A068962 -id:A068962 - cp's OEIS frontend

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

N. J. A. Sloane, J. H. Conway

This remarkable sequence is really a sequence of lists rather than numbers.

			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, ...

Zdenek Horsky, "Prazsky Orloj" ["The Astronomical Clock of Prague", in Czech], Panorama, Prague, 1988, pp. 76-78.

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).

Cf. A028354, A028356, A068962, A118382, A118383.

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 *)

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)

A028354 How the astronomical clock ("Orloj") in Prague strikes the hours (digits follow 12343212343... (A028356), n-th group adds to n).

Original entry on oeis.org

1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212, 34321, 23432, 123432, 1234321, 2343212, 3432123, 4321234, 32123432, 123432123, 43212343, 2123432123, 432123432, 1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212

Offset: 1

J. H. Conway

There is a single bell, which to indicate 5 o'clock, say, strikes thrice then twice.

Zdenek Horsky, "Prazsky Orloj" ["The Astronomical Clock of Prague", in Czech], Panorama, Prague, 1988, pp. 76-78.

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).

Cf. A028355, A028356, A068962, A118382, A118383.

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] = If[k>24, a[k-24], Table[ s[i], {i, m[k]+1, n[k]}] // FromDigits]; Array[a, 36] (* Jean-François Alcover, Mar 13 2016 *)

A321232 Length of n-th term of A321225.

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 4, 4, 5, 6, 6, 6, 8, 7, 9, 9, 9, 10, 11, 11, 11, 13, 12, 14, 14, 14, 15, 16, 16, 16, 18, 17, 19, 19, 19, 20, 21, 21, 21, 23, 22, 24, 24, 24, 25, 26, 26, 26, 28, 27, 29, 29, 29, 30, 31, 31, 31, 33, 32, 34, 34, 34, 35, 36, 36, 36, 38, 37, 39, 39, 39, 40

Offset: 1

Seiichi Manyama, Oct 31 2018

			n | A321225 | a(n)
--+---------+-----
1 |       1 |   1
2 |       2 |   1
3 |       3 |   1
4 |     211 |   3
5 |      23 |   2
6 |    2112 |   4
7 |    3211 |   4
8 |    2321 |   4
9 |   12321 |   5

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A068962, A321225.

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 1, 1, 3, 2, 4, 4, 4, 5, 6}, 100] (* Paolo Xausa, Apr 24 2024 *)

a(n) = a(n-9) + 5, for n > 9.

G.f.: x*(x^8 + 2*x^5 - x^4 + 2*x^3 + 1)/(x^10 - x^9 - x + 1). - Chai Wah Wu, Jan 06 2020

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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).

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A028354 How the astronomical clock ("Orloj") in Prague strikes the hours (digits follow 12343212343... (A028356), n-th group adds to n).

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Mathematica

A321232 Length of n-th term of A321225.

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