A231014 -id:A231014 - cp's OEIS frontend

A230995 Number of years after which a date can fall on the same day of the week, in the Gregorian calendar.

0, 5, 6, 7, 11, 12, 17, 18, 22, 23, 28, 29, 33, 34, 35, 39, 40, 45, 46, 50, 51, 56, 57, 61, 62, 63, 67, 68, 73, 74, 78, 79, 84, 85, 89, 90, 91, 95, 96, 101, 102, 106, 107, 108, 112, 113, 114, 117, 118, 119, 123, 124, 125, 129, 130, 131, 134, 135, 136, 140, 141, 142

Offset: 0

Aswini Vaidyanathan, Nov 02 2013

In the Gregorian calendar, a non-century year is a leap year if and only if it is a multiple of 4 and a century year is a leap year if and only if it is a multiple of 400.

Assuming this fact, this sequence is periodic with a period of 400.

			5 belongs to this sequence because January 1, 2008 falls on same day as January 1, 2013.
6 belongs to this sequence because January 1, 2009 falls on same day as January 1, 2015.
7 belongs to this sequence because January 1, 2097 falls on same day as January 1, 2104.

Time And Date, Repeating Calendar
Time And Date, Gregorian Calendar

Cf. A230995-A231014.

Cf. A231000 (Julian calendar).

for(i=0,400,for(y=0,400,if(((5*(y\4)+(y%4)-(y\100)+(y\400))%7)==((5*((y+i)\4)+((y+i)%4)-((y+i)\100)+((y+i)\400))%7),print1(i", ");break)))

A231000 Number of years after which a date can fall on the same day of the week, in the Julian calendar.

Original entry on oeis.org

0, 5, 6, 11, 17, 22, 23, 28, 33, 34, 39, 45, 50, 51, 56, 61, 62, 67, 73, 78, 79, 84, 89, 90, 95, 101, 106, 107, 112, 117, 118, 123, 129, 134, 135, 140, 145, 146, 151, 157, 162, 163, 168, 173, 174, 179, 185, 190, 191, 196, 201, 202, 207, 213, 218, 219, 224, 229, 230, 235

Offset: 0

Aswini Vaidyanathan, Nov 02 2013

In the Julian calendar, a year is a leap year if and only if it is a multiple of 4 and all century years are leap years.

Assuming this fact, this sequence is periodic with a period of 28.

Colin Barker, Table of n, a(n) for n = 0..1000
Time And Date, Repeating Calendar
Time And Date, Julian Calendar
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A230995-A231014.

Cf. A230995 (Gregorian calendar).

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1-x+x^2)*(5+6*x+6*x^2+6*x^3+5*x^4)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)) )); // Marius A. Burtea, Oct 17 2019

for(i=0,420,for(y=0,420,if(((5*(y\4)+(y%4))%7)==((5*((y+i)\4)+((y+i)%4))%7),print1(i", ");break)))

concat(0, Vec(x*(1 - x + x^2)*(5 + 6*x + 6*x^2 + 6*x^3 + 5*x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^40))) \\ Colin Barker, Oct 17 2019

From Colin Barker, Oct 17 2019: (Start)
G.f.: x*(1 - x + x^2)*(5 + 6*x + 6*x^2 + 6*x^3 + 5*x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-1) + a(n-7) - a(n-8) for n>7.
(End)

A231001 Number of years after which an entire year can have the same calendar, in the Julian calendar.

Original entry on oeis.org

0, 6, 11, 17, 22, 28, 34, 39, 45, 50, 56, 62, 67, 73, 78, 84, 90, 95, 101, 106, 112, 118, 123, 129, 134, 140, 146, 151, 157, 162, 168, 174, 179, 185, 190, 196, 202, 207, 213, 218, 224, 230, 235, 241, 246, 252, 258, 263, 269, 274, 280, 286, 291, 297, 302, 308, 314, 319, 325

Offset: 0

Aswini Vaidyanathan, Nov 02 2013

In the Julian calendar, a year is a leap year if and only if it is a multiple of 4 and all century years are leap years.
Assuming this fact, this sequence is periodic with a period of 28.
This is a subsequence of A231000.

Colin Barker, Table of n, a(n) for n = 0..1000
Time And Date, Repeating Calendar
Time And Date, Julian Calendar
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A230995-A231014.

Cf. A230996 (Gregorian calendar).

LinearRecurrence[{1,0,0,0,1,-1},{0,6,11,17,22,28},60] (* Harvey P. Dale, Jun 19 2023 *)

for(i=0,420,for(y=0,420,if(((5*(y\4)+(y%4))%7)==((5*((y+i)\4)+((y+i)%4))%7)&&((5*(y\4)+(y%4)-!(y%4))%7)==((5*((y+i)\4)+((y+i)%4)-!((y+i)%4))%7),print1(i", ");break)))

concat(0, Vec(x*(2 + 3*x + 2*x^2)*(3 - 2*x + 3*x^2) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Oct 17 2019

From Colin Barker, Oct 17 2019: (Start)
G.f.: x*(2 + 3*x + 2*x^2)*(3 - 2*x + 3*x^2) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.
(End)

A231010 Number of months after which a date can fall on the same day of the week, in the Julian calendar.

Original entry on oeis.org

0, 1, 3, 6, 8, 9, 11, 14, 15, 17, 18, 19, 20, 22, 23, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 45, 46, 49, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 68, 69, 71, 72, 73, 74, 75, 77, 78, 80, 81, 83, 86, 87, 89, 91, 92, 94, 95, 98, 100, 101, 103, 104, 106, 109, 110, 112

Offset: 0

Aswini Vaidyanathan, Nov 02 2013

In the Julian calendar, a year is a leap year if and only if it is a multiple of 4 and all century years are leap years.

Assuming this fact, this sequence is periodic with a period of 336.

Time And Date, Repeating Months
Time And Date, Julian Calendar

Cf. A230995-A231014.

Cf. A231005 (Gregorian calendar).

m=[0,3,3,6,1,4,6,2,5,0,3,5];n=[31,28,31,30,31,30,31,31,30,31,30,31];y=vector(336,i,(m[((i-1)%12)+1]+((5*((i-1)\48)+(((i-1)\12)%4)-!((i-1)%48)-!((i-2)%48))))%7);x=vector(336,i,n[((i-1)%12)+1]+!((i-2)%48));for(p=0,336,for(q=0,336,if(y[(q%336)+1]==y[((q+p)%336)+1],print1(p", ");break)))

A231005 Number of months after which a date can fall on the same day of the week, in the Gregorian calendar.

Original entry on oeis.org

0, 1, 3, 6, 8, 9, 11, 14, 15, 17, 18, 19, 20, 22, 23, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 45, 46, 49, 50, 52, 53, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 91, 92, 94, 95, 98, 99

Offset: 0

Aswini Vaidyanathan, Nov 02 2013

In the Gregorian calendar, a non-century year is a leap year if and only if it is a multiple of 4 and a century year is a leap year if and only if it is a multiple of 400.

Assuming this fact, this sequence is periodic with a period of 4800.

			1 belongs to this sequence because February 1, 2013 falls on the same day as March 1, 2013.
3 belongs to this sequence because December 1, 2011 falls on the same day as March 1, 2012.
6 belongs to this sequence because January 1, 2012 falls on the same day as July 1, 2012.
8 belongs to this sequence because March 1, 2013 falls on the same day as November 1, 2013.
9 belongs to this sequence because January 1, 2013 falls on the same day as October 1, 2013.
11 belongs to this sequence because December 1, 2011 falls on the same day as November 1, 2012.

Time And Date, Repeating Months
Time And Date, Gregorian Calendar

Cf. A230995-A231014.

Cf. A231010 (Julian calendar).

m=[0,3,3,6,1,4,6,2,5,0,3,5];n=[31,28,31,30,31,30,31,31,30,31,30,31];y=vector(4800,i,(m[((i-1)%12)+1]+((5*((i-1)\48)+(((i-1)\12)%4)-((i-1)\1200)+((i-1)\4800)-!((i-1)%48)+!((i-1)%1200)-!((i-1)%4800)-!((i-2)%48)+!((i-2)%1200)-!((i-2)%4800))))%7);x=vector(4800,i,n[((i-1)%12)+1]+!((i-2)%48)-!((i-2)%1200)+!((i-2)%4800));for(p=0,4800,for(q=0,4800,if(y[(q%4800)+1]==y[((q+p)%4800)+1],print1(p", ");break)))

A231236 Number of years after which it is either not possible to have a date falling on same day of the week, or the entire year can have the same calendar, in the Gregorian calendar.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Aswini Vaidyanathan, Nov 06 2013

In the Gregorian calendar, a non-century year is a leap year if and only if it is a multiple of 4 and a century year is a leap year if and only if it is a multiple of 400.
Assuming this fact, this sequence is periodic with a period of 400.
This is the complement of A230997.

Time And Date, Repeating Calendar
Time And Date, Gregorian Calendar

Cf. A230995-A231014, A231236-A231239.

Cf. A231237 (Julian calendar).

for(i=0,400,j=0;for(y=0,400,if(((5*(y\4)+(y%4)-(y\100)+(y\400))%7)==((5*((y+i)\4)+((y+i)%4)-((y+i)\100)+((y+i)\400))%7),j=1;break));for(y=0,400,if(((5*(y\4)+(y%4)-(y\100)+(y\400))%7)==((5*((y+i)\4)+((y+i)%4)-((y+i)\100)+((y+i)\400))%7)&&((5*(y\4)+(y%4)-(y\100)+(y\400)-!(y%4)+!(y%100)-!(y%400))%7)==((5*((y+i)\4)+((y+i)%4)-((y+i)\100)+((y+i)\400)-!((y+i)%4)+!((y+i)%100)-!((y+i)%400))%7),j=2;break));if(j!=1,print1(i", ")))

A231239 Number of months after which either it is not possible to have a date to fall on the same day of the week, or that it is possible to have a date falling on the same day of the week and the two months have the same number of days, in the Julian calendar.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 88, 89

Offset: 1

Aswini Vaidyanathan, Nov 06 2013

In the Julian calendar, a year is a leap year if and only if it is a multiple of 4 and all century years are leap years.
Assuming this fact, this sequence is periodic with a period of 336.
This is the complement of A231012.

Time And Date, Repeating Months
Time And Date, Julian Calendar

Cf. A230995-A231014, A231236-A231239.

Cf. A231238 (Gregorian calendar).

m=[0,3,3,6,1,4,6,2,5,0,3,5];n=[31,28,31,30,31,30,31,31,30,31,30,31];y=vector(336,i,(m[((i-1)%12)+1]+((5*((i-1)\48)+(((i-1)\12)%4)-!((i-1)%48)-!((i-2)%48))))%7);x=vector(336,i,n[((i-1)%12)+1]+!((i-2)%48));for(p=0,336,j=0;for(q=0,336,if(y[(q%336)+1]==y[((q+p)%336)+1],j=1;break));for(q=0,336,if(y[(q%336)+1]==y[((q+p)%336)+1]&&x[(q%336)+1]==x[((q+p)%336)+1],j=2;break));if(j!=1,print1(p", ")))

A230996 Number of years after which an entire year can have the same calendar, in the Gregorian calendar.

Original entry on oeis.org

0, 6, 11, 12, 17, 18, 22, 23, 28, 29, 34, 39, 40, 45, 46, 50, 51, 56, 57, 62, 67, 68, 73, 74, 78, 79, 84, 85, 90, 95, 96, 101, 102, 106, 107, 108, 112, 113, 114, 118, 119, 123, 124, 125, 129, 130, 134, 135, 136, 140, 141, 142, 146, 147, 151, 152, 153, 157, 158, 162

Offset: 0

Aswini Vaidyanathan, Nov 02 2013

In the Gregorian calendar, a non-century year is a leap year if and only if it is a multiple of 4 and a century year is a leap year if and only if it is a multiple of 400.
Assuming this fact, this sequence is periodic with a period of 400.
This is a subsequence of A230995.

			6 belongs to this sequence because year 2013 has the same calendar as year 2019.
11 belongs to this sequence because year 2002 has the same calendar as year 2013.
12 belongs to this sequence because year 2096 has the same calendar as year 2108.

Time And Date, Repeating Calendar
Time And Date, Gregorian Calendar

Cf. A230995-A231014.

Cf. A231001 (Julian calendar).

for(i=0,400,for(y=0,400,if(((5*(y\4)+(y%4)-(y\100)+(y\400))%7)==((5*((y+i)\4)+((y+i)%4)-((y+i)\100)+((y+i)\400))%7)&&((5*(y\4)+(y%4)-(y\100)+(y\400)-!(y%4)+!(y%100)-!(y%400))%7)==((5*((y+i)\4)+((y+i)%4)-((y+i)\100)+((y+i)\400)-!((y+i)%4)+!((y+i)%100)-!((y+i)%400))%7),print1(i", ");break)))

A231002 Number of years after which it is possible to have a date falling on same day of the week, but the entire year does not have the same calendar, in the Julian calendar.

Original entry on oeis.org

5, 23, 33, 51, 61, 79, 89, 107, 117, 135, 145, 163, 173, 191, 201, 219, 229, 247, 257, 275, 285, 303, 313, 331, 341, 359, 369, 387, 397, 415, 425, 443, 453, 471, 481, 499, 509, 527, 537, 555, 565, 583, 593, 611, 621, 639, 649, 667, 677, 695, 705, 723, 733, 751, 761, 779, 789

Offset: 1

Aswini Vaidyanathan, Nov 02 2013

In the Julian calendar, a year is a leap year if and only if it is a multiple of 4 and all century years are leap years.
Assuming this fact, this sequence is periodic with a period of 28.
These are the terms of A231000 not in A231001.
The statement about the period is misleading: this is the sequence of (positive) numbers congruent to 5 or -5 (mod 28). It is strictly increasing, not periodic; the sequence a(n) - 28*floor(n/2) is 2-periodic. - M. F. Hasler, Apr 14 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Time And Date, Repeating Calendar
Time And Date, Julian Calendar
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A230995-A231014.

Cf. A230997 (Gregorian calendar).

LinearRecurrence[{1,1,-1},{5,23,33},70] (* Harvey P. Dale, May 21 2021 *)

for(i=0,420,j=0;for(y=0,420,if(((5*(y\4)+(y%4))%7)==((5*((y+i)\4)+((y+i)%4))%7),j=1;break));for(y=0,420,if(((5*(y\4)+(y%4))%7)==((5*((y+i)\4)+((y+i)%4))%7)&&((5*(y\4)+(y%4)-!(y%4))%7)==((5*((y+i)\4)+((y+i)%4)-!((y+i)%4))%7),j=2;break));if(j==1,print1(i", ")))

A231002(n) = n\2*28-5*(-1)^n \\ M. F. Hasler, Apr 14 2015

Vec(x*(5 + 18*x + 5*x^2) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Oct 15 2019

a(n+1) = a(n-1)+28, for all n > 1. - M. F. Hasler, Apr 14 2015
a(2n) = 28n-5 (n>0), a(2n+1) = 28n+5 (n>=0), a(n) = 28*floor(n/2)-5*(-1)^n. - M. F. Hasler, Apr 14 2015
From Colin Barker, Oct 15 2019: (Start)
G.f.: x*(5 + 18*x + 5*x^2) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = -7 + 2*(-1)^n + 14*n.
(End)

A231006 Number of months after which a date can fall on the same day of the week, and the two months can have the same number of days, in the Gregorian calendar.

Original entry on oeis.org

0, 3, 6, 8, 9, 14, 15, 17, 18, 20, 22, 23, 26, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 46, 49, 50, 52, 54, 55, 57, 58, 60, 63, 64, 66, 67, 68, 69, 72, 75, 76, 77, 78, 80, 81, 84, 86, 87, 89, 90, 92, 94, 95, 98, 101, 103, 104, 106, 107, 109, 110, 112, 113, 114, 115, 117, 118

Offset: 0

Aswini Vaidyanathan, Nov 02 2013

In the Gregorian calendar, a non-century year is a leap year if and only if it is a multiple of 4 and a century year is a leap year if and only if it is a multiple of 400.
Assuming this fact, this sequence is periodic with a period of 4800.
This is a subsequence of A231005.

			3 belongs to this sequence because December 1, 2011 falls on the same day as March 1, 2012 and both December and March have 31 days each.
6 belongs to this sequence because January 1, 2012 falls on the same day as July 1, 2012 and both January and July have 31 days each.
8 belongs to this sequence because May 1, 2013 falls on the same day as January 1, 2014 and both May and January have 31 days each.
9 belongs to this sequence because January 1, 2013 falls on the same day as October 1, 2013 and both January and October have 31 days each.

Time And Date, Repeating Months
Time And Date, Gregorian Calendar

Cf. A230995-A231014.

Cf. A231011 (Julian calendar).

m=[0,3,3,6,1,4,6,2,5,0,3,5];n=[31,28,31,30,31,30,31,31,30,31,30,31];y=vector(4800,i,(m[((i-1)%12)+1]+((5*((i-1)\48)+(((i-1)\12)%4)-((i-1)\1200)+((i-1)\4800)-!((i-1)%48)+!((i-1)%1200)-!((i-1)%4800)-!((i-2)%48)+!((i-2)%1200)-!((i-2)%4800))))%7);x=vector(4800,i,n[((i-1)%12)+1]+!((i-2)%48)-!((i-2)%1200)+!((i-2)%4800));for(p=0,4800,for(q=0,4800,if(y[(q%4800)+1]==y[((q+p)%4800)+1]&&x[(q%4800)+1]==x[((q+p)%4800)+1],print1(p", ");break)))

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A230995 Number of years after which a date can fall on the same day of the week, in the Gregorian calendar.

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A231000 Number of years after which a date can fall on the same day of the week, in the Julian calendar.

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A231001 Number of years after which an entire year can have the same calendar, in the Julian calendar.

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A231010 Number of months after which a date can fall on the same day of the week, in the Julian calendar.

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A231005 Number of months after which a date can fall on the same day of the week, in the Gregorian calendar.

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A231236 Number of years after which it is either not possible to have a date falling on same day of the week, or the entire year can have the same calendar, in the Gregorian calendar.

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A231239 Number of months after which either it is not possible to have a date to fall on the same day of the week, or that it is possible to have a date falling on the same day of the week and the two months have the same number of days, in the Julian calendar.

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A230996 Number of years after which an entire year can have the same calendar, in the Gregorian calendar.

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