OFFSET
1,1
COMMENTS
For a blocky m, a starting k^2 in the required run of squares is obtained by taking k = a - b - (m-1)/2, where a*b = (m^2 - 1)/48.
Positive integers k such that hypergeometric([k/8, (8-k)/8], [1/2], 3/4) = 2*cos(Pi/4). - Artur Jasinski, Oct 30 2008
Numbers > 1 that are congruent to {1, 7} mod 24. - David Lovler, Aug 10 2022
LINKS
David Lovler, Table of n, a(n) for n = 1..10000
S. Marivani and others, Problem 11227: Consecutive Squares with a Square Average, Amer. Math. Monthly, 115, No. 6, 2008, 568-569.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
n is blocky if and only if n>1 and n (mod 24) = 1 or -1 or 7 or -7.
a(n) = 8*(s-1)+1 for n odd, a(n) = 8*(s-1)+7 for n even. - Artur Jasinski, Oct 30 2008
From R. J. Mathar, Nov 25 2008: (Start)
G.f.: x*(7+18*x-x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2) + 24 = 12n - 2 + 3*(-1)^n. (End)
a(n) = a(n-1) + a(n-2) - a(n-3). - Colin Barker, May 12 2012
E.g.f.: (12*x - 2)*exp(x) + 3*exp(-x) - 1. - David Lovler, Aug 09 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - (1+sqrt(2)/2)*Pi/12 - arccoth(sqrt(3))/(2*sqrt(3)) - arcsinh(sqrt(2))/(2*sqrt(6)). - Amiram Eldar, Aug 23 2022
EXAMPLE
7 is blocky because ((-3)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2)/7 = 28/7 = 4 = 2^2.
MAPLE
a:=proc(n) if `mod`(n, 24)=1 or `mod`(n, 24)=-1 or `mod`(n, 24)=7 or `mod`(n, 24) =-7 then n else end if end proc: seq(a(n), n=2..600);
MATHEMATICA
Table[12*n - 2 + 3*(-1)^n, {n, 1, 50}] (* Vaclav Kotesovec, Nov 14 2017 *)
PROG
(PARI) a(n) = 12*n - 2 + 3*(-1)^n \\ David Lovler, Aug 09 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 04 2008
STATUS
approved