A109588 -id:A109588 - cp's OEIS frontend

A110001 n followed by n^2 followed by n^3 followed by n^4.

1, 1, 1, 1, 2, 4, 8, 16, 3, 9, 27, 81, 4, 16, 64, 256, 5, 25, 125, 625, 6, 36, 216, 1296, 7, 49, 343, 2401, 8, 64, 512, 4096, 9, 81, 729, 6561, 10, 100, 1000, 10000, 11, 121, 1331, 14641, 12, 144, 1728, 20736, 13, 169, 2197, 28561, 14, 196, 2744, 38416, 15, 225

Offset: 1

Mohammad K. Azarian, Sep 02 2005

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Cf. A000463, A109588, A109594.

seq(seq(n^k, k=1..4), n=1..15); # Zerinvary Lajos, Jun 29 2007

Table[(3/8 + n/4 - (1/4) Cos[(Pi n)/2] - (1/8) Cos[Pi n] + (1/4) Sin[(Pi n)/2])^(Mod[n + 3, 4] + 1), {n, 1, 58}] (* Ilya Gutkovskiy, Dec 02 2015 *)

Vec(x*(1+x+x^2+x^3-3*x^4-x^5+3*x^6+11*x^7+3*x^8-x^9-3*x^10+11*x^11-x^12+x^13-x^14+x^15) / ((1-x)^5*(1+x)^5*(1+x^2)^5) + O(x^100)) \\ Colin Barker, Dec 02 2015

a(n) = (3/8 + n/4 - (1/4)*cos((Pi*n)/2) - (1/8)*cos(Pi*n) + (1/4)*sin((Pi*n)/2))^(((n + 3) mod 4) + 1). - Ilya Gutkovskiy, Dec 02 2015
From Colin Barker, Dec 02 2015: (Start)
a(n) = 5*a(n-4)-10*a(n-8)+10*a(n-12)-5*a(n-16)+a(n-20) for n>20.
G.f.: x*(1+x+x^2+x^3-3*x^4-x^5+3*x^6+11*x^7+3*x^8-x^9-3*x^10+11*x^11-x^12+x^13-x^14+x^15) / ((1-x)^5*(1+x)^5*(1+x^2)^5).
(End)

A110009 n followed by n^4 followed by n^2 followed by n^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 16, 4, 8, 3, 81, 9, 27, 4, 256, 16, 64, 5, 625, 25, 125, 6, 1296, 36, 216, 7, 2401, 49, 343, 8, 4096, 64, 512, 9, 6561, 81, 729, 10, 10000, 100, 1000, 11, 14641, 121, 1331, 12, 20736, 144, 1728, 13, 28561, 169, 2197, 14, 38416, 196, 2744, 15, 50625

Offset: 1

Mohammad K. Azarian, Sep 02 2005

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Cf. A000463, A109588, A109594.

map(t -> (t,t^4,t^3,t^2), [$1..100]); # Robert Israel, Aug 15 2016

Flatten[Table[{n,n^4,n^2,n^3},{n,20}]] (* or *) Flatten[ With[ {c=Range[20]}, Thread[{c,c^4,c^2,c^3}]]] (* Harvey P. Dale, Mar 28 2012 *)

Vec(x*(1+x+x^2+x^3-3*x^4+11*x^5-x^6+3*x^7+3*x^8+11*x^9-x^10-3*x^11-x^12+x^13+x^14-x^15)/((1-x)^5*(1+x)^5*(1+x^2)^5) + O(x^60)) \\ Colin Barker, Aug 15 2016

a(n) = (2*n+3-(-1)^n+2*(-1)^((2*n-3-(-1)^n)/4))*(n^3+10*n^2+28*n+88+(n^3+10*n^2-4*n-72)*(-1)^n+(n^3+2*n^2-4*n+56)*(-1)^((2*n-3-(-1)^n)/4)-(n^3+2*n^2+28*n-40)*(-1)^((2*n-1+(-1)^n)/4))/2048. - Luce ETIENNE, Aug 15 2016

G.f.: x*(1+x+x^2+x^3-3*x^4+11*x^5-x^6+3*x^7+3*x^8+11*x^9-x^10-3*x^11-x^12+x^13+x^14-x^15) / ((1-x)^5*(1+x)^5*(1+x^2)^5). - Colin Barker, Aug 15 2016

A110622 n^2 followed by n followed by n^3 followed by n^4.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 8, 16, 9, 3, 27, 81, 16, 4, 64, 256, 25, 5, 125, 625, 36, 6, 216, 1296, 49, 7, 343, 2401, 64, 8, 512, 4096, 81, 9, 729, 6561, 100, 10, 1000, 10000, 121, 11, 1331, 14641, 144, 12, 1728, 20736, 169, 13, 2197, 28561, 196, 14, 2744, 38416, 225, 15

Offset: 1

Mohammad K. Azarian, Sep 14 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..8000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Cf. A000463, A109588, A109594.

&cat[[n^2, n, n^3, n^4]: n in [1..20]]; // Vincenzo Librandi, Nov 25 2012

Flatten[Table[{n^2, n, n^3, n^4}, {n, 40}]] (* Vincenzo Librandi, Nov 25 2012 *)

lista(nn) = for(n=1, nn, print1(n^2, ", ", n, ", ", n^3, ", "n^4, ", ")); \\ Jinyuan Wang, Feb 28 2020

a(n) = 5*a(n-4) - 10*a(n-8) + 10*a(n-12) - 5*a(n-16) + a(n-20).
G.f.: -x*(1 + x + x^2 + x^3 - x^4 - 3*x^5 + 3*x^6 + 11*x^7 - x^8 + 3*x^9 - 3*x^10 + 11*x^11 + x^12 - x^13 - x^14 + x^15) / ( (x-1)^5*(1+x)^5*(x^2+1)^5 ). - R. J. Mathar, Dec 20 2010
a(n) = (2*n + 3 - (-1)^n + 2*(-1)^((2*n + 5 - (-1)^n)/4))*(n^3 + 4*n^2 + 24*n + 116 + (n^3 - 4*n^2 - 24*n + 12)*(-1)^n - (n^3 + 4*n^2 - 8*n - 108)*(-1)^((2*n + 5 - (-1)^n)/4) + (n^3 - 4*n^2 + 8*n - 20)*(-1)^((2*n + 7 + (-1)^n)/4))/2048. - Luce ETIENNE, Sep 02 2016

A110650 n^2 followed by n^4 followed by n followed by n^3.

Original entry on oeis.org

1, 1, 1, 1, 4, 16, 2, 8, 9, 81, 3, 27, 16, 256, 4, 64, 25, 625, 5, 125, 36, 1296, 6, 216, 49, 2401, 7, 343, 64, 4096, 8, 512, 81, 6561, 9, 729, 100, 10000, 10, 1000, 121, 14641, 11, 1331, 144, 20736, 12, 1728, 169, 28561, 13, 2197, 196, 38416, 14, 2744, 225, 50625

Offset: 1

Mohammad K. Azarian, Sep 14 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..8000

Cf. A000463; A109588; A109594.

Cf. A010873.

Flat(List([1..20],n->[n^2,n^4,n,n^3])); # Muniru A Asiru, Apr 08 2018

&cat[[n^2, n^4, n, n^3]: n in [1..20]]; // Vincenzo Librandi, Nov 25 2012

Flatten[Table[{n^2,n^4,n,n^3},{n,20}]] (* Harvey P. Dale, Oct 10 2012 *)

a(n) = floor((n+3)/4)^((-8*(n mod 4)^3 + 33*(n mod 4)^2 - 31*(n mod 4) + 18)/6). - Luce ETIENNE, Apr 07 2018
From Chai Wah Wu, Jan 10 2020: (Start)
a(n) = 5*a(n-4) - 10*a(n-8) + 10*a(n-12) - 5*a(n-16) + a(n-20) for n > 20.
G.f.: x*(x^15 + x^14 - x^13 - x^12 + 3*x^11 - 3*x^10 - 11*x^9 + x^8 - 3*x^7 + 3*x^6 - 11*x^5 + x^4 - x^3 - x^2 - x - 1)/((x - 1)^5*(x + 1)^5*(x^2 + 1)^5). (End)

A110003 n followed by n^3 followed by n^2 followed by n^4.

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 4, 16, 3, 27, 9, 81, 4, 64, 16, 256, 5, 125, 25, 625, 6, 216, 36, 1296, 7, 343, 49, 2401, 8, 512, 64, 4096, 9, 729, 81, 6561, 10, 1000, 100, 10000, 11, 1331, 121, 14641, 12, 1728, 144, 20736, 13, 2197, 169, 28561, 14, 2744, 196, 38416, 15, 3375

Offset: 1

Mohammad K. Azarian, Sep 02 2005

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 5, 0, 0, 0, -10, 0, 0, 0, 10, 0, 0, 0, -5, 0, 0, 0, 1).

Cf. A000463, A109588, A109594.

&cat[[n,n^3,n^2,n^4]: n in [1..15]]; // Bruno Berselli, Sep 01 2016

Table[{n, n^3, n^2, n^4}, {n, 1, 15}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n) = (2*n+3-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))*(n^3+4*n^2+32*n+96+(n^3+4*n^2-64)*(-1)^n-(n^3-4*n^2-64)*(-1)^((2*n+5-(-1)^n)/4)+(n^3-4*n^2-32*n+32)*(-1)^((2*n+7+(-1)^n)/4))/2048. - Luce ETIENNE, Sep 01 2016
From Chai Wah Wu, Jan 11 2020: (Start)
a(n) = 5*a(n-4) - 10*a(n-8) + 10*a(n-12) - 5*a(n-16) + a(n-20) for n > 20.
G.f.: x*(-x^15 - x^14 + x^13 + x^12 - 11*x^11 + x^10 + 3*x^9 - 3*x^8 - 11*x^7 + x^6 - 3*x^5 + 3*x^4 - x^3 - x^2 - x - 1)/((x - 1)^5*(x + 1)^5*(x^2 + 1)^5). (End)

A110004 n followed by n^3 followed by n^4 followed by n^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 16, 4, 3, 27, 81, 9, 4, 64, 256, 16, 5, 125, 625, 25, 6, 216, 1296, 36, 7, 343, 2401, 49, 8, 512, 4096, 64, 9, 729, 6561, 81, 10, 1000, 10000, 100, 11, 1331, 14641, 121, 12, 1728, 20736, 144, 13, 2197, 28561, 169, 14, 2744, 38416, 196, 15

Offset: 1

Mohammad K. Azarian, Sep 02 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..4000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 5, 0, 0, 0, -10, 0, 0, 0, 10, 0, 0, 0, -5, 0, 0, 0, 1).

Cf. A000463, A109588, A109594.

[&cat[[n,n^3,n^4,n^2]: n in [1..20]]]; // Vincenzo Librandi, Sep 05 2016

Table[n^{1, 3, 4, 2}, {n, 15}] // Flatten (* or *)
Table[(2 n + 3 - (-1)^n + 2 (-1)^((2 n + 5 - (-1)^n)/4)) (n^3 + 7 n^2 + 35 n + 81 - (n^3 - n^2 - 29 n + 49) (-1)^n - (n^3 - n^2 + 3 n - 79) (-1)^((2 n + 5 - (-1)^n)/4) - (n^3 + 7 n^2 + 3 n - 47) (-1)^((2 n + 7 + (-1)^n)/4))/2048, {n, 57}] (* Michael De Vlieger, Sep 03 2016 *)
LinearRecurrence[{0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1},{1,1,1,1,2,8,16,4,3,27,81,9,4,64,256,16,5,125,625,25},60] (* Harvey P. Dale, Jul 19 2024 *)

a(n) = (2*n+3-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))*(n^3+7*n^2+35*n+81-(n^3-n^2-29*n+49)*(-1)^n-(n^3-n^2+3*n-79)*(-1)^((2*n+5-(-1)^n)/4)-(n^3+7*n^2+3*n-47)*(-1)^((2*n+7+(-1)^n)/4))/2048. - Luce ETIENNE, Sep 03 2016
From Chai Wah Wu, Jan 11 2020: (Start)
a(n) = 5*a(n-4) - 10*a(n-8) + 10*a(n-12) - 5*a(n-16) + a(n-20) for n > 20.
G.f.: x*(-x^15 - x^14 + x^13 + x^12 + x^11 - 11*x^10 + 3*x^9 - 3*x^8 + x^7 - 11*x^6 - 3*x^5 + 3*x^4 - x^3 - x^2 - x - 1)/((x - 1)^5*(x + 1)^5*(x^2 + 1)^5). (End)

A110008 n followed by n^4 followed by n^3 followed by n^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 16, 8, 4, 3, 81, 27, 9, 4, 256, 64, 16, 5, 625, 125, 25, 6, 1296, 216, 36, 7, 2401, 343, 49, 8, 4096, 512, 64, 9, 6561, 729, 81, 10, 10000, 1000, 100, 11, 14641, 1331, 121, 12, 20736, 1728, 144, 13, 28561, 2197, 169, 14, 38416, 2744, 196, 15

Offset: 1

Mohammad K. Azarian, Sep 02 2005

Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Cf. A000463, A109588, A109594.

Flatten[Table[{n,n^Range[4,2,-1]},{n,15}]] (* Harvey P. Dale, Aug 11 2011 *)

s=[]; for(n=1, 100, s=concat(s, [n,n^4,n^3,n^2])); s \\ Colin Barker, Dec 13 2014

G.f.: -x*(x^15-x^14+x^13-x^12-x^11-3*x^10+11*x^9+3*x^8-x^7+3*x^6+11*x^5-3*x^4+x^3+x^2+x+1) / ((x-1)^5*(x+1)^5*(x^2+1)^5). - Colin Barker, Dec 13 2014

a(n) = (2*n+3-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))*(n^3+10*n^2+36*n+76+(n^3+2*n^2+20*n-60)*(-1)^n+(n^3+2*n^2-12*n+68)*(-1)^((2*n+5-(-1)^n)/4)-(n^3+10*n^2+4*n-52)*(-1)^((2*n+7+(-1)^n)/4))/2048. - Luce ETIENNE, Aug 27 2016

A110485 n^2 followed by n followed by n^4 followed by n^3.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 16, 8, 9, 3, 81, 27, 16, 4, 256, 64, 25, 5, 625, 125, 36, 6, 1296, 216, 49, 7, 2401, 343, 64, 8, 4096, 512, 81, 9, 6561, 729, 100, 10, 10000, 1000, 121, 11, 14641, 1331, 144, 12, 20736, 1728, 169, 13, 28561, 2197, 196, 14, 38416, 2744, 225, 15

Offset: 1

Mohammad K. Azarian, Sep 14 2005

nonn

Cf. A000463, A109588, A109594.

Flatten[Table[{n^2,n,n^4,n^3},{n,20}]] (* Harvey P. Dale, Oct 24 2013 *)

a(n) = (2*n+3-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))*(n^3+7*n^2+19*n+113-(n^3-n^2+19*n-15)*(-1)^n-(n^3+7*n^2-13*n-111)*(-1)^((2*n+5-(-1)^n)/4)-(n^3-n^2-13*n+17)*(-1)^((2*n+7+(-1)^n)/4))/2048. - Luce ETIENNE, Sep 01 2016
From Chai Wah Wu, Jan 11 2020: (Start)
a(n) = 5*a(n-4) - 10*a(n-8) + 10*a(n-12) - 5*a(n-16) + a(n-20) for n > 20.
G.f.: x*(x^15 - x^14 + x^13 - x^12 + 3*x^11 - 11*x^10 - 3*x^9 + x^8 - 3*x^7 - 11*x^6 + 3*x^5 + x^4 - x^3 - x^2 - x - 1)/((x - 1)^5*(x + 1)^5*(x^2 + 1)^5). (End)

A110651 n^2 followed by n^4 followed by n^3 followed by n.

Original entry on oeis.org

1, 1, 1, 1, 4, 16, 8, 2, 9, 81, 27, 3, 16, 256, 64, 4, 25, 625, 125, 5, 36, 1296, 216, 6, 49, 2401, 343, 7, 64, 4096, 512, 8, 81, 6561, 729, 9, 100, 10000, 1000, 10, 121, 14641, 1331, 11, 144, 20736, 1728, 12, 169, 28561, 2197, 13, 196, 38416, 2744, 14, 225

Offset: 1

Mohammad K. Azarian, Sep 14 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..4000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Cf. A000463, A109588, A109594.

&cat[[n^2, n^4, n^3, n]: n in [1..20]]; // Vincenzo Librandi, Feb 06 2013

Flatten[Table[{n^2, n^4, n^3, n}, {n, 40}]](* Vincenzo Librandi, Feb 06 2013 *)
LinearRecurrence[{0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1},{1,1,1,1,4,16,8,2,9,81,27,3,16,256,64,4,25,625,125,5},60] (* Harvey P. Dale, Sep 20 2023 *)

Vec(x*(1+x+x^2+x^3-x^4+11*x^5+3*x^6-3*x^7-x^8+11*x^9-3*x^10+3*x^11+x^12+x^13-x^14-x^15) / ((1-x)^5*(1+x)^5*(1+x^2)^5) + O(x^30)) \\ Colin Barker, Sep 02 2016

a(n) = (2*n+3-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))*(n^3+10*n^2+36*n+124+(n^3+2*n^2-12*n+20)*(-1)^n+(n^3+2*n^2+20*n-12)*(-1)^((2*n+5-(-1)^n)/4)-(n^3+10*n^2+4*n-100)*(-1)^((2*n+7+(-1)^n)/4))/2048. - Luce ETIENNE, Sep 02 2016
G.f.: x*(1+x+x^2+x^3-x^4+11*x^5+3*x^6-3*x^7-x^8+11*x^9-3*x^10+3*x^11 +x^12+x^13-x^14-x^15) / ((1-x)^5*(1+x)^5*(1+x^2)^5). - Colin Barker, Sep 02 2016
a(n) = 5*a(n-4)-10*a(n-8)+10*a(n-12)-5*a(n-16)+a(n-20). - Wesley Ivan Hurt, Jun 09 2023

A110652 n^2 followed by n^3 followed by n^4 followed by n.

Original entry on oeis.org

1, 1, 1, 1, 4, 8, 16, 2, 9, 27, 81, 3, 16, 64, 256, 4, 25, 125, 625, 5, 36, 216, 1296, 6, 49, 343, 2401, 7, 64, 512, 4096, 8, 81, 729, 6561, 9, 100, 1000, 10000, 10, 121, 1331, 14641, 11, 144, 1728, 20736, 12, 169, 2197, 28561, 13, 196, 2744, 38416, 14, 225

Offset: 1

Mohammad K. Azarian, Sep 14 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..4000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Cf. A000463, A109588, A109594.

&cat[[n^2, n^3, n^4, n]: n in [1..20]]; // Vincenzo Librandi, Feb 06 2013

Flatten[Table[{n^2, n^3, n^4, n}, {n, 40}]] (* Vincenzo Librandi, Feb 06 2013 *)

Vec(x*(1+x+x^2+x^3-x^4+3*x^5+11*x^6-3*x^7-x^8-3*x^9+11*x^10+3*x^11+x^12-x^13+x^14-x^15) / ((1-x)^5*(1+x)^5*(1+x^2)^5) + O(x^20)) \\ Colin Barker, Aug 18 2016

a(n) = (2*n+3-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))*(n^3+7*n^2+35*n+129-(n^3-n^2+3*n-31)*(-1)^n-(n^3-n^2-29*n+1)*(-1)^((2*n+5-(-1)^n)/4)-(n^3+7*n^2+3*n-95)*(-1)^((2*n+7+(-1)^n)/4))/2048. - Luce ETIENNE, Aug 17 2016

G.f.: x*(1+x+x^2+x^3-x^4+3*x^5+11*x^6-3*x^7-x^8-3*x^9+11*x^10+3*x^11+x^12-x^13+x^14-x^15) / ((1-x)^5*(1+x)^5*(1+x^2)^5). - Colin Barker, Aug 18 2016

cp's OEIS Frontend

A110001 n followed by n^2 followed by n^3 followed by n^4.

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A110009 n followed by n^4 followed by n^2 followed by n^3.

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A110622 n^2 followed by n followed by n^3 followed by n^4.

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Magma

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A110650 n^2 followed by n^4 followed by n followed by n^3.

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A110003 n followed by n^3 followed by n^2 followed by n^4.

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Magma

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A110004 n followed by n^3 followed by n^4 followed by n^2.

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Magma

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A110008 n followed by n^4 followed by n^3 followed by n^2.

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A110485 n^2 followed by n followed by n^4 followed by n^3.

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