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a(n) = floor(n*r) + floor((n-2)*r) + floor((n-4)*r) + ... + floor(k*r), where r = golden ratio = (1 + sqrt(5))/2 and k = n mod 2.
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%I #11 May 12 2024 15:35:03

%S 0,1,3,5,9,13,18,24,30,38,46,55,65,76,87,100,112,127,141,157,173,190,

%T 208,227,246,267,288,310,333,356,381,406,432,459,487,515,545,574,606,

%U 637,670,703,737,772,808,844,882,920,959,999,1039,1081,1123,1166,1210

%N a(n) = floor(n*r) + floor((n-2)*r) + floor((n-4)*r) + ... + floor(k*r), where r = golden ratio = (1 + sqrt(5))/2 and k = n mod 2.

%H Paolo Xausa, <a href="/A129230/b129230.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = Sum_{k=0..n} A129229(k).

%t Module[{nn=60,fgm},fgm=Floor[GoldenRatio*Range[nn]];Union[Flatten[ Table[ {Total[ Take[fgm,{1,n,2}]],Total[Take[fgm,{2,n,2}]]},{n,nn}]]]] (* _Harvey P. Dale_, Feb 03 2016 *)

%t Accumulate[Block[{n = 0}, NestList[Floor[++n*GoldenRatio] - # &, 0, 100]]] (* _Paolo Xausa_, May 12 2024 *)

%Y Cf. A129229.

%K nonn

%O 0,3

%A _Clark Kimberling_, Apr 04 2007