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A048460 Total of odd numbers in the generations from 2 onwards. 3

%I

%S 2,3,3,3,4,6,5,3,4,6,6,6,8,12,9,3,4,6,6,6,8,12,10,6,8,12,12,12,16,24,

%T 17,3,4,6,6,6,8,12,10,6,8,12,12,12,16,24,18,6,8,12,12,12,16,24,20,12,

%U 16,24,24,24,32,48,33,3,4,6,6,6,8,12,10,6,8,12,12,12,16,24,18,6,8,12,12,12,16

%N Total of odd numbers in the generations from 2 onwards.

%F It appears that a(n) = A105321(n)/2. - _Omar E. Pol_, May 29 2010. Proof from _Nathaniel Johnston_, Nov 07 2010: If you remove every 2nd row from Pascal's triangle then the rule for constructing the parity of the next row from the current row is the same as the rule for constructing generation n+1 of the primes from generation n: add up the previous and next term in the current row.

%F a((2*n-3)*2^p) = (2^(p-1)+1)*A001316(n-2), p >= 0 and n >= 2. - _Johannes W. Meijer_, Jan 22 2013

%e a(7)=6 because in generation 7 there are six odd numbers: 127,237,403,729,879,1109.

%p A048460 := proc(nmax) local par, c, r, prevc, prevl, cpar; par := [[],[1,1]] ; for c from 3 to nmax do prevc := op(-1,par) ; prevl := nops(prevc) ; if nops(prevc) < 2 then cpar := [0] ; else cpar := [op(2,prevc)] ; end if; for r from 2 to prevl-1 do cpar := [op(cpar),( op(r-1,prevc) + op(r+1,prevc)) mod 2] ; end do: cpar := [op(cpar), op(prevl-1,prevc),1] ; par := [op(par),cpar] ; end do: cpar := [] ; for c from 2 to nops(par) do add(r,r=op(c,par)) ; cpar := [op(cpar),%] ; end do: cpar ; end proc: A048460(120) ; # _R. J. Mathar_, Aug 07 2010

%p nmax := 86: A001316 := n -> if n <=- 1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 2 to nmax/(p+2) do a((2*n-3)*2^p) := (2^(p-1)+1)*A001316(n-2) od: od: seq(a(n), n=2..nmax); # _Johannes W. Meijer_, Jan 22 2013

%Y For "Generations" see A048448-A048455. See also A047844.

%Y Cf. A220466.

%K nonn

%O 2,1

%A _Patrick De Geest_, May 15 1999

%E More terms from _R. J. Mathar_, Aug 07 2010

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Last modified June 24 09:37 EDT 2019. Contains 324323 sequences. (Running on oeis4.)