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A361990
Numbers that are both the concatenation of a Fibonacci number and a square and the concatenation of a square and a Fibonacci number.
1
10, 11, 134, 1144, 1440, 1441, 1961, 8121, 14489, 54761, 116641, 144144, 148841, 168121, 313689, 1964181, 3249001, 14932841, 21436921, 21622521, 23164841, 84272489, 89870489, 176475025, 312033961, 591948921, 1326416489, 1392872041, 1493772841, 1877996161, 2120602521, 2129822521, 2165971689
OFFSET
1,1
Leading 0's are not allowed, so the first number concatenated cannot be 0.
Sequence based on a suggestion by ChatGPT.
EXAMPLE
a(3) = 134 is a term because it is the concatenation of A000045(7) = 13 and 2^2 = 4, and also the concatenation of 1^2 = 1 and A000045(9) = 34.
MAPLE
icat:= proc(n, m) if m = 0 then n*10 else n*10^(1+ilog10(m))+m fi end proc:
for i from 1 while length(combinat:-fibonacci(i))<9 do od:
f8:= [seq(combinat:-fibonacci(n), n=2..i-1)]:
s8:= [seq(i^2, i=1..9999)]:
f0:= [0, op(f8)]: s0:= {0, op(s8)}:
S1:= select(t -> t < 10^9, {seq(seq(icat(a, b), a=f8), b=s0)}):
S2:= select(t -> t < 10^9, {seq(seq(icat(a, b), a=s8), b=f0)}):
sort(convert(S1 intersect S2, list));
PROG
(Python)
from math import isqrt
from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import fibonacci
def A361990_gen(): # generator of terms
for l in count(2):
c = set()
for i in range(1, isqrt(10**(l-1)-1)+1):
i2 = i**2
k = 10**(l-len(str(i2))-1)
for j in count(0):
f = int(fibonacci(j))
if f>=10*k:
break
if (f==0 and k==1) or f>=k:
n = i2*10*k+f
for w in range(1, len(str(n))):
w2 = 10**(w-1)
a, b = divmod(n, w2*10)
if w==1 or b>=w2:
if (is_square(b) and (is_square(r:=5*a**2-4) or is_square(r+8))):
yield from sorted(c)
A361990_list = list(islice(A361990_gen(), 30)) # Chai Wah Wu, Apr 05 2023
CROSSREFS
Sequence in context: A063697 A058943 A222473 * A335801 A363835 A041217
KEYWORD
nonn,base
AUTHOR
Robert Israel, Apr 02 2023
EXTENSIONS
Although this was originally suggested by an AI program, it has been fully checked by the OEIS Editors - N. J. A. Sloane, Oct 19 2023
STATUS
approved

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Last modified September 20 03:21 EDT 2024. Contains 376016 sequences. (Running on oeis4.)