A024026 -id:A024026 - cp's OEIS frontend

A024040 a(n) = 4^n - n^4.

1, 3, 0, -17, 0, 399, 2800, 13983, 61440, 255583, 1038576, 4179663, 16756480, 67080303, 268397040, 1073691199, 4294901760, 17179785663, 68719371760, 274877776623, 1099511467776, 4398046316623, 17592185810160, 70368743897823, 281474976378880, 1125899906451999

Offset: 0

N. J. A. Sloane

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-30,50,-45,21,-4).

Cf. A024012, A024026, A058794. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[4^n-n^4: n in [0..30]]; // Vincenzo Librandi, May 14 2011

seq(4^n - n^4, n=0..50); # Robert Israel, Dec 29 2014

lst={}; Do[AppendTo[lst,4^n-n^4],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[4^n-n^4,{n,0,40}] (* Harvey P. Dale, Nov 10 2019 *)

G.f.: (1-6*x+3*x^2+23*x^3+48*x^4+3*x^5)/((1-4*x)*(1-x)^5).

E.g.f.: exp(4*x)-(x^4+6*x^3+7*x^2+x)*exp(x). - Robert Israel, Dec 29 2014

A024054 a(n) = 5^n - n^5.

Original entry on oeis.org

1, 4, -7, -118, -399, 0, 7849, 61318, 357857, 1894076, 9665625, 48667074, 243891793, 1220331832, 6102977801, 30516818750, 152586842049, 762938033268, 3814695376057, 19073483852026, 95367428440625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11, -45, 95, -115, 81, -31, 5).

Cf. A024012, A024026, A058794, A024040. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[5^n-n^5: n in [0..30]]; // Vincenzo Librandi, May 14 2011

lst={}; Do[AppendTo[lst,5^n-n^5],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[5^n-n^5,{n,0,20}] (* or *) LinearRecurrence[{11,-45,95,-115,81,-31,5},{1,4,-7,-118,-399,0,7849},30] (* Harvey P. Dale, Oct 15 2014 *)

a(n)=5^n-n^5 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-6*x^6 - 123*x^5 - 319*x^4 - 44*x^3 + 6*x^2 + 7*x - 1)/((x - 1)^6*(5*x - 1)). - Harvey P. Dale, Oct 15 2014

a(0)=1, a(1)=4, a(2)=-7, a(3)=-118, a(4)=-399, a(5)=0, a(6)=7849, a(n) = 11*a(n-1) - 45*a(n-2) + 95*a(n-3) - 115*a(n-4) + 81*a(n-5) - 31*a(n-6) + 5*a(n-7). - Harvey P. Dale, Oct 15 2014

A024068 a(n) = 6^n - n^6.

Original entry on oeis.org

1, 5, -28, -513, -2800, -7849, 0, 162287, 1417472, 9546255, 59466176, 361025495, 2173796352, 13055867207, 78356634560, 470173593951, 2821093130240, 16926635307167, 101559922656192, 609359692964615, 3656158376062976, 21936950554611735, 131621703728887232, 789730222905566927

Offset: 0

N. J. A. Sloane

6^n in the formula can be removed (for example) with the following Maple code: "with(gfun): rec1:={u1(0)=1,u1(n+1)=6*u1(n)}: rec2:={u2(n)=n^6}: poltorec(u1(n)-u2(n),[rec1,rec2],u1(n),u2(n)],a(n));". This yields a polynomial recurrence: {a(n+1)-5*n^6+6*n^5+15*n^4+20*n^3+15*n^2-6*a(n)+6*n+1, a(0) = 1} that can further be transformed into a linear recurrence with constant coefficients. - Georg Fischer, Feb 23 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (13,-63,161,-245,231,-133,43,-6).

Cf. A024012, A024026, A058794, A024040, A024054. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[6^n-n^6: n in [0..30]]; // Vincenzo Librandi, May 14 2011

Table[6^n-n^6,{n,0,20}] (* Harvey P. Dale, Jan 30 2019 *)

From Chai Wah Wu, Jan 26 2020: (Start)
a(n) = 13*a(n-1) - 63*a(n-2) + 161*a(n-3) - 245*a(n-4) + 231*a(n-5) - 133*a(n-6) + 43*a(n-7) - 6*a(n-8) for n > 7.
G.f.: (5*x^7 + 348*x^6 + 1734*x^5 + 1545*x^4 + 5*x^3 - 30*x^2 - 8*x + 1)/((x - 1)^7*(6*x - 1)). (End)

More terms from Georg Fischer, Feb 23 2021

A024082 7^n-n^7.

Original entry on oeis.org

1, 6, -79, -1844, -13983, -61318, -162287, 0, 3667649, 35570638, 272475249, 1957839572, 13805455393, 96826261890, 678117659345, 4747390650568, 33232662134145, 232630103648534, 1628412985690417, 11398894291501404, 79792265017612001, 558545862282195466

Offset: 0

N. J. A. Sloane

sign

a(20)=79792265017612001 and a(24)=191581231375979942977 are primes, thus terms of A123206. - M. F. Hasler, Aug 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A024012, A024026, A058794, A024040, A024054, A024068.

[7^n-n^7: n in [0..25]]; // Vincenzo Librandi, May 16 2011

Table[7^n - n^7, {n, 0, 25}] (* Stefan Steinerberger, Apr 08 2006 *)

G.f.: (1-9*x-85*x^2-407*x^3+5991*x^4+15665*x^5+8245*x^6+831*x^7+8*x^8)/((1-7*x)*(1-x)^8). - Bruno Berselli, May 16 2011

More terms from Stefan Steinerberger, Apr 08 2006

A024096 a(n) = 8^n - n^8.

Original entry on oeis.org

1, 7, -192, -6049, -61440, -357857, -1417472, -3667649, 0, 91171007, 973741824, 8375575711, 68289495040, 548940083167, 4396570722048, 35181809198207, 281470681743360, 2251792837927807, 18014387489521408, 144115171092292831

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (17, -108, 372, -798, 1134, -1092, 708, -297, 73, -8).

Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[8^n-n^8: n in [0..25]]; // Vincenzo Librandi, May 16 2011

Table[8^n - n^8, {n, 0, 20}] (* or *) LinearRecurrence[ {17, -108, 372, -798, 1134, -1092, 708, -297, 73, -8}, {1, 7, -192, -6049, -61440, -357857, -1417472, -3667649, 0, 91171007}, 20] (* Harvey P. Dale, Oct 10 2013 *)

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

G.f.: (1 - 10*x - 203*x^2 - 2401*x^3 + 18851*x^4 + 109207*x^5 + 120743*x^6 + 34061*x^7 + 1984*x^8 + 7*x^9) / ((1-8*x)*(1-x)^9). - Bruno Berselli, May 16 2011

a(0)=1, a(1)=7, a(2)=-192, a(3)=-6049, a(4)=-61440, a(5)=-357857, a(6)=-1417472, a(7)=-3667649, a(8)=0, a(9)=91171007; for n>9, a(n) = 17*a(n-1) - 108*a(n-2) + 372*a(n-3) - 798*a(n-4) + 1134*a(n-5) - 1092*a(n-6) + 708*a(n-7) - 297*a(n-8) + 73*a(n-9) - 8*a(n-10). - Harvey P. Dale, Oct 10 2013

A024110 a(n) = 9^n - n^9.

Original entry on oeis.org

1, 8, -431, -18954, -255583, -1894076, -9546255, -35570638, -91171007, 0, 2486784401, 29023111918, 277269756129, 2531261328956, 22856131408177, 205852688735274, 1852951469375105, 16677063111790072, 150094436937708753

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082, A024096. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[9^n-n^9: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

lst={}; Do[AppendTo[lst,9^n-n^9],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[9^n-n^9,{n,0,20}] (* Harvey P. Dale, Jul 27 2022 *)

From Chai Wah Wu, Jan 26 2020: (Start)
a(n) = 19*a(n-1) - 135*a(n-2) + 525*a(n-3) - 1290*a(n-4) + 2142*a(n-5) - 2478*a(n-6) + 2010*a(n-7) - 1125*a(n-8) + 415*a(n-9) - 91*a(n-10) + 9*a(n-11) for n > 10.
G.f.: (-10*x^10 - 4507*x^9 - 131015*x^8 - 779378*x^7 - 1317686*x^6 - 637664*x^5 - 43448*x^4 + 10210*x^3 + 448*x^2 + 11*x - 1)/((x - 1)^10*(9*x - 1)). (End)

A055651 Table T(m,k)=m^k-k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

Original entry on oeis.org

0, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 2, 0, -2, -1, 1, 3, 1, -1, -3, -1, 1, 4, 0, 0, 0, -4, -1, 1, 5, -7, -17, 17, 7, -5, -1, 1, 6, -28, -118, 0, 118, 28, -6, -1, 1, 7, -79, -513, -399, 399, 513, 79, -7, -1, 1, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, -1, 1, 9, -431

Offset: 0

Henry Bottomley, Jun 07 2000

Rows A000012 (offset), A023443, A024012, A024026, A024040 and diagonals A000004, A007925, A046065, A055652.

Title corrected by Sean A. Irvine, Mar 30 2022

A024124 a(n) = 10^n - n^10.

Original entry on oeis.org

1, 9, -924, -58049, -1038576, -9665625, -59466176, -272475249, -973741824, -2486784401, 0, 74062575399, 938082635776, 9862141508151, 99710745345024, 999423349609375, 9998900488372224, 99997984006099551, 999996429532773376

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082, A024096, A024110. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

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

lst={}; Do[AppendTo[lst,10^n-n^10],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[10^n-n^10,{n,0,20}] (* Harvey P. Dale, Apr 22 2018 *)

From Chai Wah Wu, Jan 26 2020: (Start)
a(n) = 21*a(n-1) - 165*a(n-2) + 715*a(n-3) - 1980*a(n-4) + 3762*a(n-5) - 5082*a(n-6) + 4950*a(n-7) - 3465*a(n-8) + 1705*a(n-9) - 561*a(n-10) + 111*a(n-11) - 10*a(n-12) for n > 11.
G.f.: (9*x^11 + 10140*x^10 + 477332*x^9 + 4504245*x^8 + 12648018*x^7 + 11793648*x^6 + 3241104*x^5 + 23538*x^4 - 37875*x^3 - 948*x^2 - 12*x + 1)/((x - 1)^11*(10*x - 1)). (End)

A024152 a(n) = 12^n - n^12.

Original entry on oeis.org

1, 11, -3952, -529713, -16756480, -243891793, -2173796352, -13805455393, -68289495040, -277269756129, -938082635776, -2395420006033, 0, 83695120256591, 1227224552173568, 15277275236695743, 184602783918325760

Offset: 0

N. J. A. Sloane

Conjecture: satisfies a linear recurrence having signature (25, -234, 1222, -4147, 9867, -17160, 22308, -21879, 16159, -8866, 3510, -949, 157, -12). - Harvey P. Dale, Jan 27 2019

The conjecture above is correct. From the general formula for {a(n)} we can see that the roots for the characteristic polynomial are one 12 and thirteen 1's, so the characteristic polynomial is (x - 12)*(x - 1)^13 = x^14 - 25*x^13 + 234*x^12 - ... + 12, with corresponding recurrence coefficients 25, -234, ..., -12. - Jianing Song, Jan 28 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (25,-234,1222,-4147,9867,-17160,22308,-21879,16159,-8866,3510,-949,157,-12).

Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082, A024096, A024110, A024124, A024138. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[12^n-n^12: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

Table[12^n-n^12,{n,0,30}] (* Harvey P. Dale, Jan 27 2019 *)

A024013 2^n-n^3.

Original entry on oeis.org

1, 1, -4, -19, -48, -93, -152, -215, -256, -217, 24, 717, 2368, 5995, 13640, 29393, 61440, 126159, 256312, 517429, 1040576, 2087891, 4183656, 8376441, 16763392, 33538807, 67091288, 134198045, 268413504, 536846523, 1073714824, 2147453857, 4294934528

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..240
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Cf. sequences of the form k^n-n^3: this sequence (k=2), A024026 (k=3), A024039 (k=4), A024052 (k=5), A024065 (k=6), A024078 (k=7), A024091 (k=8), A024104 (k=9), A024117 (k=10), A024130 (k=11), A024143 (k=12).

[2^n-n^3: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011

I:=[1,1,-4,-19,-48]; [n le 5 select I[n] else 6*Self(n-1)-14*Self(n-2)+16*Self(n-3)-9*Self(n-4)+2*Self(n-5): n in [1..35]]; // Vincenzo Librandi, Oct 06 2014

A024013:=n->2^n-n^3: seq(A024013(n), n=0..40); # Wesley Ivan Hurt, Oct 21 2014

Table[2^n-n^3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

G.f.: (-1-3*x^4-3*x^3-4*x^2+5*x)/((-1+2*x)*(x-1)^4). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009]

a(n) = 6*a(n-1)-14*a(n-2)+16*a(n-3)-9*a(n-4)+2*a(n-5) for n>4. - Vincenzo Librandi, Oct 06 2014

cp's OEIS Frontend

A024040 a(n) = 4^n - n^4.

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A024054 a(n) = 5^n - n^5.

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A024068 a(n) = 6^n - n^6.

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A024082 7^n-n^7.

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A024096 a(n) = 8^n - n^8.

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Magma

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A024110 a(n) = 9^n - n^9.

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Magma

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A055651 Table T(m,k)=m^k-k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

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A024124 a(n) = 10^n - n^10.

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