A069187 -id:A069187 - cp's OEIS frontend

A071253 a(n) = n^2*(n^2+1).

0, 2, 20, 90, 272, 650, 1332, 2450, 4160, 6642, 10100, 14762, 20880, 28730, 38612, 50850, 65792, 83810, 105300, 130682, 160400, 194922, 234740, 280370, 332352, 391250, 457652, 532170, 615440, 708122, 810900, 924482, 1049600, 1187010, 1337492, 1501850, 1680912

Offset: 0

N. J. A. Sloane, Jun 12 2002

The identity (n^5 + n^3)^2 + (n^2*(n^2 + 1))^2 = n*(n^3 + n)^3 can be written as A155977(n)^2 + a(n)^2 = n*A034262(n)^3. - Vincenzo Librandi, Aug 08 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A002522, A002378, A034262, A037270, A069187, A155977.

A071253:= func< n | 2*Binomial(n^2+1,2) >;
[A071253(n): n in [0..40]]; // G. C. Greubel, Sep 12 2024

with(combinat):seq(lcm(fibonacci(3,n),n^2),n=0..35); # Zerinvary Lajos, Apr 20 2008
a:=n->add(n+add(n+add(n, j=1..n-1),j=1..n),j=1..n):seq(a(n), n=0..41); # Zerinvary Lajos, Aug 27 2008

Table[(1/4) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n,0,40}] (* Artur Jasinski, Feb 10 2010 *)
Table[n^2*(n^2+1),{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[2 x (1+x) (1+4 x+x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

a(n)=n^2*(n^2+1) \\ Charles R Greathouse IV, Sep 24 2015

def A071253(n): return 2*binomial(n^2+1,2)
[A071253(n) for n in range(41)] # G. C. Greubel, Sep 12 2024

a(n) = A002522(n)*A000290(n). - Zerinvary Lajos, Apr 20 2008
a(n) = (1/4)*sinh(2*arcsinh(n))^2. - Artur Jasinski, Feb 10 2010
G.f.: 2*x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Jan 08 2012
a(n) = A002378(A000290(n)). - Rick L. Shepherd, Sep 22 2014
Sum_{n>=1} 1/a(n) = 0.5682... = Pi^2/6- (Pi*coth Pi-1)/2 = A013661 - A259171 [J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Oct 18 2019
a(n) = 2*A037270(n). - R. J. Mathar, Oct 18 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 1/2 + Pi*cosech(Pi)/2. - Amiram Eldar, Nov 05 2020
E.g.f.: exp(x)*x*(2 + 8*x + 6*x^2 + x^3). - Stefano Spezia, Oct 08 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 16 2023

A189883 Numbers k such that the square part of k is one greater than the squarefree part of k.

Original entry on oeis.org

12, 240, 1260, 20592, 38220, 65280, 104652, 159600, 233772, 809100, 1047552, 1335180, 1678320, 2083692, 2558400, 3109932, 7308912, 8500140, 9831360, 11313132, 12956400, 18970380, 21376752, 24005100, 26868672, 37008972, 49780080

Offset: 1

Antonio Roldán, Apr 29 2011

nonn

The complementary sequence, squarefree part of k is one greater than the square part of k, is A069187.

			1260 = 2^2*3^2*5*7, square part: 2^2*3^2 = 36, squarefree part: 5*7 = 35, and 36 = 35+1.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Antonio Roldán, hojaynumeros.blogspot.com

b:= proc() 1 end:
a:= proc(n) option remember; local i, k;
      if n>1 then a(n-1) fi;
      for k from b(n-1)+1 while 1<>mul(i[2], i=ifactors(k^2-1)[2])
      do od; b(n):=k; k^4-k^2
    end:
seq(a(n), n=1..50); # Alois P. Heinz, Apr 29 2011

okQ[n_] := Module[{p, e, sfp}, {p, e} = Transpose[FactorInteger[n]]; e = Mod[e, 2]; sfp = Times @@ (p^e); n/sfp - sfp == 1]; Select[Range[10^5], okQ] (* T. D. Noe, Apr 29 2011 *)

for(n=1,1e3,if(issquarefree(n^2-1),print1(n^4-n^2", "))) \\ Charles R Greathouse IV, Apr 29 2011

n such that A008833(n) - A007913(n) = 1.

a(n) = x^2 (x^2-1), where x = A067874(n). - T. D. Noe, Apr 29 2011

cp's OEIS Frontend

A071253 a(n) = n^2*(n^2+1).

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