A362409 -id:A362409 - 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

A362503 a(n) is the number of k such that n - A000045(k) is a square.

Original entry on oeis.org

1, 3, 3, 2, 2, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 0, 1, 4, 1, 1, 0, 2, 2, 0, 1, 2, 2, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 3, 3, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 3, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 0, 1, 0, 2, 2, 1, 1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 2, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 2

Offset: 0

Robert Israel, Apr 22 2023

nonn

Number of ways to write n as the sum of a Fibonacci number and a square, where A000045(1) and A000045(2) are counted as separate.

			a(5) = 3 because 5 = A000045(1) + 2^2 = A000045(2) + 2^2 = A000045(5) + 0^2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A108852, A362409, A362434.

N:= 100: # 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:
convert(V,list);

f(n) = my(k=1); while (fibonacci(k) <= n, k++); k; \\ A108852
a(n) = sum(k=0, f(n), issquare(n-fibonacci(k))); \\ Michel Marcus, Apr 23 2023

a(1 + A000045(6*k)^2/4) >= 4.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula

Extensions