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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. 19
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,base,easy
AUTHOR
Henry Bottomley, Jul 14 2000
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)