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A189916 Sequence for finding the day of the week for the first day of the month in leap years.

%I A189916 #9 May 17 2023 15:47:34
%S A189916 0,3,4,0,2,5,0,3,6,1,4,6
%N A189916 Sequence for finding the day of the week for the first day of the month in leap years.
%C A189916 The days of the week, starting with Sunday, have indices 0,1,..,6. The months of the year, starting with January, are numbered 1,2,...,12. The following pattern holds for leap years. Remember that the year 2000 was a leap year. See A189915 for non-leap years.
%C A189916 If Jan 01 falls on a day of the week with index I, then Feb 01 falls on the day with index I+3 (mod 7), Mar 01 falls on the day with index I+4 (mod 7), Apr 01 is again on the day with index I, etc.
%C A189916 If one uses 0->A, 1->B, 2->C, 3->D, 4->E, 5->F and 6->G then the sequence becomes A, D, E, A, C, F, A, D, G, B, E, G.
%D A189916 See A189915.
%F A189916 I(n) = I + a(n) (mod 7), n=1,2,..,12, with I the index of the day of Jan 01 in a leap year, and I(n) the index of the first day the n-th month in this year.
%F A189916 a(n) = A061251(35+n) - 1095 mod(7), n=1..12.
%F A189916 a(n) = A178054(144+n), n=1,...,12 (indices for the year 2012).
%e A189916 In the year 2008 Jan 01 has index 2 (Tuesday). Therefore, Feb 01 has index 2+3 = 5 (mod 7) (Friday), Mar 01 has index 2+4 = 6 (mod 7) (Saturday), Apr 01 falls again on a Tuesday, May 01 has index 2+2 = 4 (Thursday), Jun 01 has index 2+5= 0 (mod 7) (Sunday). Jul 01 falls again on Tuesday, etc.
%e A189916 For leap years in which Jan 01 has index 0 (Sunday) the pattern for the first days of the months is Sun, Wed, Thu, Sun, Tue, Fri, Sun, Wed, Sat, Mon, Thu, Sat.
%Y A189916 Cf. A061251, A178054 (index pattern starting with the year 2000), A189915.
%K A189916 nonn,easy,fini,full
%O A189916 1,2
%A A189916 _Wolfdieter Lang_, May 02 2011