|
|
A086541
|
|
a(1) = 1, a(2) = 4; a(n) = smallest square of the form k*a(n-1) + a(n-2), k > 0.
|
|
3
|
|
|
1, 4, 9, 49, 2116, 1104601, 1220041748025, 73506264463383837985201, 152589000107917580345020742323132226398704361, 1669769004292648133509475727812173973930459916514124965718261930975078855532062950740889
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The sequence is infinite. Proof: In a(n) = k*a(n-1)+a(n-2), One ( and the largest) value of k is a(n-1)-2{a(n-2)}^(1/2). which gives a(n) = {a(n-1)-{a(n-2)^(1/2)}^2.
If k is allowed to be 0, the sequence would be 1, 4, 1, 4, ... - Chai Wah Wu, Mar 27 2020
|
|
LINKS
|
|
|
EXAMPLE
|
a(3) = 4*2 +1 = 9, a(4) = 5*9 +4 = 49, a(5) = 43*49 + 9 = 2116= 46^2.
|
|
PROG
|
(PARI) A = vector(11); A[1] = 1; A[2] = 2; B = vector(11, i, A[i]^2); for (n = 3, 11, z = znstar(B[n - 1]); l = length(z[2]); c = vector(l, i, z[3][i]^(z[2][i]/2)); v = vector(2^l, i, A[n - 2]*prod(j = 1, l, c[j]^(i\2^(l - j)%2))); v = vecsort(lift(v)); print(B[n - 2], vector(2^l, i, v[i]^2%B[n - 1])); A[n] = if (v[1] == A[n - 2], v[2], v[1]); B[n] = A[n]^2; print(B[n])); \\ David Wasserman, Mar 21 2005
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|