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A370560 Periodic sequence S_{30} of length 28: used to compute the number that the day 30 of a month occurs on a day of the week for the Gregorian cycle of 400 years. 3
3, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
For the Gregorian Calendar (in use since Friday, October 15, 1582) of period 400, the number of occurrences of day d = 30 of a month on a day D of the week from 0 to 6 (0 for Sunday) can be given for the representative years y from 0 to 399 (for years congruent to modulo 400, and y >= 1583) by pieces of the present sequence S_{30} with period 28.
This is done with the help of the seven length 400 codes M(30, D), for D from 0..6, shown in Table 3 of the linked WL paper. The meaning of these codes and their encoding in terms of the periodic sequence of length 28, S_{30} (the S_{30} 'clock'), see Figure 3 of the linked paper is explained there and in A370558 (the case M(1, D)).
LINKS
Wolfdieter Lang, Four Sequences of Length 28 and the Gregorian Calendar, arXiv:2311.13646v2 [math.HO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1,0,0,-1,1,0,0,-1,1,0,0,-1,1,0,0,-1,1,0,0,-1,1).
FORMULA
{a(n)}_{n>=0} = repeat(3, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1,).
From Chai Wah Wu, Apr 12 2024: (Start)
a(n) = a(n-1) - a(n-4) + a(n-5) - a(n-8) + a(n-9) - a(n-12) + a(n-13) - a(n-16) + a(n-17) - a(n-20) + a(n-21) - a(n-24) + a(n-25) for n > 24.
G.f.: (-x^24 - x^22 + x^21 - 2*x^20 + x^17 - 2*x^16 - x^15 + 2*x^13 - 2*x^12 - 2*x^11 + x^10 + x^9 - 2*x^8 - x^7 + x^6 - 2*x^4 - x^3 + x^2 + x - 3)/(x^25 - x^24 + x^21 - x^20 + x^17 - x^16 + x^13 - x^12 + x^9 - x^8 + x^5 - x^4 + x - 1). (End)
EXAMPLE
1) M(30, 0) = (0)103(17)97(17)100(17)100,
decoded by the four pieces:
S_{30}, S_{30}, S_{30}, S_{30}(0..18);
S_{30}(7..27), S_{30}, S_{30}, S_{30}(0..19);
S_{30}(8..27), S_{30}, S_{30}, S_{30}(0..23);
S_{30}(12..27), S_{30}, S_{30}, S_{30}.
For example in the year 2024 == 24 mod 400 there is S_{30}(24) = 1 Sunday on the 30th day of a month (indeed this month is June).
For 2100 == 100 mod 400 there will be S_{30}(100 mod(28)) S_{30)(16) = 1 Sunday on the 30th day of a month (in May).
2) M(30, 4) = (8)100(17)100(17)(7)5(11)96(17)99,
decoded by the five pieces:
S_{30}(8..27), S_{30}, S_{30}, S_{30}(0..23);
S_{30}(12..27), S_{30}, S_{30}, S_{30};
S_{30}(6..10);
S_{30}(21..27), S_{30}, S_{30}, S_{30}, S_{30}(0..4);
S_{30}(21..27), S_{30}, S_{30}, S_{30}, S_{30}(0..7);
For example in the year 2024 == 24 mod 400 there is S_{30}(24 - 20) = S_{30}(4) = 1 Thursday on the 30th day of a month (indeed this month is May).
For 2100 == 100 mod 400 there will be S_{30}(12) = 2 Thursdays on the 30th day of a month (in September and December). This comes from the first entry of the second piece.
CROSSREFS
Sequence in context: A156352 A175191 A328928 * A342468 A324534 A248505
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 11 2024
STATUS
approved

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Last modified July 13 02:27 EDT 2024. Contains 374260 sequences. (Running on oeis4.)