Leap year starting on Saturday

A leap year starting on Saturday is any year with 366 days (i.e. it includes 29 February) that begins on Saturday, 1 January, and ends on Sunday, 31 December. Its dominical letters hence are BA, such as the years 1780, 1820, 1848, 1876, 1916, 1944, 1972, 2000, 2028, 2056, 2084, 2124, 2152, and 2180 in the Gregorian calendar or, likewise, 2012 and 2040 in the obsolete Julian calendar. In the Gregorian calendar all centennial leap years start on Saturday; the next such year will be 2400, see below for more.[1]

Any leap year that starts on Tuesday, Friday or Saturday has only one Friday the 13th; the only Friday the 13th in this leap year occurs in October. Common years starting on Sunday share this characteristic, but also have another in January. From August of the year that precedes this year until October in this type of year is also the longest period (14 months) that occurs without a Friday the 13th. Any common year starting on Tuesday shares this characteristic, from July of the year that precedes it to September in that type of year. There are two other ways this 14 month interval can be formed, one involving a common year followed by a leap year and the other a leap year followed by a common year.

Calendars

Calendar for any leap year starting on Saturday,
presented as common in many English-speaking areas

ISO 8601-conformant calendar with week numbers for
any leap year starting on Saturday (dominical letter BA)

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	January
52						01	02
01	03	04	05	06	07	08	09
02	10	11	12	13	14	15	16
03	17	18	19	20	21	22	23
04	24	25	26	27	28	29	30
05	31

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	February
05		01	02	03	04	05	06
06	07	08	09	10	11	12	13
07	14	15	16	17	18	19	20
08	21	22	23	24	25	26	27
09	28	29

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	March
09			01	02	03	04	05
10	06	07	08	09	10	11	12
11	13	14	15	16	17	18	19
12	20	21	22	23	24	25	26
13	27	28	29	30	31

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	April
13						01	02
14	03	04	05	06	07	08	09
15	10	11	12	13	14	15	16
16	17	18	19	20	21	22	23
17	24	25	26	27	28	29	30

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	May
18	01	02	03	04	05	06	07
19	08	09	10	11	12	13	14
20	15	16	17	18	19	20	21
21	22	23	24	25	26	27	28
22	29	30	31

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	June
22				01	02	03	04
23	05	06	07	08	09	10	11
24	12	13	14	15	16	17	18
25	19	20	21	22	23	24	25
26	26	27	28	29	30

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	July
26						01	02
27	03	04	05	06	07	08	09
28	10	11	12	13	14	15	16
29	17	18	19	20	21	22	23
30	24	25	26	27	28	29	30
31	31

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	August
31		01	02	03	04	05	06
32	07	08	09	10	11	12	13
33	14	15	16	17	18	19	20
34	21	22	23	24	25	26	27
35	28	29	30	31

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	September
35					01	02	03
36	04	05	06	07	08	09	10
37	11	12	13	14	15	16	17
38	18	19	20	21	22	23	24
39	25	26	27	28	29	30

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	October
39							01
40	02	03	04	05	06	07	08
41	09	10	11	12	13	14	15
42	16	17	18	19	20	21	22
43	23	24	25	26	27	28	29
44	30	31

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	November
44			01	02	03	04	05
45	06	07	08	09	10	11	12
46	13	14	15	16	17	18	19
47	20	21	22	23	24	25	26
48	27	28	29	30

Wk	Mo	Tu	We	Th	Fr	Sa	Su
	December
48					01	02	03
49	04	05	06	07	08	09	10
50	11	12	13	14	15	16	17
51	18	19	20	21	22	23	24
52	25	26	27	28	29	30	31

Applicable years

Gregorian Calendar

Leap years that begin on Saturday, along with those that start on Monday or Thursday, occur least frequently: 13 out of 97 (≈ 13.402%) total leap years in a 400-year cycle of the Gregorian calendar. Their overall occurrence is thus 3.25% (13 out of 400).

Gregorian leap years starting on Saturday[1]
Decade	1st	2nd	3rd	5th	6th	8th	9th	10th
16th century	prior to first adoption (proleptic)							1600
17th century			1628		1656		1684
18th century			1724		1752	1780
19th century		1820		1848		1876
20th century		1916		1944		1972		2000
21st century			2028		2056		2084
22nd century			2124		2152	2180
23rd century		2220		2248		2276
24th century		2316		2344		2372		2400
25th century			2428		2456		2484
26th century			2524		2552	2580
27th century		2620		2648		2676

Century leap years are always leap years starting on Saturday and February 29 is always on Tuesday.

Julian Calendar

Like all leap year types, the one starting with 1 January on a Saturday occurs exactly once in a 28-year cycle in the Julian calendar, i.e. in 3.57% of years. As the Julian calendar repeats after 28 years that means it will also repeat after 700 years, i.e. 25 cycles. The year's position in the cycle is given by the formula ((year + 8) mod 28) + 1).

Julian leap years starting on Saturday
Decade	1st	2nd	3rd	4th	5th	6th	7th	8th	9th	10th
15th century			1424			1452		1480
16th century	1508			1536			1564			1592
17th century		1620			1648			1676
18th century	1704			1732		1760			1788
19th century		1816			1844			1872		1900
20th century			1928			1956			1984
21st century		2012		2040			2068			2096
22nd century			2124			2152		2180

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References

Robert van Gent (2017). "The Mathematics of the ISO 8601 Calendar". Utrecht University, Department of Mathematics. Retrieved 20 July 2017.

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[math-1] Robert van Gent (2017). "The Mathematics of the ISO 8601 Calendar". Utrecht University, Department of Mathematics. Retrieved 20 July 2017.