A362503 -id:A362503 - cp's OEIS frontend

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

Robert Israel, Apr 21 2023

nonn

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

			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.

Cf. A000045, A362409, A362503.

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]);

A362503(n) = my(f, s=0); for(i=0, oo, if(nsqrtint(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

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.

More terms from Jinyuan Wang, Apr 23 2023

A362409 a(n) is the least number that is the sum of a Fibonacci number and a square in exactly n ways.

Original entry on oeis.org

15, 7, 3, 1, 17

Offset: 0

Robert Israel, Apr 21 2023

a(n) is the least k such that there are n pairs (i,j) of nonnegative integers such that A000045(i) + j^2 = k.
We count A000045(1) and A000045(2) separately, even though both are 1.
a(5) > 10^20, if it exists. - Martin Ehrenstein, May 01 2023

			a(0) = 15 because 15 is not the sum of a Fibonacci number and a square.
a(1) = 7 because 7 = A000045(4) + 2^2 is the sum of a Fibonacci number and a square in one way.
a(2) = 3 because 3 = A000045(3) + 1^2 = A000045(4) + 0^2.
a(3) = 1 because 1 = A000045(0) + 1^2 = A000045(1) + 0^2 = A000045(2) + 0^2.
a(4) = 17 because 17 = A000045(1) + 4^2 = A000045(2) + 4^2 = A000045(6) + 3^2 = A000045(7) + 2^2.

Cf. A000045, A362434, A362503.

N:= 10^8: # to get terms <= N
V:= Array(0..N):
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:
W:= Array(0..max(V)):
for i from 1 to N do
  w:= V[i];
  if W[w] = 0 then W[w]:= i fi
od:
convert(W,list);

cp's OEIS Frontend

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

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