A028355 How the astronomical clock ("Orloj") in Prague would strike 1,2,3,...,24,25,.. (digits follow 12343212343... (A028356), n-th group adds to n).
1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212, 34321, 23432, 123432, 1234321, 2343212, 3432123, 4321234, 32123432, 123432123, 43212343, 2123432123, 432123432, 1234321234, 32123432123, 43212343212
Offset: 1
Examples
1, 2, 3, 4, 3+2=5, 1+2+3=6, 4+3=7, 2+1+2+3=8, 4+3+2=9, 1+2+3+4=10, 3+2+1+2+3=11, 4+3+2+1+2=12, 3+4+3+2+1=13, 2+3+4+3+2=14, 1+2+3+4+3+2=15, ...
References
- Zdenek Horsky, "Prazsky Orloj" ["The Astronomical Clock of Prague", in Czech], Panorama, Prague, 1988, pp. 76-78.
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..2500
- Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.
- N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Programs
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Mathematica
s[i_] := {1, 2, 3, 4, 3, 2}[[Mod[i, 6, 1]]]; m[k_] := If[k == 1, 0, For[m0 = 1, True, m0++, If[k (k - 1)/2 == Sum[s[i], {i, 1, m0}], Return[m0]]]]; n[k_] := For[n0 = m[k] + 1, True, n0++, If[Sum[s[i], {i, m[k] + 1, n0}] == k, Return[n0]]]; a[k_] := a[k] = Table[s[i], {i, m[k] + 1, n[k]}] // FromDigits; Array[a, 27] (* Jean-François Alcover, Mar 14 2016 *)
Formula
Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = 1000001*a(n-15) - 1000000*a(n-30) for n > 30.
G.f.: x*(100000*x^28 + 200000*x^27 + 300000*x^26 + 400000*x^25 + 320000*x^24 + 123000*x^23 + 430000*x^22 + 212300*x^21 + 432000*x^20 + 123400*x^19 + 321230*x^18 + 432120*x^17 + 343210*x^16 + 234320*x^15 + 123432*x^14 + 23432*x^13 + 34321*x^12 + 43212*x^11 + 32123*x^10 + 1234*x^9 + 432*x^8 + 2123*x^7 + 43*x^6 + 123*x^5 + 32*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(1000000*x^30 - 1000001*x^15 + 1). (End)
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