A339057 a(n) = (-1)^(n + 1)*3^(2*n + 1)*Euler(2*n + 1, 1/3)*2^(valuation_{2}(2*(n + 1))), the Steinhaus-Euler sequence S_{3}(n).
1, 13, 121, 18581, 305071, 61203943, 4353296221, 6669149100757, 206772189255571, 128970681211645873, 24697503335329725121, 45583359018138184284551, 6235055851689626935206871, 7982707567621372702411448803, 2955418704408380517540605162821, 40101878131071637461151318174173269
Offset: 0
Keywords
Examples
The array of the general case S_{k}(n) starts:
[k]
[1] -1, -1, -1, -17, -31, -691, -5461, ... [-A002425]
[2] 0, 0, 0, 0, 0, 0, 0, ...
[3] 1, 13, 121, 18581, 305071, 61203943, 4353296221, ... [this seq.]
[4] 2, 44, 722, 196888, 5746082, 2049374444, 259141449842, ...
[5] 3, 99, 2523, 1074243, 48982293, 27296351769, 5393115879063, ...
...
Links
- Sandor Csörgö, Gordon Simons, On Steinhaus' resolution of the St. Petersburg paradox, Probab. Math. Statist. 14 (1993), 157-172. MR1321758 (96b:60017).
Programs
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Maple
GenEuler := k -> (n -> (-1)^n*(-k)^(2*n+1)*euler(2*n+1, 1/k)): Steinhaus := n -> 2^padic[ordp](2*(n+1), 2): seq(Steinhaus(n)*GenEuler(3)(n), n = 0..15);
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Mathematica
GenEuler[n_, k_] := (-1)^n (-k)^(2 n + 1) EulerE[2 n + 1, 1/k] ; Steinhaus[n_] := 2^IntegerExponent[2*(n+1), 2]; a[n_] := GenEuler[n, 3] Steinhaus[n]; Table[a[n], {n, 0, 15}]