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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Aaron Kastel

Aaron Kastel's wiki page.

Aaron Kastel has authored 4 sequences.

A256668 a(n) = 3*B*C*(n mod A) + 5*A*C*(n mod B) + 2*A*B*(n mod C) with A=7, B=11, C=17.

Original entry on oeis.org

1310, 2620, 3930, 5240, 6550, 7860, 5243, 6553, 7863, 9173, 3938, 5248, 6558, 3941, 5251, 6561, 5253, 6563, 7873, 9183, 6566, 1331, 2641, 3951, 5261, 6571, 7881, 5264, 6574, 7884, 9194, 10504, 5269, 3961, 1344, 2654, 3964, 5274, 6584, 7894
Offset: 1

Views

Author

Aaron Kastel, Apr 07 2015

Keywords

Comments

After 0 it cycles again from 1310 (a(1309)=0 so there are 1309 (A*B*C) terms).
This is another variation on A256496, where a(n) = B*C*(n mod A) + A*C*(n mod B) + A*B*(n mod C), modified to take the values A=7, B=11, C=17 and still maintain the equivalence a(n) mod ABC = n mod ABC.
Here some modification is required (to maintain that equivalence) so that 'BC' + 'AC' + 'AB' = ABC + 1 where 'BC', 'AC' and 'AB' are the coefficients. Therefore, a(n) = 3B*C*(n mod A) + 5A*C*(n mod B) + 2A*B*(n mod C) so that 3*11*17 + 5*7*17 + 2*7*11 =7*11*17 + 1 = 561 + 595 + 154 = 1310.
This is an example with 3 modifications.
a(n) = n for n = 154, 308, 462, 561, 595, 616, 715, 749, 770, 869, 903, 924, 1023, 1057, 1078, 1122, 1156, 1177, 1190, 1211, 1232, 1276.

Links

Crossrefs

Cf. A255818 for an example with 1 modification and A256643 for 2 modifications.

Programs

  • Magma
    A:=7; B:=11; C:=17; [3*B*C*(n mod A)+5*A*C*(n mod B)+2*A*B*(n mod C): n in [1..60]]; // Bruno Berselli, Apr 14 2015
  • Mathematica
    Table[561*Mod[n,7]+595*Mod[n,11]+154*Mod[n,17],{n,40}] (* or *) LinearRecurrence[{-2,-3,-4,-5,-6,-7,-7,-7,-7,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7,7,7,7,7,6,5,4,3,2,1},{1310,2620,3930,5240,6550,7860,5243,6553,7863,9173,3938,5248,6558,3941,5251,6561,5253,6563,7873,9183,6566,1331,2641,3951,5261,6571,7881,5264,6574,7884,9194,10504,5269},40]  (* Harvey P. Dale, May 03 2023 *)
  • PARI
    my(A=7, B=11, C=17, nn = A*B*C); vector(nn, n, 3*B*C*(n % A) + 5*A*C*(n % B) + 2*A*B*(n % C)) \\ Michel Marcus, Apr 14 2015

Formula

G.f.: -x*(11780*x^31 +34030*x^30 +65440*x^29 +104700*x^28 +150500*x^27 +201530*x^26 +256480*x^25 +306187*x^24 +350651*x^23 +389872*x^22 +423850*x^21 +447350*x^20 +461682*x^19 +468156*x^18 +468082*x^17 +462770*x^16 +453530*x^15 +432510*x^14 +403638*x^13 +368224*x^12 +327578*x^11 +283010*x^10 +235830*x^9 +187348*x^8 +144109*x^7 +106113*x^6 +73360*x^5 +45850*x^4 +26200*x^3 +13100*x^2 +5240*x +1310) / ((x -1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^10 +x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^16 +x^15 +x^14 +x^13 +x^12 +x^11 +x^10 +x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Apr 14 2015

A255818 a(n) = 2*B*C*(n mod A) + A*C*(n mod B) + A*B*(n mod C) with A=3, B=5, C=7.

Original entry on oeis.org

106, 212, 108, 214, 215, 111, 112, 218, 114, 115, 221, 117, 223, 224, 15, 121, 227, 123, 229, 230, 21, 127, 233, 129, 130, 236, 132, 133, 239, 30, 136, 242, 138, 244, 140, 36, 142, 248, 144, 145, 251, 42, 148, 254, 45, 151, 257, 153, 154, 155, 51, 157, 263, 159
Offset: 1

Views

Author

Aaron Kastel, Apr 07 2015

Keywords

Comments

After 0 it cycles again from 106 (a(105)=0 so there are 105 (A*B*C) terms).
This is another variation on A256496, where a(n) = B*C*(n mod A) + A*C*(n mod B) + A*B*(n mod C), modified to take the values A=3, B=5, C=7 and still maintain the equivalence a(n) mod ABC = n mod ABC.
Here modification is required (to maintain that equivalence) so that 'BC' + 'AC' + 'AB' = ABC + 1 where 'BC', 'AC' and 'AB' are the coefficients. Therefore, a(n)= 2B*C*(n mod A) + A*C*(n mod B) + A*B*(n mod C) so that 2*5*7 + 3*7 + 3*5 = 3*5*7 = 70 + 21 + 15 = 106.
This is an example with 1 modification.
a(n) = n for n: 15, 21, 30, 36, 42, 45, 51, 57, 60, ..., 314. - Robert G. Wilson v, Apr 07 2015

Links

Crossrefs

Cf. A256643 for an example with 2 modifications and A256668 for 3 modifications.

Programs

  • Magma
    A:=3; B:=5; C:=7; [2*B*C*(n mod A)+A*C*(n mod B)+A*B*(n mod C): n in [1..60]]; // Bruno Berselli, Apr 14 2015
  • Mathematica
    f[n_] := 70 Mod[n, 3] + 21 Mod[n, 5] + 15 Mod[n, 7]; Array[f, 105] (* Robert G. Wilson v, Apr 07 2015 *)
LinearRecurrence[{-2,-3,-3,-3,-2,-1,1,2,3,3,3,2,1},{106,212,108,214,215,111,112,218,114,115,221,117,223},60] (* Harvey P. Dale, Sep 21 2024 *)

Formula

G.f.: -x*(314*x^11 +836*x^10 +1460*x^9 +1976*x^8 +2384*x^7 +2475*x^6 +2355*x^5 +1921*x^4 +1384*x^3 +850*x^2 +424*x +106) / ((x -1)*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Apr 14 2015

A256643 a(n) = B*C*(n mod A) + 2*A*C*(n mod B) + 3*A*B*(n mod C) with A=3, B=5, C=11.

Original entry on oeis.org

166, 332, 333, 499, 335, 336, 502, 668, 669, 505, 176, 177, 343, 509, 180, 346, 512, 513, 679, 515, 516, 187, 353, 354, 190, 356, 357, 523, 689, 360, 526, 692, 198, 364, 200, 201, 367, 533, 534, 370, 536, 537, 703, 374, 45, 211, 377, 378, 544, 380, 381, 547
Offset: 1

Views

Author

Aaron Kastel, Apr 07 2015

Keywords

Comments

After 0 it cycles again from 166 (a(165)=0 so there are 165 (A*B*C) terms).
This is another variation on A256496, where a(n) = B*C*(n mod A) + A*C*(n mod B) + A*B*(n mod C), modified to take the values A=3, B=5, C=11 and still maintain the equivalence a(n) mod ABC = n mod ABC.
Here modification is required (to maintain that equivalence) so that 'BC' + 'AC' + 'AB' = ABC + 1 where 'BC', 'AC' and 'AB' are the coefficients. Therefore, a(n)= B*C*(n mod A) + 2A*C*(n mod B) + 3A*B*(n mod C) so that 5*11 + 2*3*11 + 3*3*5 = 3*5*11 = 55 + 66 + 45 = 166.
This is an example with 2 modifications.

Links

Crossrefs

Cf. A255818 for an example with 1 modification and A256668 for 3 modifications.

Programs

  • Magma
    A:=3; B:=5; C:=11; [B*C*(n mod A)+2*A*C*(n mod B)+3*A*B*(n mod C): n in [1..165]]; // Bruno Berselli, Apr 14 2015

Formula

G.f.: -x*(824*x^15 +2306*x^14 +4280*x^13 +5921*x^12 +7229*x^11 +7710*x^10 +7530*x^9 +6855*x^8 +6180*x^7 +5505*x^6 +4830*x^5 +3826*x^4 +2659*x^3 +1495*x^2 +664*x +166) / ((x -1)*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)*(x^10 +x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Apr 14 2015

Extensions

Definition corrected by Bruno Berselli, Apr 14 2015

A256496 a(n) = 15(n mod 2) + 10(n mod 3) + 6(n mod 5).

Original entry on oeis.org

31, 32, 33, 34, 35, 6, 37, 38, 39, 10, 41, 12, 43, 44, 15, 16, 47, 18, 49, 20, 21, 22, 53, 24, 25, 26, 27, 28, 59, 0, 31, 32, 33, 34, 35, 6, 37, 38, 39, 10, 41, 12, 43, 44, 15, 16, 47, 18, 49, 20, 21, 22, 53, 24, 25, 26, 27, 28, 59, 0, 31, 32, 33, 34, 35, 6
Offset: 1

Views

Author

Aaron Kastel, Mar 31 2015

Keywords

Comments

After 0 it cycles again from 31.
This is the simplest example of a(n) = b*c*(n mod a) + a*c*(n mod b) + a*b*(n mod c) with a=2, b=3, c=5.
a(n) mod abc = n mod abc.
Other values for a,b,c require modification so that bc + ac + ab = abc + 1. For example, for a=3, b=5, c=7, a(n)= 2b*c*(n mod a) + a*c*(n mod b) + a*b*(n mod c) so that 2*5*7 + 3*7 + 3*5 = 3*5*7 = 70 + 21 + 15 = 106.
This expression (with a=3, b=5, c=7) (A255818) is found in the Kol Bo (Hebrew: כלבו), a book of religious Jewish law from the 13th to 14th centuries. It is given there as a method for calculating a person's age without anyone saying it explicitly.
a(n) = n for n = 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 53, 59.

Links

Crossrefs

Cf. A255818 for an example with 1 modification, A256643 for 2 modifications and A256668 for 3 modifications.

Programs

  • Mathematica
    Table[15 Mod[n, 2] + 10 Mod[n, 3] + 6 Mod[n, 5], {n, 60}] (* Michael De Vlieger, Mar 31 2015 *)
LinearRecurrence[{-2,-2,-1,0,1,2,2,1},{31,32,33,34,35,6,37,38},70] (* or *) PadRight[ {},70,{31,32,33,34,35,6,37,38,39,10,41,12,43,44,15,16,47,18,49,20,21,22,53,24,25,26,27,28,59,0}] (* Harvey P. Dale, Oct 31 2016 *)
  • PARI
    vector(30, n, 15*(n%2) + 10*(n%3) + 6*(n%5)) \\ Michel Marcus, Mar 31 2015

Formula

a(n) = 15(n mod 2) + 10(n mod 3) + 6(n mod 5).
G.f.: -x*(59*x^6+146*x^5+201*x^4+195*x^3+159*x^2+94*x+31) / ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Apr 07 2015
a(n) = -2*a(n-1) - 2*a(n-2) - a(n-3) + a(n-5) + 2*a(n-6) + 2*a(n-7) + a(n-8), for n>=9. - Vaclav Kotesovec, Apr 07 2015

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