A000463 -id:A000463 - cp's OEIS frontend

A188653 Second differences of A000463; first differences of A188652.

1, 1, -3, 7, -11, 17, -23, 31, -39, 49, -59, 71, -83, 97, -111, 127, -143, 161, -179, 199, -219, 241, -263, 287, -311, 337, -363, 391, -419, 449, -479, 511, -543, 577, -611, 647, -683, 721, -759, 799, -839, 881, -923, 967, -1011, 1057, -1103, 1151, -1199, 1249, -1299, 1351, -1403, 1457, -1511, 1567, -1623, 1681, -1739, 1799, -1859, 1921, -1983, 2047, -2111, 2177, -2243, 2311, -2379, 2449, -2519, 2591, -2663, 2737, -2811

Offset: 1

Reinhard Zumkeller, Apr 13 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A056220, A142463, A047838.

Cf. A000463, A188652.

a188653 n = a188653_list !! (n-1)
a188653_list = zipWith (-) (tail a188652_list) a188652_list

LinearRecurrence[{-2, 0, 2, 1}, {1, 1, -3, 7}, 75] (* Jean-François Alcover, Dec 16 2021 *)
Differences[Flatten[Table[{n,n^2},{n,50}]],2] (* Harvey P. Dale, Aug 03 2025 *)

a(2*n) = a(2*n-1)+4*n^2-2*n-2, a(2*n+1) = -a(2*n)-2*n.
a(2*n) = A056220(n), a(2*n-1) = -A142463(n).
Abs(a(n)) = A047838(n) for n > 1.
a(n) = A188652(n+1)-A188652(n) = A000463(n+2)-2*A000463(n+1)+A000463(n).
G.f.: x*(-1-3*x+x^2+x^3) / ((x-1)*(1+x)^3). - R. J. Mathar, Apr 14 2011
a(n) = a(-n) = ((2*n^2-5)*(-1)^n+1)/4. - Bruno Berselli, Sep 14 2011
E.g.f.: 1 + ((x^2 - x - 2)*cosh(x) - (x^2 - x - 3)*sinh(x))/2. - Stefano Spezia, Jul 08 2023
Sum_{n>=1} 1/a(n) = 1/2 - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)) - tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, May 11 2025

A188652 First differences of A000463.

Original entry on oeis.org

0, 1, 2, -1, 6, -5, 12, -11, 20, -19, 30, -29, 42, -41, 56, -55, 72, -71, 90, -89, 110, -109, 132, -131, 156, -155, 182, -181, 210, -209, 240, -239, 272, -271, 306, -305, 342, -341, 380, -379, 420, -419, 462, -461, 506, -505, 552, -551, 600, -599, 650, -649, 702, -701, 756, -755, 812, -811, 870, -869, 930, -929, 992, -991, 1056, -1055, 1122, -1121, 1190, -1189, 1260, -1259, 1332, -1331, 1406

Offset: 1

Reinhard Zumkeller, Apr 13 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-1,2,2,-1,-1).

Cf. A188653 (first differences).

Cf. A000463, A002378, A165900.

a188652 n = a188652_list !! (n-1)
a188652_list = zipWith (-) (tail a000463_list) a000463_list

Differences[Flatten[Array[{#,#^2}&,40]]] (* Harvey P. Dale, Aug 04 2012 *)

a(2n) = 1 - a(2n-1), a(2n+1) = 2*n + 1 - a(2n).
a(n) = A000463(n+1) - A000463(n).
a(2n-1) = A002378(n-1), a(2n) = - A165900(n).
G.f.: -x^2*(-1-3*x+x^2+x^3) / ( (x-1)^2*(1+x)^3 ). - R. J. Mathar, Apr 14 2011
a(n) = (2*n+3-(2*n^2-2*n-5)*(-1)^n)/8. - Luce ETIENNE, Dec 18 2014
E.g.f.: ((4 + x - x^2)*cosh(x) - (1 - x - x^2)*sinh(x) - 4)/4. - Stefano Spezia, Jul 08 2023
Sum_{n>=2} 1/a(n) = 1 + (sqrt(5)-5)*Pi*tan(sqrt(5)*Pi/2)/(5*(sqrt(5)-1)). - Amiram Eldar, May 11 2025

A159693 Partial sums of A000463.

Original entry on oeis.org

1, 2, 4, 8, 11, 20, 24, 40, 45, 70, 76, 112, 119, 168, 176, 240, 249, 330, 340, 440, 451, 572, 584, 728, 741, 910, 924, 1120, 1135, 1360, 1376, 1632, 1649, 1938, 1956, 2280, 2299, 2660, 2680, 3080, 3101, 3542, 3564, 4048, 4071, 4600, 4624, 5200, 5225, 5850

Offset: 1

Gerald Hillier, Apr 20 2009

nonn

Sum of integers followed by squares.

			For n=9, a(n) = 1+1+2+4+3+9+4+16+5 = 45.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A000290, A000463.

a159693 n = a159693_list !! (n-1)
a159693_list = scanl1 (+) a000463_list -- Reinhard Zumkeller, Nov 08 2015

S:=&cat[ [ n, n^2 ]: n in [1..25] ]; [ n eq 1 select S[1] else Self(n-1)+S[n]: n in [1..#S] ]; // Klaus Brockhaus, Apr 20 2009

seq((2*n^3+9*n^2+19*n+9+3*(n^2-n-3)*(-1)^n)/48, n=1..100); # Robert Israel, Dec 30 2014

CoefficientList[Series[x*(1+x-x^2+x^3)/((1+x)^3*(x-1)^4), {x, 0, 50}], x] (* or *) Table[(2*n^3+9*n^2+19*n+9+3*(n^2-n-3)*(-1)^n)/48, {n,0,50}] (* G. C. Greubel, Jun 02 2018 *)
Accumulate[Flatten[{#,#^2}&/@Range[30]]] (* Harvey P. Dale, Nov 30 2019 *)

a(n) = (n^3+3*n^2+8*n+r(n))/24, where r(n) = 3*n+9 if n is odd, 3*n^2 if n is even.
G.f.: x*(1+x-x^2+x^3)/((1+x)^3*(x-1)^4). - R. J. Mathar, Apr 20 2009
a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). - R. J. Mathar, Apr 20 2009
a(n) = (2*n^3+9*n^2+19*n+9+3*(n^2-n-3)*(-1)^n)/48. - Luce ETIENNE, Dec 29 2014
E.g.f.: (2*x^3+15*x^2+30*x+9)*exp(x)/48 +(x^2-3)*exp(-x)/16. - Robert Israel, Dec 30 2014

More terms from R. J. Mathar and Klaus Brockhaus, Apr 20 2009

A109588 n followed by n^2 followed by n^3.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 3, 9, 27, 4, 16, 64, 5, 25, 125, 6, 36, 216, 7, 49, 343, 8, 64, 512, 9, 81, 729, 10, 100, 1000, 11, 121, 1331, 12, 144, 1728, 13, 169, 2197, 14, 196, 2744, 15, 225, 3375, 16, 256, 4096, 17, 289, 4913, 18, 324, 5832, 19, 361, 6859, 20, 400, 8000

Offset: 1

Mohammad K. Azarian, Aug 30 2005

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-6,0,0,4,0,0,-1).

Cf. A000463.

Flat(List([1..20],n->[n,n^2,n^3])); # Muniru A Asiru, Sep 12 2018

seq(seq(n^k, k=1..3), n=1..20); # Zerinvary Lajos, Jun 29 2007

CoefficientList[Series[(1 + x + x^2 - 2*x^3 + 4*x^5 + x^6 - x^7 + x^8)/((1 - x)^4*(1 + x + x^2)^4), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)
Table[{n,n^2,n^3},{n,20}]//Flatten (* or *) LinearRecurrence[{0,0,4,0,0,-6,0,0,4,0,0,-1},{1,1,1,2,4,8,3,9,27,4,16,64},60] (* Harvey P. Dale, Jan 10 2020 *)

From R. J. Mathar, Mar 30 2009: (Start)
a(n) = 4*a(n-3) - 6*a(n-6) + 4*a(n-9) - a(n-12).
a(3*k+1) = k+1, a(3*k+2) = A000290(k+1), a(3*k+3) = A000578(k+1).
G.f.: x*(1 + x + x^2 - 2*x^3 + 4*x^5 + x^6 - x^7 + x^8)/((1 - x)^4*(1 + x + x^2)^4). (End)
a(n) = floor((n + 2)/3)*((1 - (-1)^(2^(n + 2 - 3*floor((n + 2)/3))))/2 + floor((n + 2)/3)*(1 - (-1)^(2^(n + 1 - 3*floor((n + 1)/3))))/2 + (floor((n + 2)/3))^2*(1 - (-1)^(2^(n - 3*floor(n/3))))/2). - Luce ETIENNE, Dec 16 2014
E.g.f.: ((2*x^3 + 3*x^2 + 8*x - 21)*exp(-x/2)*cos(sqrt(3)*x/2) + (3*x^2 + 8*x + 15)*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2) + (x^3 + 6*x^2 + 19*x + 21)*exp(x))/81. - Robert Israel, Dec 17 2014

A109594 n followed by n^3 followed by n^2.

Original entry on oeis.org

1, 1, 1, 2, 8, 4, 3, 27, 9, 4, 64, 16, 5, 125, 25, 6, 216, 36, 7, 343, 49, 8, 512, 64, 9, 729, 81, 10, 1000, 100, 11, 1331, 121, 12, 1728, 144, 13, 2197, 169, 14, 2744, 196, 15, 3375, 225, 16, 4096, 256, 17, 4913, 289, 18, 5832, 324, 19, 6859, 361, 20, 8000, 400

Offset: 1

Mohammad K. Azarian, Aug 30 2005

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-6,0,0,4,0,0,-1).

Cf. A000463, A010872.

map(t -> (t,t^3,t^2), [$1..100]); # Robert Israel, Dec 17 2017

Flatten[Table[{n,n^3,n^2},{n,20}]] (* Harvey P. Dale, Jul 22 2012 *)

Vec(x*(1+x+x^2-2*x^3+4*x^4+x^6+x^7-x^8)/((1-x)^4*(1+x+x^2)^4) + O(x^100)) \\ Colin Barker, Dec 02 2015

From Colin Barker, Dec 02 2015: (Start)
a(n) = 4*a(n-3) - 6*a(n-6) + 4*a(n-9) - a(n-12) for n > 12.
G.f.: x*(1+x+x^2-2*x^3+4*x^4+x^6+x^7-x^8) / ((1-x)^4*(1+x+x^2)^4).
(End)
a(n) = floor((n+2)/3)^((3*(n mod 3)^2-5*(n mod 3)+4)/2). - Luce ETIENNE, Dec 17 2017

A137442 n^2 followed by smallest integer not yet listed.

Original entry on oeis.org

1, 2, 4, 3, 9, 5, 16, 6, 25, 7, 36, 8, 49, 10, 64, 11, 81, 12, 100, 13, 121, 14, 144, 15, 169, 17, 196, 18, 225, 19, 256, 20, 289, 21, 324, 22, 361, 23, 400, 24, 441, 26, 484, 27, 529, 28, 576, 29, 625, 30, 676, 31, 729, 32, 784, 33, 841, 34, 900, 35, 961, 37, 1024, 38

Offset: 1

Andy Martin, Apr 18 2008

Sequence is a permutation of the positive integers.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000463.

f[s_List] := Block[{k = 1}, While[ MemberQ[s, k], k++ ]; Flatten@ Append[s, {((2 + Length@s)/2)^2, k}]]; Nest[f, {1, 2}, 33] (* Robert G. Wilson v, May 31 2009 *)
Module[{nn=40,sq,int,len},sq=Range[nn]^2;int=Complement[Range[nn],sq];len=Min[Length[int],nn];Riffle[Take[sq,len],Take[int,len]]](* Harvey P. Dale, Nov 05 2013 *)

lista(nn) = {for (n=1, nn, print1(n^2, ", ", n+round(sqrt(n)), ", "););} \\ Michel Marcus, Nov 02 2014

a(n) = if (n % 2, ((n+1)/2)^2, (n/2)+round(sqrt(n/2))); \\ Michel Marcus, Nov 02 2014

# correct to any term:
sk_ct = 2
skip = 4
at = 1
(1..(1.0/0)).each{ |i|
if (at+=1) == skip
at+=1
sk_ct +=1
skip = sk_ct * sk_ct
end
print i*i, " ", at, " "
}

# Simpler Ruby code, correct until i is so large that floating point rounding causes errors. I estimate this will be before i reaches 10000000000000000
(1..(1.0/0)).each{ |i|
print i*i, " ", i + (Math.sqrt(i) + 0.5).to_i, " "
}

Formula, generating two terms for every m: m^2, m + round(sqrt(m)).

IFTE(n mod 2 ==1, ((n+1)/2)^2, (n/2)+round(sqrt(n/2),0)). - Gerald Hillier, Nov 15 2010

More terms from Robert G. Wilson v, May 31 2009

A109614 n^3 followed by n followed by n^2.

Original entry on oeis.org

1, 1, 1, 8, 2, 4, 27, 3, 9, 64, 4, 16, 125, 5, 25, 216, 6, 36, 343, 7, 49, 512, 8, 64, 729, 9, 81, 1000, 10, 100, 1331, 11, 121, 1728, 12, 144, 2197, 13, 169, 2744, 14, 196, 3375, 15, 225, 4096, 16, 256, 4913, 17, 289, 5832, 18, 324, 6859, 19, 361, 8000, 20, 400

Offset: 1

Mohammad K. Azarian, Aug 30 2005

Reinhard Zumkeller, Table of n, a(n) for n = 1..3000
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-6,0,0,4,0,0,-1).

Cf. A000463.
Cf. A000578, A000027, A000290.
Cf. A010872.

import Data.List (transpose)
a109614 n = a109614_list !! (n-1)
a109614_list = concat $ transpose
               [tail a000578_list, a000027_list, tail a000290_list]
-- Reinhard Zumkeller, Mar 14 2014

Table[{n^3,n,n^2},{n,20}]//Flatten (* or *) LinearRecurrence[{0,0,4,0,0,-6,0,0,4,0,0,-1},{1,1,1,8,2,4,27,3,9,64,4,16},60] (* Harvey P. Dale, Feb 25 2016 *)

From R. J. Mathar, Jun 26 2009: (Start)
a(n) = 4*a(n-3) - 6*a(n-6) + 4*a(n-9) - a(n-12).
G.f.: -(-1-x-x^2-4*x^3+2*x^4-x^6-x^7+x^8)/((x-1)^4*(1+x+x^2)^4). (End)
a(n) = floor((n+2)/3)^((-3*(n mod 3)^2 + 5*(n mod 3) + 4)/2). - Luce ETIENNE, Mar 01 2018

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

Original entry on oeis.org

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

cp's OEIS Frontend

A188653 Second differences of A000463; first differences of A188652.

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A188652 First differences of A000463.

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A159693 Partial sums of A000463.

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A109588 n followed by n^2 followed by n^3.

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A109594 n followed by n^3 followed by n^2.

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A137442 n^2 followed by smallest integer not yet listed.

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A109614 n^3 followed by n followed by n^2.

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