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Sequence for finding the day of the week for the first day of the month in a common (non-leap) year.
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%I #17 Nov 23 2019 04:03:21

%S 0,3,3,6,1,4,6,2,5,0,3,5

%N Sequence for finding the day of the week for the first day of the month in a common (non-leap) year.

%C 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 common (non-leap) years.

%C See A189916 for leap years.

%C If Jan 01 falls on a day of the week with index I, then Feb 01 is on the day with index I+3 (mod 7), Mar 01 is also on the day with index I+3 (mod 7), Apr 01 is on the day with index I+6 (mod 7), etc.

%C If one uses 0->A, 1->B, 2->C, 3->D, 4->E, 5->F and 6->G the sequence becomes

%C A, D, D, G, B, E, G, C, F, A, D, F.

%C A mnemonic rhyme for this sequence is in English:

%C At Dover dwells George Brown, Esquire,

%C Good Christopher Fitch and David Frier.

%C In German (attributed to Thomas Brown, Oxford):

%C Allvater, der du gnaedig bist,

%C Ein gesetzestreuer Christ

%C Fordert Ablauf dieser Frist.

%C See the L. Holford-Strevens reference pp. 106-7 (German translation).

%D L. Holford-Strevens, The History of Time. A Very Short Introduction, Oxford University Press, 2005.

%D L. Holford-Strevens, Kleine Geschichte der Zeitrechnung und des Kalenders, Reclams Universalbibliothek Nr.18483, Stuttgart, 2008 (German translation).

%F I(n) = I + a(n) (mod 7), n=1,...,12, with I the index of January 01 in a common (non-leap) year, and I(n) the index of the day of the week of the first day of the n-th month in this year.

%F a(n) = A061251(n) (mod 7), n=0,..,11.

%F a(n) = A178054(72+n), n=1..12.

%e In the year 2011 Jan 01 has index 6 (Saturday). Therefore, Feb 01 has index 6+3 = 2 (mod 7) (Tuesday), Mar 01 also has index 2, Apr 01 has index 6+6 = 5 (mod 7) (Friday), etc.

%e In common years with Jan 01 on a Sunday (index 0) the sequence gives the index of the first day of the n-th month of this year. E.g., in the year 2006 (but not in the leap year 2012).

%Y Cf. A178054 (indices starting with Jan 01 2000), A061251.

%K nonn,easy,fini,full

%O 1,2

%A _Wolfdieter Lang_, May 02 2011