A361990 Numbers that are both the concatenation of a Fibonacci number and a square and the concatenation of a square and a Fibonacci number.

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

Comments

Leading 0's are not allowed, so the first number concatenated cannot be 0.

Sequence based on a suggestion by ChatGPT.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..93

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))):
                                c.add(n)
        yield from sorted(c)
A361990_list = list(islice(A361990_gen(), 30)) # Chai Wah Wu, Apr 05 2023

Crossrefs

Cf. A000045, A000290.

Sequence in context: A063697 A058943 A222473 * A335801 A363835 A041217

Adjacent sequences: A361987 A361988 A361989 * A361991 A361992 A361993

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