A024050 -id:A024050 - cp's OEIS frontend

A024037 a(n) = 4^n - n.

1, 3, 14, 61, 252, 1019, 4090, 16377, 65528, 262135, 1048566, 4194293, 16777204, 67108851, 268435442, 1073741809, 4294967280, 17179869167, 68719476718, 274877906925, 1099511627756, 4398046511083, 17592186044394, 70368744177641, 281474976710632, 1125899906842599

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), this sequence (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Cf. A140660 (first differences).

[4^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011

I:=[1, 3, 14]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

Table[4^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 5 x^2) / ((1 - 4 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)

a(n)=4^n-n \\ Charles R Greathouse IV, Sep 24 2015

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1 - 3*x + 5*x^2)/((1 - 4*x)*(1 - x)^2).
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). (End)
E.g.f.: exp(x)*(exp(3*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024024 a(n) = 3^n - n.

Original entry on oeis.org

1, 2, 7, 24, 77, 238, 723, 2180, 6553, 19674, 59039, 177136, 531429, 1594310, 4782955, 14348892, 43046705, 129140146, 387420471, 1162261448, 3486784381, 10460353182, 31381059587, 94143178804, 282429536457, 847288609418, 2541865828303, 7625597484960, 22876792454933

Offset: 0

N. J. A. Sloane

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

Cf. A000325, A024050.

[3^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011

I:=[1, 2, 7]; [n le 3 select I[n] else 5*Self(n-1)-7*Self(n-2)+3*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Mar 06 2013

Table[3^n - n, {n, 0, 40}] (* or *) CoefficientList[Series[(1 - 3 x + 4 x^2)/((1 - 3 x)(1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 06 2013 *)

a(n)=3^n-n \\ Charles R Greathouse IV, Sep 24 2015

From Vincenzo Librandi, Mar 06 2013: (Start)
G.f.: (1-3*x+4*x^2)/((1-3*x)*(1-x)^2).
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). (End)
E.g.f.: exp(x)*(exp(2*x) - x). - Elmo R. Oliveira, Sep 06 2024

A024115 a(n) = 10^n - n.

Original entry on oeis.org

1, 9, 98, 997, 9996, 99995, 999994, 9999993, 99999992, 999999991, 9999999990, 99999999989, 999999999988, 9999999999987, 99999999999986, 999999999999985, 9999999999999984, 99999999999999983, 999999999999999982, 9999999999999999981, 99999999999999999980, 999999999999999999979

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. numbers of the form k^n-n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), this sequence (k=10), A024128 (k=11), A024141 (k=12).

[10^n-n: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

I:=[1, 9, 98]; [n le 3 select I[n] else 12*Self(n-1)-21*Self(n-2)+10*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013

Table[10^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 11 x^2) / ((1 - 10 x) (1 - x)^2),{x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{12,-21,10},{1,9,98},20] (* Harvey P. Dale, Jul 18 2020 *)

a(n)=10^n-n \\ Charles R Greathouse IV, Oct 07 2015

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1-3*x+11*x^2)/((1-10*x)*(1-x)^2).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 2. (End)
E.g.f.: exp(x)*(exp(9*x) - x). - Elmo R. Oliveira, Sep 06 2024

A024063 a(n) = 6^n - n.

Original entry on oeis.org

1, 5, 34, 213, 1292, 7771, 46650, 279929, 1679608, 10077687, 60466166, 362797045, 2176782324, 13060694003, 78364164082, 470184984561, 2821109907440, 16926659444719, 101559956668398, 609359740010477, 3656158440062956, 21936950640377835, 131621703842267114, 789730223053602793

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), this sequence (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

[6^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 03 2011

I:=[1, 5, 34]; [n le 3 select I[n] else 8*Self(n-1)-13*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

Table[6^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 7 x^2) / ((1 - 6 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)

a(n)=6^n-n \\ Charles R Greathouse IV, Oct 07 2015

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1-3*x+7*x^2)/((1-6*x)*(1-x)^2).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3). (End)
E.g.f.: exp(x)*(exp(5*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024076 a(n) = 7^n - n.

Original entry on oeis.org

1, 6, 47, 340, 2397, 16802, 117643, 823536, 5764793, 40353598, 282475239, 1977326732, 13841287189, 96889010394, 678223072835, 4747561509928, 33232930569585, 232630513987190, 1628413597910431, 11398895185373124, 79792266297611981, 558545864083283986, 3909821048582988027

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), this sequence (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Cf. A198688 (first differences).

[7^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 03 2011

I:=[1, 6, 47]; [n le 3 select I[n] else 9*Self(n-1)-15*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

A024076:=n->7^n-n; seq(A024076(n), n=0..30); # Wesley Ivan Hurt, Jan 24 2014

Table[7^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 8 x^2) / ((1 - 7 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)

a(n)=7^n-n \\ Charles R Greathouse IV, Oct 07 2015

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1-3*x+8*x^2)/((1-7*x)*(1-x)^2).
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3). (End)
E.g.f.: exp(x)*(exp(6*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024089 a(n) = 8^n - n.

Original entry on oeis.org

1, 7, 62, 509, 4092, 32763, 262138, 2097145, 16777208, 134217719, 1073741814, 8589934581, 68719476724, 549755813875, 4398046511090, 35184372088817, 281474976710640, 2251799813685231, 18014398509481966, 144115188075855853, 1152921504606846956, 9223372036854775787

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), this sequence (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Cf. A198855 (first differences).

[8^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011

I:=[1,7,62]; [n le 3 select I[n] else 10*Self(n-1)-17*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013

Table[8^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 9 x^2) / ((1 - 8 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{10,-17,8},{1,7,62},30] (* Harvey P. Dale, Sep 28 2017 *)

a(n)=8^n-n \\ Charles R Greathouse IV, Oct 07 2015

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1-3*x+9*x^2)/((1-8*x)*(1-x)^2).
a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3). (End)
E.g.f.: exp(x)*(exp(7*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024102 a(n) = 9^n - n.

Original entry on oeis.org

1, 8, 79, 726, 6557, 59044, 531435, 4782962, 43046713, 387420480, 3486784391, 31381059598, 282429536469, 2541865828316, 22876792454947, 205891132094634, 1853020188851825, 16677181699666552, 150094635296999103, 1350851717672992070, 12157665459056928781, 109418989131512359188

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), this sequence (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Cf. A198966 (first differences).

[9^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 06 2011

I:=[1, 8, 79]; [n le 3 select I[n] else 11*Self(n-1)-19*Self(n-2)+9*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013

Table[9^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 10 x^2) / ((1 - 9 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{11,-19,9},{1,8,79},30] (* Harvey P. Dale, Dec 25 2024 *)

a(n)=9^n-n \\ Charles R Greathouse IV, Oct 07 2015

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1-3*x+10*x^2)/((1-9*x)(1-x)^2).
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). (End)
E.g.f.: exp(x)*(exp(8*x) - x). - Elmo R. Oliveira, Sep 09 2024

A024128 a(n) = 11^n - n.

Original entry on oeis.org

1, 10, 119, 1328, 14637, 161046, 1771555, 19487164, 214358873, 2357947682, 25937424591, 285311670600, 3138428376709, 34522712143918, 379749833583227, 4177248169415636, 45949729863572145, 505447028499293754, 5559917313492231463, 61159090448414546272, 672749994932560009181

Offset: 0

N. J. A. Sloane

Smallest prime of this form is a(18) = 5559917313492231463. - Bruno Berselli, Jun 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (13,-23,11).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), this sequence (k=11), A024141 (k=12).

Cf. A199030 (first differences).

[11^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011

I:=[1, 10, 119]; [n le 3 select I[n] else 13*Self(n-1)-23*Self(n-2)+11*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013

Table[11^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 12 x^2) / ((1 - 11 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{13,-23,11},{1,10,119},20] (* Harvey P. Dale, Aug 02 2017 *)

a(n)=11^n-n \\ Charles R Greathouse IV, Jul 01 2011

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1-3*x+12*x^2)/((1-11*x) (1-x)^2).
a(n) = 13*a(n-1) - 23*a(n-2) + 11*a(n-3). (End)
E.g.f.: exp(x)*(exp(10*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024141 a(n) = 12^n - n.

Original entry on oeis.org

1, 11, 142, 1725, 20732, 248827, 2985978, 35831801, 429981688, 5159780343, 61917364214, 743008370677, 8916100448244, 106993205379059, 1283918464548850, 15407021574586353, 184884258895036400, 2218611106740436975, 26623333280885243886, 319479999370622926829

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (14,-25,12).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), this sequence (k=12).

[12^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011

I:=[1, 11, 142]; [n le 3 select I[n] else 14*Self(n-1)-25*Self(n-2)+12*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013

Table[12^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 13 x^2) / ((1 - 12 x) (1 - x)^2),{x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{14,-25,12},{1,11,142},20] (* Harvey P. Dale, Nov 03 2024 *)

a(n)=12^n-n \\ Charles R Greathouse IV, Jul 01 2011

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1 - 3*x + 13*x^2)/((1-12*x)*(1-x)^2).
a(n) = 14*a(n-1) - 25*a(n-2) + 12*a(n-3). (End)
E.g.f.: exp(x)*(exp(11*x) - x). - Elmo R. Oliveira, Sep 10 2024

A057897 Numbers which can be written as m^k-k, with m, k > 1.

Original entry on oeis.org

2, 5, 7, 12, 14, 23, 24, 27, 34, 47, 58, 61, 62, 77, 79, 98, 119, 121, 122, 142, 167, 194, 213, 223, 238, 248, 252, 254, 287, 322, 340, 359, 398, 439, 482, 503, 509, 527, 574, 621, 623, 674, 723, 726, 727, 782, 839, 898, 959, 997, 1014, 1019, 1022, 1087, 1154

Offset: 1

Henry Bottomley, Sep 26 2000

nonn

It may be that positive integers can be written as m^k-k (with m and k > 1) in at most one way [checked up to 10000].

All numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x-y for x > y > 1 and b > a > 1, which appears to have no solutions. - T. D. Noe, Oct 06 2004

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000325, A008865, A024024, A024037, A024050, A057895, A057898, A057899.

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).

nLim=1000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst] (* T. D. Noe, Oct 06 2004 *)

ok(n)={my(e=2); while(2^e <= n+e, if(ispower(n+e, e), return(1)); e++); 0} \\ Andrew Howroyd, Oct 20 2020

upto(lim)={my(p=logint(lim,2)); while(logint(lim+p+1,2)>p, p++); Vec(Set(concat(vector(p-1, e, e++; vector(sqrtnint(lim+e,e)-1, m, (m+1)^e-e)))))} \\ Andrew Howroyd, Oct 20 2020

cp's OEIS Frontend

A024037 a(n) = 4^n - n.

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Magma

Magma

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A024024 a(n) = 3^n - n.

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Magma

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A024115 a(n) = 10^n - n.

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Magma

Magma

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A024063 a(n) = 6^n - n.

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Magma

Magma

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A024076 a(n) = 7^n - n.

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Magma

Magma

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A024089 a(n) = 8^n - n.

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Magma

Magma

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A024102 a(n) = 9^n - n.

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