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A269449 The first of 33 consecutive positive integers the sum of the squares of which is a square.

%I A269449 #24 Oct 18 2020 16:54:41
%S A269449 7,27,60,181,227,612,1085,1985,3492,9047,11161,28860,50607,91987,
%T A269449 161276,416685,513883,1327652,2327541,4230121,7415908,19159167,
%U A269449 23628161,61043836,107016983,194494283,340971196,880905701,1086382227,2806689508,4920454381,8942507601
%N A269449 The first of 33 consecutive positive integers the sum of the squares of which is a square.
%C A269449 Positive integers y in the solutions to 2*x^2-66*y^2-2112*y-22880 = 0.
%C A269449 All sequences of this type (i.e. sequences with fixed offset k, and a discernible pattern: k=0...32 for this sequence, k=0..1 for A001652, k=0...10 for A106521) can be extended using a formula such as x(n) = a*x(n-p) - x(n-2p) + b, where a and b are various constants, and p is the period of the series. Alternatively 'p' can be considered the number of concurrent series. - _Daniel Mondot_, Aug 08 2016
%C A269449 Numbers x such that 11440+33*x*(32+x)is a square. - _Harvey P. Dale_, Oct 18 2020
%H A269449 Colin Barker, <a href="/A269449/b269449.txt">Table of n, a(n) for n = 1..1000</a>
%H A269449 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,46,-46,0,0,0,0,-1,1).
%F A269449 G.f.: x*(7 +20*x +33*x^2 +121*x^3 +46*x^4 +385*x^5 +151*x^6 -20*x^7 -11*x^8 -11*x^9 -2*x^10 -11*x^11 -4*x^12) / ((1 -x)*(1 -46*x^6 +x^12)).
%F A269449 a(1)=7, a(2)=27, a(3)=60, a(4)=181, a(5)=227, a(6)=612, a(7)=1085, a(8)=1985, a(9)=3492, a(10)=9047, a(11)=11161, a(12)=28860, a(n)=46*a(n-6)-a(n-12)+704. - _Daniel Mondot_, Aug 08 2016
%e A269449 7 is in the sequence because sum(k=7, 39, k^2) = 20449 = 143^2.
%t A269449 Rest@ CoefficientList[Series[x (7 + 20 x + 33 x^2 + 121 x^3 + 46 x^4 + 385 x^5 + 151 x^6 - 20 x^7 - 11 x^8 - 11 x^9 - 2 x^10 - 11 x^11 - 4 x^12)/((1 - x) (1 - 46 x^6 + x^12)), {x, 0, 32}], x] (* _Michael De Vlieger_, Aug 08 2016 *)
%t A269449 LinearRecurrence[{1,0,0,0,0,46,-46,0,0,0,0,-1,1},{7,27,60,181,227,612,1085,1985,3492,9047,11161,28860,50607},50] (* _Harvey P. Dale_, Oct 18 2020 *)
%o A269449 (PARI) Vec(x*(7 +20*x +33*x^2 +121*x^3 +46*x^4 +385*x^5 +151*x^6 -20*x^7 -11*x^8 -11*x^9 -2*x^10 -11*x^11 -4*x^12) / ((1 -x)*(1 -46*x^6 +x^12)) + O(x^40))
%Y A269449 Cf. A001032, A001652, A094196, A106521, A257767, A269447, A269448, A269451.
%K A269449 nonn,easy
%O A269449 1,1
%A A269449 _Colin Barker_, Feb 27 2016