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A064526 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n). 13
0, 1, 2, 3, 5, 13, 49, 529, 21121, 10369921, 213952189441, 2214253468601687041, 473721461635593679669210030081, 1048939288228833101089604217183056027094304481281 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Every nonzero term is relatively prime to all others (which proves that there are infinitely many primes). See A236394 for the primes that appear.
LINKS
Michael Somos and R. Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(6) (2003), 539-540.
FORMULA
a(n) = (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)) for n >= 2.
a(n) ~ c^(phi^n), where c = 1.2364241784241086061606568429916822975882631646194967549068405592472125928485... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015
MATHEMATICA
Flatten[{0, 1, RecurrenceTable[{a[n]==(a[n-1]^2 + a[n-2]^2 - a[n-1]*a[n-2] * (1+a[n-2]))/(1-a[n-2]), a[2]==2, a[3]==3}, a, {n, 2, 15}]}] (* Vaclav Kotesovec, May 21 2015 *)
PROG
(PARI) {a(n) = local(v); if( n<3, max(0, n), v = [1, 1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[1] + v[2])}
(PARI) {a(n) = if( n<4, max(0, n), (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)))}
CROSSREFS
See A236394 for the primes that are produced.
Sequence in context: A215102 A110364 A111288 * A261192 A103594 A042695
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Oct 07 2001
STATUS
approved

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Last modified June 12 19:52 EDT 2024. Contains 373360 sequences. (Running on oeis4.)