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A079394
Compare t(n) and t(n+1) where t = Ramanujan's tau function (A000594); a(n) = p concatenated with q where p and q are given below.
0
21, 31, 21, 31, 21, 11, 31, 21, 11, 31, 22, 11, 32, 41, 42, 21, 32, 41, 22, 12, 11, 31, 41, 21, 32, 21, 32, 41, 22, 11, 11, 32, 41, 22, 31, 21, 11, 12, 31, 42, 42, 22, 11, 12, 12, 31, 42, 21, 32, 21, 32, 21, 31, 41, 22, 31, 21, 11, 12, 31, 42, 41, 41, 21, 11, 11, 32, 42, 42
OFFSET
1,1
COMMENTS
p = 1 if t(n) < 0, t(n+1) < 0; p = 2 if t(n) >= 0, t(n+1) < 0; p = 3 if t(n) < 0, t(n+1) >= 0; p = 4 if t(n) >= 0, t(n+1) >= 0.
q = 1 if |t(n)| <= |t(n+1)|; q = 2 if |t(n)| > |t(n+1)|. - Corrected by Sean A. Irvine, Aug 12 2025
EXAMPLE
A000594(1)=1, A000594(2)=-24. Hence the sign changes from + to - and the modulus increasing. Consulting the key, this gives a(1)=21.
MATHEMATICA
f[i_, j_] := Which[i < 0 && j < 0, 1, i >= 0 && j < 0, 2, i < 0 && j >= 0, 3, i >= 0 && j >= 0, 4]; g[i_, j_] := If[Abs[i] <= Abs[j], 1, 2]; h[i_, j_] := FromDigits[{f[i, j], g[i, j]}]; h @@@ Partition[Array[RamanujanTau, 70], 2, 1] (* Amiram Eldar, Jan 10 2025 *)
PROG
(PARI) T(n)=n*(n+1)/2;
rtau3(n)=local(y, j); y=0; j=1; while (T(j-1)<n, j++); j--; for (i=1, j, y=y-(-1)^i*(2*i-1)*x^(T(i-1))); y=y^8; polcoeff(y, n-1);
for (n=1, 100, r3=rtau3(n); r31=rtau3(n+1); rn=(sign(r3)+1)/2; rn1=sign(r31)+1; print1((rn+rn1+1), "", (abs(r3)>abs(r31))+1, ", "))
CROSSREFS
Cf. A000594.
Sequence in context: A116096 A116116 A115433 * A112375 A376294 A067599
KEYWORD
nonn,base
AUTHOR
Jon Perry, Jan 06 2003
EXTENSIONS
Offset corrected by Amiram Eldar, Jan 10 2025
STATUS
approved