A351354 Numbers k such that the k-th centered 40-gonal numbers (A195317) is a square.

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

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

Table of n, a(n) for n=1..26.

Index entries for linear recurrences with constant coefficients, signature (1,18,-18,-1,1).

Formula

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

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).

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

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

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

Example

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

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

Sequence in context: A359046 A041349 A041016 * A365572 A301324 A184339

Adjacent sequences: A351351 A351352 A351353 * A351355 A351356 A351357

Keyword

nonn

Author

Lamine Ngom, Feb 08 2022

Status

approved