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

17, 5185, 1669265, 537497857, 173072640401, 55728852710977, 17944517500293905, 5778078906241926145, 1860523463292399924497, 599082777101246533761601, 192902793703138091471310737, 62114100489633364207228295425, 20000547454868240136636039815825, 6440114166367083690632597592399937

Offset

1,1

Comments

A000045(1) and A000045(2) are counted separately, even though they both are 1.

Links

Table of n, a(n) for n=1..14.

Formula

With a = A000045(6*k-1) and b = A000045(6*k) and x = 1 + b^2/4, we have

x = A000045(1) + (b/2)^2

= A000045(2) + (b/2)^2

= A000045(6*k) + (b/2 - 1)^2

= A000045(12*k-6) + (3*b/2 - a)^2.

Conjecture: a(n) = 1 + A000045(6*n)^2/4.

Example

17 = A000045(1) + 4^2 = A000045(2) + 4^2 = A000045(6) + 3^2 = A000045(7) + 2^2.
5185 = A000045(1) + 72^2 = A000045(2) + 72^2 = A000045(12) + 71^2 = A000045(18) + 51^2.
1669265 = A000045(1) + 1292^2 = A000045(2) + 1292^2 = A000045(18) + 1291^2 = A000045(30) + 915^2.
537497857 = A000045(1) + 23184^2 = A000045(2) + 23184^2 = A000045(24) + 23183^2 = A000045(42) + 16419^2.

Maple

N:= 10^6: # to get terms <= N
V:= Array(0..N, datatype=integer[1]):
for i from 0 do
 f:= combinat:-fibonacci(i);
 if f > N then break fi;
 s:= floor(sqrt(N-f));
 J:=[seq(f+i^2, i=0..s)];
 V[J]:= V[J] +~ 1;
od:
select(i -> V[i] >= 4, [$1..N]);

Prog

(PARI) A362503(n) = my(f, s=0); for(i=0, oo, if(n<f=fibonacci(i), return(s), if(issquare(n-f), s++)));
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

Crossrefs

Cf. A000045, A362409, A362503.

Sequence in context: A238610 A052286 A079687 * A062659 A195813 A344148

Adjacent sequences: A362431 A362432 A362433 * A362435 A362436 A362437

Keyword

nonn

Author

Robert Israel, Apr 21 2023

Extensions

More terms from Jinyuan Wang, Apr 23 2023

Status

approved