A058794 -id:A058794 - cp's OEIS frontend

A007754 Array (a frieze pattern) defined by a(n,k) = (a(n-1,k)*a(n-1,k+1) - 1) / a(n-2,k+1), read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 5, 2, 1, 4, 11, 18, 7, 1, 5, 19, 52, 85, 33, 1, 6, 29, 110, 301, 492, 191, 1, 7, 41, 198, 751, 2055, 3359, 1304, 1, 8, 55, 322, 1555, 5898, 16139, 26380, 10241, 1, 9, 71, 488, 2857, 13797, 52331, 143196, 234061, 90865, 1, 10, 89, 702

Offset: 0

N. J. A. Sloane, Nov 28 2000

Let u be a sequence with u(0)=p, u(1)=q, and u(i)^(i+k) = u(i-1)*u(i+1). Then u(n)= q^a(n-1,k)/p^a(n-2,k+1). - Example for k=1, u(5)=q^7/p^18 and for k=2, u(5)=q^85/p^52. - Olivier Gérard, Sep 19 2016

			Array begins:
  1   1   1   1   1    1    1   1 ...
    1   2   3   4   5    6    7 ...
      1   5  11  19   29   41 ...
        2  18  52  110  198 ...
          7  85  301  751 ...

Email from James Propp, Nov 28 2000.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Row 0-3: A000012, A000027(n+1), A028387, A058794-A058796. Columns 0-2: A058797-A058799.

Main diagonal gives A099933.

a(n, k) = (n+k)*a(n-1, k)-a(n-2, k) with a(0, k)=1 and a(-1, k)=0. - Henry Bottomley, Feb 28 2001
a(n, k) = Pi*(BesselJ(n+k+1, 2)*BesselY(k, 2) - BesselY(n+k+1, 2)*BesselJ(k, 2)). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 21 2005
Column asymptotics (i.e. for fixed k and n -> infinity): a(n, k) ~ BesselJ(k, 2)*(n+k)!. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 21 2005

More terms from Christian G. Bower, Dec 02 2000

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

Original entry on oeis.org

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

A121670 a(n) = n^3 - 3*n.

Original entry on oeis.org

0, -2, 2, 18, 52, 110, 198, 322, 488, 702, 970, 1298, 1692, 2158, 2702, 3330, 4048, 4862, 5778, 6802, 7940, 9198, 10582, 12098, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 32672, 35838, 39202, 42770, 46548, 50542, 54758, 59202, 63880, 68798, 73962

Offset: 0

Gary W. Adamson, Aug 14 2006

Previous name was: Real part of (n + i)^3, companion to A080663.
Reversing the order of terms in (n + i)^3 to (1 + ni)^3 generates the terms of A080663. E.g, A080663(4) = 47 since (1 + 4i)^3 = (-47 - 52i). Or, (n + i)^3 = (a(n) + A080663(a)i) and (1 + ni)^3 = (-A080663(n) - a(n)i).
Also, numbers n such that the polynomial x^6 - n*x^3 + 1 is reducible. - Ralf Stephan, Oct 24 2013

			a(4) = 52 since (4 + i)^3 = (52 + 47i); where 47 = A080663(4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A058794, A080663.

CoefficientList[Series[-2 x (x^2 - 5 x + 1)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 11 2014 *)
Table[n^3-3n,{n,0,60}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,-2,2,18},60] (* Harvey P. Dale, Nov 30 2021 *)

Vec(-2*x*(x^2-5*x+1)/(x-1)^4 + O(x^100)); \\ Colin Barker, Oct 16 2013

a(n) = Re( (n + i)^3 ).
a(n) = n^3-3*n. a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: -2*x*(x^2-5*x+1) / (x-1)^4. - Colin Barker, Oct 16 2013
a(n)^2 = A028872(n)^3 + 3*A028872(n)^2 for n>1. - Bruno Berselli, May 03 2018
a(n) = A058794(n-2) for n>1. - Altug Alkan, May 03 2018

Terms corrected, new name, and more terms from Colin Barker, Oct 16 2013

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)

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 *)

cp's OEIS Frontend

A007754 Array (a frieze pattern) defined by a(n,k) = (a(n-1,k)*a(n-1,k+1) - 1) / a(n-2,k+1), read by antidiagonals.

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

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

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Magma

Mathematica

PARI

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

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Magma

Mathematica

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A121670 a(n) = n^3 - 3*n.

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Mathematica

PARI

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A024082 7^n-n^7.

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Magma

Mathematica

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

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Magma

Mathematica