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.

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

1,1

Comments

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

Links

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).

Formula

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)

Mathematica

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

Prog

(PARI) 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", ")))
(PARI) A231002(n) = n\2*28-5*(-1)^n \\ M. F. Hasler, Apr 14 2015
(PARI) Vec(x*(5 + 18*x + 5*x^2) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Oct 15 2019

Crossrefs

Cf. A230995-A231014.

Cf. A230997 (Gregorian calendar).

Sequence in context: A082283 A071199 A238195 * A329160 A337436 A050906

Adjacent sequences: A230999 A231000 A231001 * A231003 A231004 A231005

Keyword

nonn,easy

Author

Aswini Vaidyanathan, Nov 02 2013

Status

approved