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Search: a024037 -id:a024037
Displaying 1-10 of 14 results found.
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A024050
a(n) = 5^n - n.
+10
13
1, 4, 23, 122, 621, 3120, 15619, 78118, 390617, 1953116, 9765615, 48828114, 244140613, 1220703112, 6103515611, 30517578110, 152587890609, 762939453108, 3814697265607, 19073486328106, 95367431640605, 476837158203104, 2384185791015603, 11920928955078102, 59604644775390601
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).
FORMULA
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
G.f.: (1 - 3*x + 6*x^2)/((1-5*x)*(1-x)^2). - Vincenzo Librandi, Jun 16 2013
E.g.f.: exp(x)*(exp(4*x) - x). - Elmo R. Oliveira, Sep 10 2024
MAPLE
g:=1/(1-5*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-n, n=0..31); # Zerinvary Lajos, Jan 09 2009
MATHEMATICA
Table[5^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 6 x^2) / ((1 - 5 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)
LinearRecurrence[{7, -11, 5}, {1, 4, 23}, 30] (* Harvey P. Dale, Mar 03 2022 *)
PROG
(Magma) [5^n-n: n in [0..35]]; // Vincenzo Librandi, Jun 12 2011
CROSSREFS
Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), this sequence (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A024115
a(n) = 10^n - n.
+10
11
1, 9, 98, 997, 9996, 99995, 999994, 9999993, 99999992, 999999991, 9999999990, 99999999989, 999999999988, 9999999999987, 99999999999986, 999999999999985, 9999999999999984, 99999999999999983, 999999999999999982, 9999999999999999981, 99999999999999999980, 999999999999999999979
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).
FORMULA
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
MATHEMATICA
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 *)
PROG
(Magma) [10^n-n: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011
(Magma) 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
(PARI) a(n)=10^n-n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A024063
a(n) = 6^n - n.
+10
10
1, 5, 34, 213, 1292, 7771, 46650, 279929, 1679608, 10077687, 60466166, 362797045, 2176782324, 13060694003, 78364164082, 470184984561, 2821109907440, 16926659444719, 101559956668398, 609359740010477, 3656158440062956, 21936950640377835, 131621703842267114, 789730223053602793
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).
FORMULA
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
MATHEMATICA
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 *)
PROG
(Magma) [6^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 03 2011
(Magma) 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
(PARI) a(n)=6^n-n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A024076
a(n) = 7^n - n.
+10
10
1, 6, 47, 340, 2397, 16802, 117643, 823536, 5764793, 40353598, 282475239, 1977326732, 13841287189, 96889010394, 678223072835, 4747561509928, 33232930569585, 232630513987190, 1628413597910431, 11398895185373124, 79792266297611981, 558545864083283986, 3909821048582988027
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).
FORMULA
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
MAPLE
A024076:=n->7^n-n; seq(A024076(n), n=0..30); # Wesley Ivan Hurt, Jan 24 2014
MATHEMATICA
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 *)
PROG
(Magma) [7^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 03 2011
(Magma) 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
(PARI) a(n)=7^n-n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A024089
a(n) = 8^n - n.
+10
10
1, 7, 62, 509, 4092, 32763, 262138, 2097145, 16777208, 134217719, 1073741814, 8589934581, 68719476724, 549755813875, 4398046511090, 35184372088817, 281474976710640, 2251799813685231, 18014398509481966, 144115188075855853, 1152921504606846956, 9223372036854775787
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).
FORMULA
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
MATHEMATICA
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 *)
PROG
(Magma) [8^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011
(Magma) 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
(PARI) a(n)=8^n-n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A024102
a(n) = 9^n - n.
+10
9
1, 8, 79, 726, 6557, 59044, 531435, 4782962, 43046713, 387420480, 3486784391, 31381059598, 282429536469, 2541865828316, 22876792454947, 205891132094634, 1853020188851825, 16677181699666552, 150094635296999103, 1350851717672992070, 12157665459056928781, 109418989131512359188
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).
FORMULA
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
MATHEMATICA
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 *)
PROG
(Magma) [9^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 06 2011
(Magma) 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
(PARI) a(n)=9^n-n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A024128
a(n) = 11^n - n.
+10
9
1, 10, 119, 1328, 14637, 161046, 1771555, 19487164, 214358873, 2357947682, 25937424591, 285311670600, 3138428376709, 34522712143918, 379749833583227, 4177248169415636, 45949729863572145, 505447028499293754, 5559917313492231463, 61159090448414546272, 672749994932560009181
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Smallest prime of this form is a(18) = 5559917313492231463. - Bruno Berselli, Jun 17 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (13,-23,11).
FORMULA
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
MATHEMATICA
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 *)
PROG
(Magma) [11^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011
(PARI) a(n)=11^n-n \\ Charles R Greathouse IV, Jul 01 2011
(Magma) 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
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A024141
a(n) = 12^n - n.
+10
9
1, 11, 142, 1725, 20732, 248827, 2985978, 35831801, 429981688, 5159780343, 61917364214, 743008370677, 8916100448244, 106993205379059, 1283918464548850, 15407021574586353, 184884258895036400, 2218611106740436975, 26623333280885243886, 319479999370622926829
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (14,-25,12).
FORMULA
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
MATHEMATICA
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 *)
PROG
(Magma) [12^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011
(PARI) a(n)=12^n-n \\ Charles R Greathouse IV, Jul 01 2011
(Magma) 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
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A057897
Numbers which can be written as m^k-k, with m, k > 1.
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..10000
MATHEMATICA
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 *)
PROG
(PARI) ok(n)={my(e=2); while(2^e <= n+e, if(ispower(n+e, e), return(1)); e++); 0} \\ Andrew Howroyd, Oct 20 2020
(PARI) 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
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
Henry Bottomley, Sep 26 2000
STATUS
approved
A224451
Primes of the form 4^n - n.
+10
4
3, 61, 1019, 1180591620717411303389, 1569275433846670190958947355801916604025588861116008628129
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Corresponding n are given in A057908.
The next term has 797 digits. - Harvey P. Dale, Mar 16 2015
LINKS
Table of n, a(n) for n=1..5.
MATHEMATICA
Select[Table[4^n-n, {n, 200}], PrimeQ] (* Harvey P. Dale, Mar 16 2015 *)
CROSSREFS
Cf. A057908 (n such that 4^n - n is prime).
Cf. A024037, A129963, A224420.
KEYWORD
nonn
AUTHOR
Alex Ratushnyak, Apr 06 2013
STATUS
approved
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