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A097105 Gregorian years containing "blue" Islamic New Year Days. The boundary of a calendrical period is hereby called "blue" w.r.t. a similarly named period in another calendar when the shorter one does not contain the boundaries of the longer one. Gregorian calendar prior to 1582 is extrapolated according to the calculator in References. +20
1
640, 672, 705, 738, 770, 803, 835, 868, 900, 933, 966, 998, 1031, 1063, 1096, 1129, 1161, 1194, 1226, 1259, 1291, 1324, 1357, 1389, 1422, 1454, 1487, 1520, 1552, 1585, 1617, 1650, 1682, 1715, 1748, 1780, 1813, 1845, 1878, 1911, 1943, 1976, 2008, 2041 (list; graph; listen)
OFFSET

1,1
COMMENT

The ratio of Gregorian to Islamic Year is 365.2425/354.36666... = 438291/425240. The interesting approximating continuous fractions are 403/391, 638/619, 1041/1010 and a very long sequence of (1041+403*n)/(1010+391*n), ending with 7489/7266, so the 403/391 pattern will remain for thousands of years.

LINKS

Calendar Converter

EXAMPLE

1396-1-1 A.H. = 1976-1-3 C.E.

1397-1-1 A.H. = 1976-12-22 C.E. therefore 1976 is listed.

CROSSREFS

Sequence in context: A061623 A043483 A027885 this_sequence A170774 A160203 A105130

Adjacent sequences: A097102 A097103 A097104 this_sequence A097106 A097107 A097108

KEYWORD

nonn

AUTHOR

Leonid Broukhis (leob(AT)mailcom.com), Sep 15 2004

A057353 Floor(3n/4). +10
19
0, 0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52, 53, 54 (list; graph; listen)
OFFSET

0,4
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

Index entries for sequences related to Beatty sequences

FORMULA

G.f.: (1+x+x^2)*x^2/((1-x)*(1-x^4)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002

For all m>=0 a(4m)=0 mod 3; a(4m+1)=0 mod 3; a(4m+2)= 1 mod 3; a(4m+3) = 2 mod 3

a(n)=-1+Sum{k=0..n}{(1/8)*((k mod 4)+((k+1) mod 4)-((k+2) mod 4)+3*((k+3) mod 4)} [From Paolo P. Lava (ppl(AT)spl.at), Nov 17 2008]

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A086525 A120503 A083544 this_sequence A076539 A074184 A093700

Adjacent sequences: A057350 A057351 A057352 this_sequence A057354 A057355 A057356

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057354 floor(2n/5). +10
15
0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30 (list; graph; listen)
OFFSET

0,6
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: x^3(1 + x^3)/((1 - x)(1 - x^5)).

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A061375 A029920 A100719 this_sequence A097508 A109964 A025778

Adjacent sequences: A057351 A057352 A057353 this_sequence A057355 A057356 A057357

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057355 Floor(3n/5). +10
15
0, 0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 42, 42, 43, 43 (list; graph; listen)
OFFSET

0,5
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: (1+x^2+x^3)*x^2/((1-x)*(1-x^5)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002

for all m>=0 a(5m)=0 mod 3; a(5m+1)=0 mod 3; a(5m+2)= 1 mod 3; a(5m+3) = 1 mod 3; a(5m+4) = 2 mod 3

a(n)=-1+Sum{k=0..n}{(1/50)*(3*(k mod 5)-7*((k+1) mod 5)+13*((k+2) mod 5)-7*((k+3) mod 5)+13*((k+4) mod 5)} [From Paolo P. Lava (ppl(AT)spl.at), Nov 17 2008]

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A156261 A071823 A139338 this_sequence A160511 A055930 A079952

Adjacent sequences: A057352 A057353 A057354 this_sequence A057356 A057357 A057358

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057356 floor(2n/7). +10
15
0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22 (list; graph; listen)
OFFSET

0,8
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: x^4(1 + x^4)/((1 - x)(1 - x^7)).

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A034973 A066927 A060065 this_sequence A020913 A025787 A057810

Adjacent sequences: A057353 A057354 A057355 this_sequence A057357 A057358 A057359

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057357 Floor(3n/7). +10
15
0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32 (list; graph; listen)
OFFSET

0,6
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: (1+x^2+x^4)*x^3/((1-x)*(1-x^7)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002

for all m>=0 a(7m)=0 mod 3; a(7m+1)=0 mod 3; a(7m+2)= 0 mod 3; a(7m+3) = 1 mod 3; a(5m+4) = 1 mod 3; a(7m+5) = 2 mod 3; a(7m+6) = 2 mod 3 - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002

a(n)=-1+Sum{k=0..n}{(1/49)*(-6*(k mod 7)+8*((k+1) mod 7)-6*((k+2) mod 7)+8*((k+3) mod 7)-6*((k+4) mod 7)+((k+5) mod 7)+8*((k+6) mod 7)} [From Paolo P. Lava (ppl(AT)spl.at), Nov 17 2008]

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A028827 A083055 A121828 this_sequence A029123 A025777 A145703

Adjacent sequences: A057354 A057355 A057356 this_sequence A057358 A057359 A057360

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057358 floor(4n/7). +10
15
0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 38, 39, 40, 40, 41, 41, 42 (list; graph; listen)
OFFSET

0,5
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: x^2(1 + x^2 + x^4 + x^6)/((1 - x)(1 - x^7)).

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A089575 A023190 A047783 this_sequence A038128 A097337 A163464

Adjacent sequences: A057355 A057356 A057357 this_sequence A057359 A057360 A057361

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057359 Floor(5n/7). +10
15
0, 0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47, 48, 49, 50, 50, 51 (list; graph; listen)
OFFSET

0,4
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: x^2(1 + x^2 + x^3 + x^5 + x^6)/((1 - x)(1 - x^7)).

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A156351 A057561 A064726 this_sequence A076538 A138466 A066530

Adjacent sequences: A057356 A057357 A057358 this_sequence A057360 A057361 A057362

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057360 floor(3n/8). +10
15
0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28 (list; graph; listen)
OFFSET

0,7
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: x^3(1 + x^3 + x^6)/((1 - x)(1 - x^8)).

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A084520 A084510 A053620 this_sequence A057364 A060144 A107347

Adjacent sequences: A057357 A057358 A057359 this_sequence A057361 A057362 A057363

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

A057361 Floor(5n/8). +10
15
0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 41, 42, 43, 43, 44, 45, 45 (list; graph; listen)
OFFSET

0,5
COMMENT

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

REFERENCES

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

LINKS

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

FORMULA

G.f.: x^2(1 + x^2 + x^4 + x^5 + x^7)/((1 - x)(1 - x^8)).

CROSSREFS

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A086335 A123387 A123070 this_sequence A136409 A039729 A074065

Adjacent sequences: A057358 A057359 A057360 this_sequence A057362 A057363 A057364

KEYWORD

nonn,easy

AUTHOR

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

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