cp's OEIS Frontend

A181318 a(n) = A060819(n)^2.

%I A181318 #79 Jan 18 2025 20:15:35
%S A181318 0,1,1,9,1,25,9,49,4,81,25,121,9,169,49,225,16,289,81,361,25,441,121,
%T A181318 529,36,625,169,729,49,841,225,961,64,1089,289,1225,81,1369,361,1521,
%U A181318 100,1681,441,1849,121,2025,529,2209,144,2401,625,2601,169,2809,729
%N A181318 a(n) = A060819(n)^2.
%C A181318 The first sequence, p=0, of the family A060819(n)*A060819(n+p).
%C A181318 Hence array
%C A181318 p=0:  0, 1, 1,  9,  1, 25,  9,  49,    a(n)=A060819(n)^2,
%C A181318 p=1:  0, 1, 3,  3,  5, 15, 21,  14,     A064038(n),
%C A181318 p=2:  0, 3, 1, 15,  3, 35,  6,  63,     A198148(n),
%C A181318 p=3:  0, 1, 5,  9,  7, 10, 27,  35,     A160050(n),
%C A181318 p=4:  0, 5, 3, 21,  2, 45, 15,  77,     A061037(n),
%C A181318 p=5:  0, 3, 7,  6,  9, 25, 33,  21,     A178242(n),
%C A181318 p=6:  0, 7, 2, 27,  5, 55,  9,  91,     A217366(n),
%C A181318 p=7:  0, 2, 9, 15, 11, 15, 39,  49,     A217367(n),
%C A181318 p=8:  0, 9, 5, 33,  3, 65, 21, 105,     A180082(n).
%C A181318 Compare columns 2, 3 and 5, columns 4 and 7 and columns 6 and 8.
%C A181318 From _Peter Bala_, Feb 19 2019: (Start)
%C A181318 We make some general remarks about the sequence a(n) = numerator(n^2/(n^2 + k^2)) = (n/gcd(n,k))^2 for k a fixed positive integer (we suppress the dependence of a(n) on k). The present sequence corresponds to the case k = 4.
%C A181318 a(n) is a quasi-polynomial in n. In fact, a(n) = n^2/b(n) where b(n) = gcd(n^2,k^2) is a purely periodic sequence in n.
%C A181318 In addition to being multiplicative these sequences are also strong divisibility sequences, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m).
%C A181318 By the multiplicativeness and strong divisibility property of the sequence a(n) it follows that if gcd(n,m) = 1 then a( a(n)*a(m) ) = a(a(n)) * a(a(m)), a( a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.
%C A181318 The sequence a(n) has the rational generating function Sum_{d divides k} f(d)*F(x^d), where F(x) = x*(1 + x)/(1 - x)^3 = x + 4*x^2 + 9*x^3 + 16*x^4 + ... is the o.g.f. for the squares A000290, and where f(n) is the Dirichlet inverse of the Jordan totient function J_2(n) - see A007434. The function f(n) is multiplicative and is defined on prime powers p^k by f(p^k) = (1 - p^2). See A046970. Cf. A060819. (End)
%C A181318 a(n-4) is the constant needed to complete the n-polygonal numbers into squares (see A377851); a(-1) = 1, which completes the triangle numbers, is not shown in the data. - _Jonathan Dushoff_, Nov 12 2024
%H A181318 G. C. Greubel, <a href="/A181318/b181318.txt">Table of n, a(n) for n = 0..10000</a>
%H A181318 Wikipedia, <a href="https://en.wikipedia.org/wiki/Completing_the_square">Completing the square</a>.
%H A181318 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,-3,0,0,0,1).
%F A181318 a(2*n) = A168077(n), a(2*n+1) = A016754(n).
%F A181318 a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
%F A181318 G.f.: x*(1 + x + 9*x^2 + x^3 + 22*x^4 + 6*x^5 + 22*x^6 + x^7 + 9*x^8 + x^9 + x^10)/(1-x^4)^3. - _R. J. Mathar_, Mar 10 2011
%F A181318 From _Peter Bala_, Feb 19 2019: (Start)
%F A181318 a(n) = numerator(n^2/(n^2 + 16)) = n^2/(gcd(n^2,16)) = (n/gcd(n,4))^2.
%F A181318 a(n) = n^2/b(n), where b(n) = [1, 4, 1, 16, 1, 4, 1, 16, ...] is a purely periodic sequence of period 4.
%F A181318 a(n) is a quasi-polynomial in n: a(4*n) = n^2; a(4*n + 1) = (4*n + 1)^2; a(4*n + 2) = (2*n + 1)^2; a(4*n + 3) = (4*n + 3)^2.
%F A181318 O.g.f.: Sum_{d divides 4} A046970(d)*x^d*(1 + x^d)/(1 - x^d)^3 = x*(1 + x)/(1 - x)^3 - 3*x^2*(1 + x^2)/(1 - x^2)^3 - 3*x^4*(1 + x^4)/(1 - x^4)^3. (End)
%F A181318 Sum_{n>=1} 1/a(n) = 5*Pi^2/12. - _Amiram Eldar_, Aug 12 2022
%F A181318 From _Amiram Eldar_, Nov 25 2022: (Start)
%F A181318 Multiplicative with a(2^e) = 4^max(0, e-2), and a(p^e) = p^(2*e) for p > 2.
%F A181318 Dirichlet g.f.: zeta(s-2)*(1 - 3/2^s - 3/4^s).
%F A181318 Sum_{k=1..n} a(k) ~ (37/192) * n^3. (End)
%F A181318 a(n) = (37 - 27*(-1)^n - 3*(-1)^(n*(n-1)/2) - 3*(-1)^(n*(n+1)/2)) * n^2/64. - _Vaclav Kotesovec_, Nov 14 2024
%p A181318 a:=n->n^2/gcd(n,4)^2: seq(a(n),n=0..60); # _Muniru A Asiru_, Feb 20 2019
%t A181318 Table[n^2/GCD[n,4]^2, {n,0,100}] (* _G. C. Greubel_, Sep 19 2018 *)
%t A181318 LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{0,1,1,9,1,25,9,49,4,81,25,121},60] (* _Harvey P. Dale_, Jan 18 2025 *)
%o A181318 (PARI) a(n)=n^2/gcd(n,4)^2 \\ _Charles R Greathouse IV_, Dec 21 2011
%o A181318 (Magma) [n^2/GCD(n,4)^2: n in [0..100]]; // _G. C. Greubel_, Sep 19 2018
%o A181318 (Sage) [n^2/gcd(n, 4)^2 for n in (0..100)] # _G. C. Greubel_, Feb 20 2019
%Y A181318 Cf. A181829, A046970, A007434, A060819, A168077, A377851.
%K A181318 nonn,mult,easy
%O A181318 0,4
%A A181318 _Paul Curtz_, Jan 26 2011
%E A181318 Edited by _Jean-François Alcover_, Oct 01 2012 and Jan 15 2013
%E A181318 More terms from _Michel Marcus_, Jun 09 2014