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A057945 Number of triangular numbers needed to represent n with greedy algorithm.

%I A057945 #16 Mar 10 2019 08:35:50
%S A057945 0,1,2,1,2,3,1,2,3,2,1,2,3,2,3,1,2,3,2,3,4,1,2,3,2,3,4,2,1,2,3,2,3,4,
%T A057945 2,3,1,2,3,2,3,4,2,3,4,1,2,3,2,3,4,2,3,4,3,1,2,3,2,3,4,2,3,4,3,2,1,2,
%U A057945 3,2,3,4,2,3,4,3,2,3,1,2,3,2,3,4,2,3,4,3,2,3,4,1,2,3,2,3,4,2,3,4,3,2,3,4,3
%N A057945 Number of triangular numbers needed to represent n with greedy algorithm.
%C A057945 a(n) = sum of digits of A000462(n). - _Reinhard Zumkeller_, Mar 27 2011
%C A057945 The length of (number of moves in) Simon Norton's game in A006019 starting with an initial heap of n if both players always take, never put. - _R. J. Mathar_, May 13 2016
%H A057945 Reinhard Zumkeller, <a href="/A057945/b057945.txt">Table of n, a(n) for n = 0..10000</a>
%F A057945 a(0)=0, otherwise a(n)=a(A002262(n))+1.
%e A057945 a(35)=3 since 35=28+6+1
%p A057945 A057945 := proc(n)
%p A057945     local a,x;
%p A057945     a := 0 ;
%p A057945     x := n ;
%p A057945     while x > 0 do
%p A057945         x := x-A057944(x) ;
%p A057945         a := a+1 ;
%p A057945     end do:
%p A057945     a ;
%p A057945 end proc: # _R. J. Mathar_, May 13 2016
%t A057945 A057944[n_] := With[{k = Floor[Sqrt[8n+1]]}, Floor[(k-1)/2]* Floor[(k+1)/2]/2];
%t A057945 a[n_] := Module[{k = 0, x = n}, While[x>0, x = x - A057944[x]; k++]; k];
%t A057945 Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Mar 10 2019, after _R. J. Mathar_ *)
%o A057945 (Haskell)
%o A057945 a057945 n = g n $ reverse $ takeWhile (<= n) $ tail a000217_list where
%o A057945    g 0 _      = 0
%o A057945    g x (t:ts) = g r ts + a where (a,r) = divMod x t
%o A057945 -- _Reinhard Zumkeller_, Mar 27 2011
%Y A057945 Cf. A000217, A002262, A056944, A057944. See A006893 for records.
%K A057945 nonn
%O A057945 0,3
%A A057945 _Henry Bottomley_, Oct 05 2000