A140972 Numbers k such that arithmetic mean of squares of first k Lucas numbers is an integer.
1, 10, 12, 24, 36, 48, 60, 72, 96, 108, 120, 144, 168, 180, 192, 216, 240, 250, 288, 300, 324, 336, 360, 384, 432, 442, 480, 504, 540, 550, 552, 576, 600, 612, 624, 648, 660, 672, 684, 720, 768, 840, 864, 900, 960, 972, 1008, 1068, 1080, 1104, 1152, 1176, 1200
Offset: 1
Examples
k = 10: (L(0)^2+...+L(9)^2)/10 = 935.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Programs
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Mathematica
With[{nn=1200},Transpose[Select[Thread[{Range[nn],Accumulate[ LucasL[ Range[0,nn-1]]^2]}],IntegerQ[Last[#]/First[#]]&]][[1]]] (* Harvey P. Dale, Jul 15 2012 *) -
PARI
list(lim) = {my(L1 = 2, L2 = 1, s = 5); print1(1, ", "); for(k = 3, lim, L3 = L1 + L2; s += L3^2; if(!(s % k), print1(k, ", ")); L1 = L2; L2 = L3);} \\ Amiram Eldar, Jul 04 2025
Formula
k such that (L(0)^2+ L(1)^2+ ... + L(k-1)^2) / k is an integer, where L(i) is the i-th Lucas number.
Extensions
a(1) inserted and a(48) onwards added by R. J. Mathar, Aug 04 2008
Comments