A055808 a(n) and floor(a(n)/4) are both squares; i.e., squares that remain squares when written in base 4 and last digit is removed.

0, 1, 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056, 7396, 7744, 8100

Offset

0,3

Comments

Let A(x) = (1 + k*x + (k-1)*x^2). Then the coefficients of (A(x))^2 sum to 4*k^2, where k = (n - 1). Examples: if k = 3 we have (1 + 3*x + 2*x^2)^2 = (1 + 6*x + 13x^2 + 12*x^3 + 4*x^4), and ( 1 + 6 + 13 + 12 + 4) = 36. If k = 4 we have (1 + 4*x + 3*x^2)^2 = (1 + 8*x + 22*x^2 + 24*x^3 + 9*x^4), and (1 + 8 + 22 + 24 + 9) = 64 = a(5). - Gary W. Adamson, Aug 02 2015

For n>0, a(n) are the Engel expansion of A197036. - Benedict W. J. Irwin, Dec 15 2016

Links

Table of n, a(n) for n=0..46.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = A004275(n)^2. - M. F. Hasler, Jan 16 2012

a(n) = 4*(-1+n)^2 for n>1; a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>4; G.f.: x*(x^3-7*x^2-x-1) / (x-1)^3. - Colin Barker, Sep 15 2014

Example

36 is in the sequence because 36 = 6^2 = 210 base 3 and 21 base 4 = 9 = 3^2.

Mathematica

Join[{0, 1}, LinearRecurrence[{3, -3, 1}, {4, 16, 36}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

Prog

(PARI) concat(0, Vec(x*(x^3-7*x^2-x-1)/(x-1)^3 + O(x^100))) \\ Colin Barker, Sep 15 2014
(PARI) is_ok(n)=issquare(n) && issquare(floor(n/4));
first(m)=my(v=vector(m), r=0); for(i=1, m, while(!is_ok(r), r++); v[i]=r; r++; ); v; /* Anders Hellström, Aug 08 2015 */
(Magma) [Floor((2*n^2)/(1 + n))^2: n in [0..60]]; // Vincenzo Librandi, Aug 03 2015

Crossrefs

Cf. A023110. Essentially A016742 with one addition.

Sequence in context: A281795 A063540 A349223 * A016742 A221285 A121317

Adjacent sequences: A055805 A055806 A055807 * A055809 A055810 A055811

Keyword

nonn,base,easy

Author

Henry Bottomley, Jul 14 2000

Status

approved