OFFSET
0,2
COMMENTS
For the Gregorian Calendar (in use since Friday, October 15, 1582) of period 400, the number of occurrences of day d = 29 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_{29} with period 28.
This is done with the help of the seven length 400 codes M(29,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_{29} (the S_{29} 'clock'), see Figure 2 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 (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).
FORMULA
{a(n)}_{n>=0} = repeat(1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1,).
G.f.: (-x^27 - 2*x^26 - x^25 - 2*x^24 - 2*x^23 - x^22 - x^21 - 2*x^20 - 2*x^19 - 2*x^18 - 2*x^17 - x^16 - 2*x^15 - x^14 - 2*x^13 - 3*x^12 - x^11 - x^10 - 2*x^9 - x^8 - 2*x^7 - 2*x^6 - x^5 - 2*x^4 - x^3 - 2*x^2 - 2*x - 1)/(x^28 - 1). - Chai Wah Wu, Apr 12 2024
EXAMPLE
Let S_{29}(i..j) denote the subsequence S_{29}(i), S_{29}(i+1), ..., S_{29}(j), i.e., a(i), a(1+1), ..., a(j).
1) M(29, 0) = (0)101(17)99(17)101(17)99 (by coincidence the same as M(1, 0)), is decoded by the four pieces:
S_{29}, S_{29}, S_{29}, S_{29}(0..16);
S_{29}(5..27), S_{29}, S_{29}, S_{29}(0..19);
S_{29}(8..27), S_{29}, S_{29}, S_{29}(0..24);
S_{29}(13..27), S_{29}, S_{29}, S_{29}.
For example in the year 2024 == 24 mod 400 there are S_{29}(24) = 2 Sundays on a 29th (namely in September and December).
For 2100 == 100 mod 400 there will be S_{29}(16) = 1 Sunday on a 29th, (namely August).
2) M(29, 1) = (16)102(17)98(17)100(7)4(11)96 decoded by the five pieces:
S_{29}(16..27), S_{29}, S_{29}, S_{29}, S_{29}(0..5);
S_{29}(22..27), S_{29}, S_{29}, S_{29}, S_{29}(0..7);
S_{29}(24..27), S_{29}, S_{29}, S_{29}, S_{29}(0..114);
S_{29}(18..21);
S_{29}(4..27), S_{29}, S_{29}, S_{29}(0..15).
For the year 2024 there are S_{29}(16 + 24, mod 28) = S_{29}(12) = 3 Mondays on the 29th (namely January, April, July).
For the year 2102 there will be S_{29}(22) = 1 Monday on a 29th (namely in May). This comes from the first entry of the second piece. The first piece has 102 entries with offset 0.
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Feb 22 2024
STATUS
approved