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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 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 HellstrÃ¶m, 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: A044065 A281795 A063540 * A016742 A221285 A121317 Adjacent sequences:  A055805 A055806 A055807 * A055809 A055810 A055811 KEYWORD nonn,base,easy AUTHOR Henry Bottomley, Jul 14 2000 STATUS approved

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