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A351354
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Numbers k such that the k-th centered 40-gonal numbers (A195317) is a square.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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)) = A000032(n)*(A000032(2*n) - 1)/2, if n is even.
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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LinearRecurrence[{1, 18, -18, -1, 1}, {1, 3, 7, 45, 117}, 30] (* Amiram Eldar, Feb 08 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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