A362434 -id:A362434 - cp's OEIS frontend

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

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.

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);

A362528 Numbers that can be written in at least 3 ways as the sum of a Lucas number (A000032) and a square.

Original entry on oeis.org

11, 27, 488, 683, 852, 907, 964, 1372, 1445, 3971, 5947, 6563, 8587, 40003, 70803, 111603, 116285, 129603, 133958, 291607, 465125, 1229884, 1555208, 2088027, 37442165, 89629867, 93896107, 149768645, 197712043, 287946964, 298391123

Offset: 1

Robert Israel, Apr 23 2023

nonn

Numbers k such that k = A000032(x) + y^2 for x, y >= 0 has at least 3 solutions.

Conjecture: there are never more than 3 solutions.

			a(1) = 11 = A000032(0) + 3^2 = A000032(4) + 2^2 = A000032(5) + 0^2.
a(2) = 27 = A000032(0) + 5^2 = A000032(5) + 4^2 = A000032(6) + 3^2.
a(3) = 488 = A000032(3) + 22^2 = A000032(8) + 21^2 = A000032(11) + 17^2.

Cf. A000032, A362434.

N:= 3*10^8: # for terms <= N
luc:= n -> combinat:-fibonacci(n-1) + combinat:-fibonacci(n+1):
S:= {}:
for x from 1 to floor(sqrt(N)) do
  s:= x^2;
  for i from 2 do
    l:= luc(i);
    if s+l > N then break fi;
    v:= f(s+l);
    if v >= 3 and not member(s+l,S) then S:= S union {s+l}; fi
od od:
sort(convert(S,list));

cp's OEIS Frontend

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

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

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A362528 Numbers that can be written in at least 3 ways as the sum of a Lucas number (A000032) and a square.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple