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A362434 Numbers that can be written as A000045(i) + j^2 for i,j>=0 in 4 ways.

%I A362434 #29 May 20 2025 12:43:53
%S A362434 17,5185,1669265,537497857,173072640401,55728852710977,
%T A362434 17944517500293905,5778078906241926145,1860523463292399924497,
%U A362434 599082777101246533761601,192902793703138091471310737,62114100489633364207228295425,20000547454868240136636039815825,6440114166367083690632597592399937
%N A362434 Numbers that can be written as A000045(i) + j^2 for i,j>=0 in 4 ways.
%C A362434 A000045(1) and A000045(2) are counted separately, even though they both are 1.
%F A362434 With a = A000045(6*k-1) and b = A000045(6*k) and x = 1 + b^2/4, we have
%F A362434   x = A000045(1) + (b/2)^2
%F A362434     = A000045(2) + (b/2)^2
%F A362434     = A000045(6*k) + (b/2 - 1)^2
%F A362434     = A000045(12*k-6) + (3*b/2 - a)^2.
%F A362434 Conjecture: a(n) = 1 + A000045(6*n)^2/4.
%e A362434 17 = A000045(1) + 4^2 = A000045(2) + 4^2 = A000045(6) + 3^2 = A000045(7) + 2^2.
%e A362434 5185 = A000045(1) + 72^2 = A000045(2) + 72^2 = A000045(12) + 71^2 = A000045(18) + 51^2.
%e A362434 1669265 = A000045(1) + 1292^2 = A000045(2) + 1292^2 = A000045(18) + 1291^2 = A000045(30) + 915^2.
%e A362434 537497857 = A000045(1) + 23184^2 = A000045(2) + 23184^2 = A000045(24) + 23183^2 = A000045(42) + 16419^2.
%p A362434 N:= 10^6: # to get terms <= N
%p A362434 V:= Array(0..N, datatype=integer[1]):
%p A362434 for i from 0 do
%p A362434  f:= combinat:-fibonacci(i);
%p A362434  if f > N then break fi;
%p A362434  s:= floor(sqrt(N-f));
%p A362434  J:=[seq(f+i^2, i=0..s)];
%p A362434  V[J]:= V[J] +~ 1;
%p A362434 od:
%p A362434 select(i -> V[i] >= 4, [$1..N]);
%o A362434 (PARI) A362503(n) = my(f, s=0); for(i=0, oo, if(n<f=fibonacci(i), return(s), if(issquare(n-f), s++)));
%o A362434 lista(nn) = my(v=List([]), x, y, t); for(i=3, log(sqrt(5)*nn+1.5)\log((1+sqrt(5))/2), x=fibonacci(i); for(j=1, i-1, y=x-fibonacci(j); fordiv(y, d, if(d>sqrtint(y), break); t=y/d-d; if(t%2==0, for(k=0, j-1, if(issquare(t^2+4*(x-fibonacci(k))), listput(v, x+t^2/4))))))); v=Set(v); for(i=1, #v, if(v[i]>nn, break); if(A362503(v[i])==4, print1(v[i], ", "))); \\ _Jinyuan Wang_, Apr 24 2023
%Y A362434 Cf. A000045, A362409, A362503.
%K A362434 nonn
%O A362434 1,1
%A A362434 _Robert Israel_, Apr 21 2023
%E A362434 More terms from _Jinyuan Wang_, Apr 23 2023