cp's OEIS Frontend

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: Xavier Roulleau

Xavier Roulleau's wiki page.

Xavier Roulleau has authored 6 sequences.

A381351 Number of subsets of 9 integers between 1 and n such that their sum is 3 modulo n.

Original entry on oeis.org

1, 5, 19, 55, 143, 335, 715, 1430, 2703, 4862, 8398, 14000, 22610, 35530, 54484, 81719, 120175, 173592, 246675, 345345, 476913, 650325, 876525, 1168710, 1542684, 2017356, 2615103, 3362260, 4289780, 5433736, 6835972, 8544965, 10616489, 13114465
Offset: 10

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Author

Xavier Roulleau, Feb 21 2025

Keywords

Comments

For s an integer such that GCD(s,9)=3, this is also the number of subsets of 9 integers between 1 and n such that their sum is s modulo n.

Examples

			For n=10, there are a(10)=1 order 9 subsets of Z/10Z with sum equal to 3 mod 10.

Links

Crossrefs

Cf. A011796, A031164, A032194, A381289, A381290, A381291.

Formula

G.f.: x^10*(1 - x + 4*x^2 - 6*x^3 + 15*x^4 - 17*x^5 + 15*x^6 - 6*x^7 + 4*x^8 - x^9 + x^10)/((1 - x)^6*(1 - x^3)^2*(1 - x^9)).

A381350 Number of subsets of 8 integers between 1 and n such that their sum is 2 modulo n.

Original entry on oeis.org

1, 5, 15, 42, 99, 217, 429, 808, 1430, 2438, 3978, 6308, 9690, 14550, 21318, 30664, 43263, 60115, 82225, 111038, 148005, 195143, 254475, 328752, 420732, 534076, 672452, 840648, 1043460, 1287036, 1577532, 1922736, 2330445, 2810385, 3372291, 4028178, 4790071, 5672645
Offset: 9

Views

Author

Xavier Roulleau, Feb 21 2025

Keywords

Comments

For s an integer such that GCD(s,8)=2, this is also the number of subsets of 8 integers between 1 and n such that their sum is s modulo n.

Examples

			For n=10, there are a(10)=5 order 8 subsets of Z/10Z with sum equal to 2 mod 10.

Links

Crossrefs

Cf. A011796, A031164, A056594, A381289, A381290, A381291.

Formula

G.f.: x^9*(1 + x - x^2 + 6*x^3 + 2*x^5 + 6*x^7 - x^8 + x^9 + x^10)/((1 - x)^4*(1 - x^2)^2*(1 - x^4)*(1 - x^8)).
a(n) = (n - 4)*(2520 - 24*(281 + 35*(-1)^n)*n + 5*(1039 + 21*(-1)^n)*n^2 - 2112*n^3 + 452*n^4 - 48*n^5 + 2*n^6 - 2520*A056594(n))/80640. - Stefano Spezia, Feb 21 2025

A381289 Number of subsets of 6 integers between 1 and n such that their sum is 0 modulo n.

Original entry on oeis.org

1, 3, 10, 20, 42, 76, 132, 212, 335, 497, 728, 1028, 1428, 1932, 2586, 3384, 4389, 5601, 7084, 8844, 10966, 13442, 16380, 19780, 23751, 28301, 33566, 39536, 46376, 54086, 62832, 72624, 83661, 95931, 109668, 124872, 141778, 160398, 181006
Offset: 7

Views

Author

Xavier Roulleau and David Broadhurst, Feb 19 2025

Keywords

Comments

For an integer s multiple of 6, this is also the number of subsets of 6 integers between 1 and n such that their sum is s modulo n.

Examples

			For n=8, there are a(8)=3 order 6 subsets of Z/8Z with sum equal to 0 mod 8.

Links

Crossrefs

Cf. A011796.

Formula

G.f.: x^7*(1 + x + 3*x^2 + 2*x^4 + 4*x^5 + x^6 - x^7 + x^8)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)*(1 - x^6)). - David Broadhurst, Feb 19 2025

A381291 Number of subsets of 8 integers between 1 and n such that their sum is 0 modulo n.

Original entry on oeis.org

1, 5, 15, 43, 99, 217, 429, 809, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30666, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328755, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922740, 2330445, 2810385, 3372291, 4028183, 4790071
Offset: 9

Views

Author

Xavier Roulleau and David Broadhurst, Feb 19 2025

Keywords

Comments

For an integer s multiple of 8, this is also the number of subsets of 8 integers between 1 and n such that their sum is s modulo n.

Examples

			For n=10, there are a(10)=5 order 8 subsets of Z/10Z with sum equal to 0 mod 10.

Links

Crossrefs

Cf. A381290, A381289, A011796.

Formula

G.f.: x^9*(1 + x - x^2 + 7*x^3 - 4*x^4 + 6*x^5 + 4*x^6 - 4*x^7 + 3*x^8 + 5*x^9 - 3*x^10 + x^11)/((1 - x)^4*(1 - x^2)^2*(1 - x^4)*(1 - x^8)).

A381290 Number of subsets of 6 integers between 1 and n such that their sum is 1 modulo n.

Original entry on oeis.org

1, 4, 9, 22, 42, 78, 132, 217, 333, 504, 728, 1035, 1428, 1944, 2583, 3399, 4389, 5616, 7084, 8866, 10962, 13468, 16380, 19806, 23751, 28336, 33561, 39576, 46376, 54126, 62832, 72675, 83655, 95988, 109668, 124929, 141778, 160468, 180999
Offset: 7

Views

Author

Xavier Roulleau and David Broadhurst, Feb 19 2025

Keywords

Comments

For s an integer such that GCD(s,6)=1, this is also the number of subsets of 6 integers between 1 and n such that their sum is s modulo n.

Examples

			For n=7, a(7)=1 since the set {0,1,2,3,4,5} is the unique order 6 subset of Z/7Z with sum equal to 1 mod 7.

Links

Crossrefs

Cf. A381289, A011796.

Formula

G.f.: x^7*(1 + 2*x + 3*x^3 + 2*x^4 + 2*x^5 + x^6 + x^7)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)*(1 - x^6)).

A379920 Number of irreducible conic curves containing 6 points of a cyclic order n-torsion subgroup of an elliptic curve.

Original entry on oeis.org

1, 2, 7, 13, 36, 67, 113, 196, 312, 455, 693, 984, 1353, 1869, 2508, 3261, 4284, 5478, 6898, 8684, 10780, 13174, 16146, 19516, 23381, 27976, 33201, 39041, 45936, 53601, 62187, 72048, 83028, 95109, 108927, 124068, 140749, 159467, 179998, 202321, 227304, 254380, 283844, 316360
Offset: 9

Views

Author

Xavier Roulleau, Jan 17 2025

Keywords

Comments

For n < 9, there are no such curves.
There are precisely 7 primes in this sequence, namely
a(10)=2, a(11)=7, a(12)=13, a(14)=67, a(15)=113, a(36)=39041, a(63)=907237.

Examples

			For n=9, there is a unique irreducible conic that contains 6 points in a cyclic order n torsion subgroup of an elliptic curve, and for n=11 there are 7 such conics.

Links

Programs

  • Magma
    sq:=[];
for NN in [9..30] do
G:=Integers(NN);
SG:={q: q in G};
QNT:=Subsets(SG,5);
QNT:={q join {-(&+ q)} : q in QNT | not -(&+ q) in q};
TRS:=Subsets(SG,3);
TRS:={q : q in TRS|&+q eq 0};
QNT:={q :q in QNT| not #{u : u in TRS| u subset q} ge 1};
Append(~sq,#QNT);
end for;
sq;
  • PARI
    {a(n)=[(n-6)*(n^4-19*n^3+121*n^2-384*n+840),(n-1)*(n-4)*(n-5)*(n-7)*(n-8),(n-2)*(n-4)*(n-8)*(n^2-11*n+25),(n-3)*(n^4-22*n^3+169*n^2-588*n+1200)][gcd(n,6)%6+1]/6!;} \\ David Broadhurst, Jan 17 2025

Formula

G.f.: x^9*(1 + x + 3*x^2 + 2*x^3 + 12*x^4 + 14*x^5 - 3*x^6 - x^7 + 7*x^8)/((1 - x)*(1 - x^2)^2*(1 - x^3)^2*(1 - x^6)) \\ David Broadhurst, Jan 17 2025

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