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A104257 Square array T(a,n) read by antidiagonals: replace 2^i with a^i in binary representation of n, where a,n>=2. 17

%I

%S 2,3,3,4,4,4,5,5,9,5,6,6,16,10,6,7,7,25,17,12,7,8,8,36,26,20,13,8,9,9,

%T 49,37,30,21,27,9,10,10,64,50,42,31,64,28,10,11,11,81,65,56,43,125,65,

%U 30,11,12,12,100,82,72,57,216,126,68,31,12,13,13,121,101,90,73,343

%N Square array T(a,n) read by antidiagonals: replace 2^i with a^i in binary representation of n, where a,n>=2.

%C Sums of distinct powers of a. Numbers having only {0,1} in a-ary representation.

%F T(a, n) = Sum{j>=1, [(n+2^(j-1))/2^j] * ((a-2)*a^(j-1) + 1) }/(a-1).

%F T(a, n) = Sum{j=1..n, ((a-2)*a^A007814(j) + 1)/(a-1) }.

%F G.f. of a-th row: 1/(1-x) * Sum{k>=0, a^k*x^2^k/(1+x^2^k) }.

%F Recurrence: T(a, 2n) = aT(a, n), T(a, 2n+1) = aT(a, n) + 1, T(a, 0)=0.

%e 2,3,4,5, 6, 7, 8, 9,

%e 3,4,9,10,12,13,27,28,

%e 4,5,16,17,20,21,64,65,

%e 5,6,25,26,30,31,125,126,

%e 6,7,36,37,42,43,216,217,

%e 7,8,49,50,56,57,343,344,

%e 8,9,64,65,72,73,512,513,

%e 9,10,81,82,90,91,729,730,

%Y Rows include (essentially) A005836, A000695, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A033050, A033051, A033052.

%Y Columns include A000290, A002522, A002378, A000578, A001093, A034262, A071568, A011379, A098547, A027444.

%Y Main diagonal is A104258.

%K nonn,tabl

%O 2,1

%A _Ralf Stephan_, Mar 05 2005

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Last modified October 16 13:12 EDT 2018. Contains 316263 sequences. (Running on oeis4.)