

A351354


Numbers k such that the kth centered 40gonal numbers (A195317) is a square.


1



1, 3, 7, 45, 117, 799, 2091, 14329, 37513, 257115, 673135, 4613733, 12078909, 82790071, 216747219, 1485607537, 3889371025, 26658145587, 69791931223, 478361013021, 1252365390981, 8583840088783, 22472785106427, 154030760585065, 403257766524697, 2763969850442379
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Corresponding square roots are listed in A351353.
3 and 7 are the unique primes in this sequence, a(2*n+1) and a(2*n) always sharing common factors that are closely linked to Fibonacci (A000045) and Lucas (A000032) numbers (detailed in formula section).
In addition, the ratio a(2*n+1)/a(2*n) converges to 2.618033988 ... = golden ratio squared: A104457.


LINKS



FORMULA

a(1)=1, a(2)=3, a(3)=7, a(4)=45, a(5)=117 and a(n) = a(n1) + 18*a(n2)  18*a(n3)  a(n4) + a(n5).
gcd(a(2*n+1), a(2*n)) = A000032(n)*(A000032(2*n)  1)/2, if n is even.


EXAMPLE

45 is in the sequence because the 45th centered 40gonal number is 39601, which is a square: 199^2 = A000032(11)^2.
799 is in the sequence because the 799th centered 40gonal number is 12752041, which is a square: 3571^2 = A000032(17)^2.


MAPLE

a[1] := 1: a[2] := 3: a[3] := 7: a[4] := 45: a[5] := 117:
for n from 6 to 30 do a[n] := a[n  1] + 18*a[n  2]  18*a[n  3]  a[n  4] + a[n  5]: od:
seq(a[n], n = 1 .. 30);


MATHEMATICA

LinearRecurrence[{1, 18, 18, 1, 1}, {1, 3, 7, 45, 117}, 30] (* Amiram Eldar, Feb 08 2022 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



