A007892 A Kutz sequence.

1, 4, 9, 1, 4, 9, 16, 4, 9, 16, 25, 9, 16, 25, 36, 16, 25, 36, 49, 25, 36, 49, 64, 36, 49, 64, 81, 49, 64, 81, 100, 64, 81, 100, 121, 81, 100, 121, 144, 100, 121, 144, 169, 121, 144, 169, 196, 144, 169, 196, 225, 169, 196, 225, 256, 196, 225, 256, 289, 225

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The pattern is obvious: after the initial three terms, we have four successive squares.

Another description of the same sequence: array read by rows, with four columns, in which row n lists n^2, (n+1)^2, (n+2)^2, n^2. - Omar E. Pol, Sep 28 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• R. E. Kutz, Two unusual sequences, Two-Year College Mathematics Journal, 12 (1981), 316-319.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Programs

Magma

[(n-3*Floor(n/4))^2: n in [1..60]]; // Vincenzo Librandi, Sep 28 2011

Mathematica

Table[(n - 3*Floor[n/4])^2, {n, 60}] (* Arkadiusz Wesolowski, Sep 29 2011 *)
Rest[Flatten[Table[Range[n,n+3]^2,{n,0,20}]]] (* Harvey P. Dale, Oct 24 2015 *)

Maxima

makelist(((2*n-6*(-1)^((n-1)*n/2)-3*(-1)^n+9)/8)^2,n,1,60); /* Bruno Berselli, Sep 28 2011 */

PARI

a(n)=(floor((-1)^n+(n+5)/2)-3*floor((n+6)/4))^2 \\ Charles R Greathouse IV, Sep 28 2011

Formula

a(n) = (floor((-1)^n+(n+5)/2)-3*floor((n+6)/4))^2. [Arkadiusz Wesolowski, Sep 27 2011]

a(n) = (n-3*floor(n/4))^2. [Arkadiusz Wesolowski, Sep 28 2011]

G.f.: x*(1+3*x+5*x^2-8*x^3+x^4-x^5-3*x^6+4*x^7)/((1-x)^3*(1+x+x^2+x^3)^2). a(n) = (A110657(n-1)+1)^2 = ((2*n-6*(-1)^((n-1)*n/2)-3*(-1)^n+9)/8)^2. [Bruno Berselli, Sep 28 2011]