|
|
A321214
|
|
a(n) = ((2 + sqrt(5))^p + (2 - sqrt(5))^p - 2^(p+1))/p where p = prime(n).
|
|
1
|
|
|
5, 20, 260, 3460, 716100, 10877380, 2678663940, 43007216580, 11439823225220, 52423583379994820, 880012516784503300, 4260164250933079388740, 1237929447780495036788100, 21180545285375859022020420, 6239638330555928133105753860
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
This is an integer sequence. For odd primes p, (2 + sqrt(5))^p + (2 - sqrt(5))^p - 2^(p+1) = binomial(p, 2)*2^(p-1)*5 + binomial(p, 4)*2^(p-3)*5^2 + ... + binomial(p, p-1)*2^2*5^((p-1)/2), and p divides binomial(p, k) for 1 <= k <= p - 1.
For n > 1, a(n) is divisible by 20.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k=1..(p-1)/2} (binomial(p, 2*k)/p)*2^(p-2*k+1)*5^k with p = A000040(n), for n > 1.
|
|
MATHEMATICA
|
Table[Floor[(2+Sqrt[5])^(Prime[n]) + (2-Sqrt[5])^(Prime[n]) - 2^(Prime[n]+1)]/Prime[n], {n, 1, 10}]
|
|
PROG
|
(PARI) a(n) = my(p=prime(n)); (floor((2*quadgen(5)+1)^p+(-2*quadgen(5)+3)^p+.) - 2^(p+1))/p; \\ Michel Marcus, Nov 04 2018
(PARI) a(n) = my(p=prime(n)); (([1, 1; 1, 0]^(3*p)*[1; 2])[2, 1] - 2^(p+1))/p \\ Jianing Song, Dec 22 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|