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A374490 Greatest common divisor of sums of n consecutive cubes.

%I A374490 #44 Jul 11 2024 13:23:40
%S A374490 1,1,9,4,5,9,7,8,27,5,11,36,13,7,45,16,17,27,19,20,63,11,23,72,25,13,
%T A374490 81,28,29,45,31,32,99,17,35,108,37,19,117,40,41,63,43,44,135,23,47,
%U A374490 144,49,25,153,52,53,81,55,56,171,29,59,180,61
%N A374490 Greatest common divisor of sums of n consecutive cubes.
%C A374490 A quasipolynomial of order 12 and degree 2. - _Charles R Greathouse IV_, Jul 11 2024
%H A374490 Charles R Greathouse IV, <a href="/A374490/b374490.txt">Table of n, a(n) for n = 1..10000</a>
%H A374490 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer-valued_polynomial">Integer-valued polynomial</a>
%H A374490 Kaleb Williams, <a href="https://drive.google.com/file/d/1gbZ_2EohAXhPuI7-E0ZCUmpB__YwZJDn/view?usp=drivesdk">Proof that 3*n^3 + 15*n is divisible 9 by induction</a>
%H A374490 <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).
%F A374490 From _Stefano Spezia_, Jul 10 2024: (Start)
%F A374490 G.f.: x*(1 + x + 9*x^2 + 4*x^3 + 5*x^4 + 9*x^5 + 7*x^6 + 8*x^7 + 27*x^8 + 5*x^9 + 11*x^10 + 36*x^11 + 11*x^12 + 5*x^13 + 27*x^14 + 8*x^15 + 7*x^16 + 9*x^17 + 5*x^18 + 4*x^19 + 9*x^20 + x^21 + x^22)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2*(1 - x + x^2)^2*(1 + x + x^2)^2*(1 - x^2 + x^4)^2).
%F A374490 a(A359380(n)) = A359380(n). (End)
%F A374490 a(n) = n/2 if n is 2 or 10 mod 12; a(n) = 3n if n is 0, 3, or 9 mod 12; a(n) = 3n/2 if n = 6 mod 12; and a(n) = n otherwise (if n is 1, 4, 5, 7, 8, or 11 mod 12). In particular, n/2 <= a(n) <= 3n. - _Charles R Greathouse IV_, Jul 11 2024
%e A374490 For n=3, the sum of 3 consecutive cubes is S(x) = x^3 + (x+1)^3 + (x+2)^3 which has S(0) = 9 and thereafter remains a multiple of 9 since S(x) - S(x-1) = 9*(x^2 + x + 1), so that the GCD of all S(x) is a(3) = 9.
%o A374490 (PARI) f(n,x='x)=n*x^3 + (3/2*n^2 - 3/2*n)*x^2 + (n^3 - 3/2*n^2 + 1/2*n)*x + (1/4*n^4 - 1/2*n^3 + 1/4*n^2)
%o A374490 Polya(P)=my(x=variable(P),D=poldegree(P),f=D!,t=0); forstep(d=D,0,-1, my(c=polcoef(P,d,x)*d!); P-=c*binomial(x,d); t=gcd(t,c); f/=max(d,1)); t
%o A374490 a(n)=Polya(f(n)) \\ _Charles R Greathouse IV_, Jul 09 2024
%Y A374490 Cf. A026741 (for consecutive integers), A060789 (for consecutive squares).
%Y A374490 Cf. A359380.
%K A374490 nonn,easy
%O A374490 1,3
%A A374490 _Kaleb Williams_, Jul 09 2024
%E A374490 a(41)-a(61) from _Charles R Greathouse IV_, Jul 09 2024