = if(n>2, 1, n);
```

a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.
a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).
a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).
a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.
A005803(a(n)) = A005096(a(n)) = A000007(n).
A107583(a(n)) = A103775(n+5).
a(n+1) = 2^A185012(n+1), with a(0) = 0.
a(n) = A163985(n) mod A004278(n+1).
a(n) = A157928(n) + A171386(n+1).
a(n) = A063524(n) + A157928(n) + A185012(n).
a(n) = A010701(n) - A141044(n) - A179184(n).
G.f.: (x + x^2 - x^3)/(1 - x).
E.g.f.: (2*exp(x) - 2 + x^2)/2.

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

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[6^n-6*n: n in [0..25]];
```

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 *)

a(n)=6^n-6*n \\ Charles R Greathouse IV, Jul 06 2017

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

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[7^n-7*n: n in [0..25]];
```

CoefficientList[Series[(1 - 9*x + 50*x^2)/((1 - 7*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)

a(n)=7^n-7*n \\ Charles R Greathouse IV, Jul 06 2017

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

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[8^n-8*n: n in [0..25]];
```

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 *)

a(n)=8^n-8*n \\ Charles R Greathouse IV, Jul 06 2017

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

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[9^n-9*n: n in [0..25]];
```

CoefficientList[Series[(1 - 11*x + 82*x^2)/((1 - 9*x)*(1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)

a(n)=9^n-9*n \\ Charles R Greathouse IV, Jul 06 2017

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

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[10^n-10*n: n in [0..25]];
```

CoefficientList[Series[(1-12*x+101*x^2)/((1-10*x)*(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)

a(n)=10^n-10*n \\ Charles R Greathouse IV, Jul 06 2017

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

A165624 a(n) = 3^(3^n)/(3^n)^3.

Original entry on oeis.org

3, 1, 27, 387420489, 834385168331080533771857328695283, 6076396096647706909168138770838836135530328017648434830996201971201776350890241322455818405320466786549738961

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 22 2009

nonn

			a(0) = 3^1/1^3 = 3. a(1) = 3^3/3^3 = 1. a(2) = 3^9/9^3 = 27.

Cf. A007758, A165623

Table[3^(3^n)/(3^n)^3,{n,0,6}] (* Harvey P. Dale, Sep 17 2019 *)

a(n) = 3^A107583(n). [R. J. Mathar, Oct 07 2009]

Definition simplified by R. J. Mathar, Oct 07 2009

cp's OEIS Frontend

A107584 a(n) = 4^n - 4*n.

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

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A221905 a(n) = 3^n + 3*n.

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A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

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

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Magma

Mathematica

PARI

Formula

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

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Magma

Mathematica

PARI

Formula

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

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