A217855 Numbers m such that 16*m*(3*m+1)+1 is a square.

0, 2, 30, 420, 5852, 81510, 1135290, 15812552, 220240440, 3067553610, 42725510102, 595089587820, 8288528719380, 115444312483502, 1607931846049650, 22395601532211600, 311930489604912752, 4344631252936566930, 60512907051507024270, 842836067468161772852

Offset

0,2

Comments

If S(h) is the sum of the squares of the integers from 1 to h, then S(h)/h = (h+1)*(2*h+1)/6. If S(h)/h is a square, then h+1 is even and 2*h+1 is a multiple of 3, so that (h+1)/2 and (2*h+1)/3 must be odd squares. The numbers h that satisfy this condition are in A084231. Let h = 1+24*k+72*k^2; for any natural k we have (h+1)/2 = (6*k+1)^2, but only if 16*k*(3*k+1)+1 is a square also (2*h+1)/3 is a square. Searching what integer numbers k satisfy this condition leads to this sequence.

Links

Bruno Berselli, Table of n, a(n) for n = 0..500

Forum Rudi Mathematici, Un po' di calcoli...un po' di logica, post #923 (in Italian)

Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Formula

a(n) = ((2 + sqrt(3))^(2*n + 1) + (2 - sqrt(3))^(2*n + 1) - 4)/24.

a(n) = (cosh((2*n + 1)*log(2 + sqrt(3))) - 2)/12.

a(n) = a(-n-1) = 15*a(n-1) - 15*a(n-2) + a(n-3).

a(n) = 14*a(n-1) - a(n-2) + 2.

a(n) = 2*A076139(n).

72*a(n)^2 + 24*a(n)+1 = A084231(n+1).

16*a(n)*(3*a(n) + 1) + 1 = A028230(n+1)^2.

G.f.: 2*x/((1 - x)*(1 - 14*x + x^2)). - Bruno Berselli, Oct 16 2012

a(n) = 2*A076139(n) = 1/2*A123480(n) = 1/4*A045899(n) = 2/3*A076140(n). - Peter Bala, Dec 31 2012

Example

a(0) = ((2+sqrt(3))^1+(2-sqrt(3))^1-4)/24 = 0,
a(1) = ((2+sqrt(3))^3+(2-sqrt(3))^3-4)/24 = 2,
a(2) = ((2+sqrt(3))^5+(2-sqrt(3))^5-4)/24 = 30;
a(2) = 14*a(1)-a(0)+2 = 30,
a(3) = 14*a(2)-a(1)+2 = 420,
a(4) = 14*a(3)-a(2)+2 = 5852;
a(3) = 15*a(2)-15*a(1)+a(0) = 420,
a(4) = 15*a(3)-15*a(2)+a(1) = 5852.
Regarding to the comment, a(3) = 420 and so 72*a(3)^2+24*a(3)+1 = A084231(4) = 12710881, therefore Sum_{i=1..12710881} i^2/12710881 = 7338631^2 = A084232(3)^2. - Bruno Berselli, Oct 17 2012

Mathematica

LinearRecurrence[{15, -15, 1}, {0, 2, 30}, 20] (* Bruno Berselli, Oct 16 2012 *)

Prog

(Maxima) makelist(expand(((2+sqrt(3))^(2*n+1)+(2-sqrt(3))^(2*n+1)-4)/24), n, 0, 19); /* Bruno Berselli, Oct 16 2012 */
(Magma) m:=19; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2/((1-x)*(1-14*x+x^2)))); // Bruno Berselli, Oct 16 2012
(PARI) a=vector(20); a[1]=0; a[2]=2; a[3]=30; for(i=4, #a, a[i]=15*a[i-1]-15*a[i-2]+a[i-3]); a \\ Bruno Berselli, Oct 16 2012

Crossrefs

Cf. A028230, A076139, A084231, A084232. A045899, A076140, A123480.

Sequence in context: A211906 A077517 A060042 * A285298 A134362 A296980

Adjacent sequences: A217852 A217853 A217854 * A217856 A217857 A217858

Keyword

nonn,easy

Author

Arnaldo Vicentini, Oct 12 2012

Status

approved