A059837 -id:A059837 - cp's OEIS frontend

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

0, 0, 0, 2, 1, 4, 4, 6, 4, 8, 8, 10, 9, 12, 12, 14, 11, 16, 16, 18, 17, 20, 20, 22, 20, 24, 24, 26, 25, 28, 28, 30, 26, 32, 32, 34, 33, 36, 36, 38, 36, 40, 40, 42, 41, 44, 44, 46, 43, 48, 48, 50, 49, 52, 52, 54, 52, 56, 56, 58, 57, 60, 60, 62, 57, 64, 64, 66, 65, 68

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

			G.f. = 2*x^3 + x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + ... - _Michael Somos_, Jan 25 2020

a(n) = n - A007814(n) - 1 = A093048(n) - 1, n>0.

a(n) is the exponent of 2 in A001761(n+1), A002105(n), A002682(n-1), A006963(n), A036770(n-1), A059837(n), A084623(n), |A003707(n)|, |A011859(n)|.

a[ n_] := If[ n == 0, 0, n - 1 - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2-1,n-1))

{a(n) = if( n, n - 1 - valuation(n, 2))}; /* Michael Somos, Jan 25 2020 */

def A093049(n): return n-1-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n.

G.f.: sum(k>=0, t^3(t+2)/(1-t^2)^2, t=x^2^k).

A059836 Triangle T(s,t), s >= 1, 1 <= t <= s (see formula line).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 18, 1, 4, 16, 48, 144, 1, 5, 25, 100, 400, 1200, 1, 6, 36, 180, 900, 3600, 14400, 1, 7, 49, 294, 1764, 8820, 44100, 176400, 1, 8, 64, 448, 3136, 18816, 112896, 564480, 2822400, 1, 9, 81, 648, 5184, 36288, 254016, 1524096, 9144576

Offset: 1

N. J. A. Sloane, Feb 25 2001

			Triangle begins:
  1;
  1,1;
  1,2,4;
  1,3,9,18;
  ...

S. G. Mikhlin, Constants in Some Inequalities of Analysis, Wiley, NY, 1986, see p. 59.

Cf. A059837.

T := proc(s,t) option remember: if s=1 or t=1 then RETURN(1) fi: if t>1 and t mod 2 = 1 then RETURN(product((s-i)^2, i=1..(t-1)/2)) else RETURN((s-t/2)*product((s-i)^2, i=1..t/2-1)) fi: end: for s from 1 to 15 do for t from 1 to s do printf(`%d,`, T(s,t)) od:od:

T[s_, t_] := If[OddQ[t], Times @@ (s - Range[(t - 1)/2])^2, Times @@ (s - Range[t/2 - 1])^2*(s - t/2)];
Table[T[s, t], {s, 1, 15}, {t, 1, s}] // Flatten (* Jean-François Alcover, Apr 29 2023 *)

T(s, t) = (s-1)^2*(s-2)^2*...*(s-(t-1)/2)^2 if t odd, else (s-1)^2*(s-2)^2*...*(s-t/2+1)^2*(s-t/2).

More terms from James Sellers, Feb 26 2001 and from Larry Reeves (larryr(AT)acm.org), Feb 26 2001

cp's OEIS Frontend

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

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