%I #20 Mar 23 2022 13:07:21

%S 1,3,7,45,117,799,2091,14329,37513,257115,673135,4613733,12078909,

%T 82790071,216747219,1485607537,3889371025,26658145587,69791931223,

%U 478361013021,1252365390981,8583840088783,22472785106427,154030760585065,403257766524697,2763969850442379

%N Numbers k such that the k-th centered 40-gonal numbers (A195317) is a square.

%C Corresponding square roots are listed in A351353.

%C 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).

%C In addition, the ratio a(2*n+1)/a(2*n) converges to 2.618033988 ... = golden ratio squared: A104457.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,18,-18,-1,1).

%F a(n) = A077259(n-1) + 1.

%F a(1)=1, a(2)=3, a(3)=7, a(4)=45, a(5)=117 and a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5).

%F gcd(a(2*n+1), a(2*n)) = A000045(n)*(A000032(2*n) - 1)/2, if n is odd.

%F gcd(a(2*n+1), a(2*n)) = A000032(n)*(A000032(2*n) - 1)/2, if n is even.

%F A195317(a(n)) = A000032(A007310(n))^2 = A351353(n)^2.

%e 45 is in the sequence because the 45th centered 40-gonal number is 39601, which is a square: 199^2 = A000032(11)^2.

%e 799 is in the sequence because the 799th centered 40-gonal number is 12752041, which is a square: 3571^2 = A000032(17)^2.

%p a[1] := 1: a[2] := 3: a[3] := 7: a[4] := 45: a[5] := 117:

%p 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:

%p seq(a[n], n = 1 .. 30);

%t LinearRecurrence[{1, 18, -18, -1, 1}, {1, 3, 7, 45, 117}, 30] (* _Amiram Eldar_, Feb 08 2022 *)

%Y Cf. A000032, A000045, A007310, A077259, A104457, A195317, A351353.

%K nonn

%O 1,2

%A _Lamine Ngom_, Feb 08 2022