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A140716 Blocky integers, i.e., integers m > 1 such that there is a run of m consecutive integer squares the average of which is a square.

%I A140716 #41 Aug 31 2022 14:31:44
%S A140716 7,25,31,49,55,73,79,97,103,121,127,145,151,169,175,193,199,217,223,
%T A140716 241,247,265,271,289,295,313,319,337,343,361,367,385,391,409,415,433,
%U A140716 439,457,463,481,487,505,511,529,535,553,559,577,583
%N A140716 Blocky integers, i.e., integers m > 1 such that there is a run of m consecutive integer squares the average of which is a square.
%C A140716 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.
%C A140716 Positive integers k such that hypergeometric([k/8, (8-k)/8], [1/2], 3/4) = 2*cos(Pi/4). - _Artur Jasinski_, Oct 30 2008
%C A140716 Numbers > 1 that are congruent to {1, 7} mod 24. - _David Lovler_, Aug 10 2022
%H A140716 David Lovler, <a href="/A140716/b140716.txt">Table of n, a(n) for n = 1..10000</a>
%H A140716 S. Marivani and others, <a href="http://www.jstor.org/stable/27642547">Problem 11227: Consecutive Squares with a Square Average</a>, Amer. Math. Monthly, 115, No. 6, 2008, 568-569.
%H A140716 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A140716 n is blocky if and only if n>1 and n (mod 24) = 1 or -1 or 7 or -7.
%F A140716 a(n) = 8*(s-1)+1 for n odd, a(n) = 8*(s-1)+7 for n even. - _Artur Jasinski_, Oct 30 2008
%F A140716 From _R. J. Mathar_, Nov 25 2008: (Start)
%F A140716 G.f.: x*(7+18*x-x^2)/((1+x)*(1-x)^2).
%F A140716 a(n) = a(n-2) + 24 = 12n - 2 + 3*(-1)^n. (End)
%F A140716 a(n) = a(n-1) + a(n-2) - a(n-3). - _Colin Barker_, May 12 2012
%F A140716 E.g.f.: (12*x - 2)*exp(x) + 3*exp(-x) - 1. - _David Lovler_, Aug 09 2022
%F A140716 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
%e A140716 7 is blocky because ((-3)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2)/7 = 28/7 = 4 = 2^2.
%p A140716 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);
%t A140716 Table[12*n - 2 + 3*(-1)^n, {n, 1, 50}] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%o A140716 (PARI) a(n) = 12*n - 2 + 3*(-1)^n \\ _David Lovler_, Aug 09 2022
%Y A140716 Cf. A146502-A146522. - _Artur Jasinski_, Oct 30 2008
%K A140716 nonn,easy
%O A140716 1,1
%A A140716 _Emeric Deutsch_, Jun 04 2008