A068009 -id:A068009 - cp's OEIS frontend

A068012 Number of subsets of {1,2,3,...,n} that sum to 0 mod 6.

1, 1, 1, 2, 3, 6, 12, 22, 44, 88, 172, 344, 688, 1368, 2736, 5472, 10928, 21856, 43712, 87392, 174784, 349568, 699072, 1398144, 2796288, 5592448, 11184896, 22369792, 44739328, 89478656, 178957312, 357914112, 715828224, 1431656448, 2863311872, 5726623744

Offset: 0

Antti Karttunen, Feb 11 2002

6th row of A068009.

b:= proc(n, s) option remember; `if`(n=0, `if`(s=0, 1, 0),
      b(n-1, s)+b(n-1, irem(s+n, 6)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, May 02 2025

b[n_, s_] := b[n, s] = If[n == 0, If[s == 0, 1, 0], b[n-1, s] + b[n-1, Mod[s+n, 6]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Sep 04 2025, after Alois P. Heinz *)

Empirical g.f.: (x-1)*(2*x^4+3*x^3+x^2-1) / ((2*x-1)*(2*x^3-1)). - Colin Barker, Dec 22 2012

A068013 Number of subsets of {1,2,3,...,n} that sum to 0 mod 7.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 10, 20, 38, 74, 146, 294, 586, 1172, 2344, 4684, 9364, 18724, 37452, 74900, 149800, 299600, 599192, 1198376, 2396744, 4793496, 9586984, 19173968, 38347936, 76695856, 153391696, 306783376, 613566768, 1227133520, 2454267040

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

7th row of A068009.

Empirical G.f.: -(2*x^7+x^5-x^4+x^3+x^2+x-1) / ((2*x-1)*(2*x^7-1)). [Colin Barker, Dec 22 2012]

A068032 Number of subsets of {1,2,3,...,n} that sum to 0 mod 11.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 6, 12, 24, 47, 94, 188, 374, 746, 1490, 2978, 5958, 11916, 23832, 47664, 95326, 190652, 381304, 762604, 1525204, 3050404, 6100804, 12201612, 24403224, 48806448, 97612896, 195225788, 390451576, 780903152, 1561806296

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

11th row of A068009.

Empirical G.f.: -(2*x^11+x^9-x^5+x^4+x^3+x^2+x-1) / ((2*x-1)*(2*x^11-1)). [Colin Barker, Dec 22 2012]

A068033 Number of subsets of {1,2,3,...,n} that sum to 0 mod 12.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 6, 11, 22, 44, 86, 172, 344, 684, 1368, 2736, 5464, 10928, 21856, 43696, 87392, 174784, 349536, 699072, 1398144, 2796224, 5592448, 11184896, 22369664, 44739328, 89478656, 178957056, 357914112, 715828224, 1431655936

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

12th row of A068009.

Empirical G.f.: -(2*x^8-x^7-2*x^6-3*x^5-x^4+3*x^3+x^2+x-1) / ((2*x-1)*(2*x^3-1)). [Colin Barker, Dec 22 2012]

A068034 Number of subsets of {1,2,3,...,n} that sum to 0 mod 13.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 10, 20, 39, 79, 158, 316, 632, 1262, 2522, 5042, 10082, 20164, 40330, 80660, 161320, 322638, 645278, 1290556, 2581112, 5162224, 10324444, 20648884, 41297764, 82595524, 165191048, 330382100, 660764200, 1321528400

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

13th row of A068009.

Table[Count[Subsets[Range[n]], ?(Divisible[Total[#], 13]&)], {n,20}] (* _Harvey P. Dale, Feb 14 2012 *)

Empirical G.f.: -(2*x^13-x^10+x^9-x^6+x^4+x^3+x^2+x-1) / ((2*x-1)*(2*x^13-1)). [Colin Barker, Dec 22 2012]

A068035 Number of subsets of {1,2,3,...,n} that sum to 0 mod 14.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 10, 19, 37, 73, 147, 293, 586, 1172, 2342, 4682, 9362, 18726, 37450, 74900, 149800, 299596, 599188, 1198372, 2396748, 4793492, 9586984, 19173968, 38347928, 76695848, 153391688, 306783384, 613566760, 1227133520, 2454267040

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

14th row of A068009.

Empirical G.f.: -(2*x^13+x^12-3*x^11-x^10-x^9-x^8+2*x^7-x^6+x^4+x^3+x^2+x-1) / ((2*x-1)*(2*x^7-1)). [Colin Barker, Dec 22 2012]

A068036 Number of subsets of {1,2,3,...,n} that sum to 0 mod 15.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 9, 18, 36, 70, 138, 276, 548, 1096, 2192, 4374, 8744, 17486, 34958, 69916, 139830, 279630, 559260, 1118520, 2236988, 4473964, 8947920, 17895736, 35791472, 71582944, 143165660, 286331296, 572662588, 1145324764, 2290649528

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

15th row of A068009.

Empirical G.f.: -(8*x^23 +4*x^21 -8*x^20 +4*x^19 -4*x^18 +2*x^17 -4*x^16 +2*x^15 -8*x^14 +2*x^12 +8*x^11 +4*x^10 +4*x^9 -6*x^8 -3*x^7 -5*x^6 -x^4 +3*x^3 +x^2 +x -1) / ((2*x-1) * (2*x^3-1) * (2*x^5-1) * (2*x^15-1)). - Colin Barker, Dec 22 2012

A068037 Number of subsets of {1,2,3,...,n} that sum to 0 mod 16.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 9, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Antti Karttunen, Feb 11 2002

Index entries for linear recurrences with constant coefficients, signature (2).

16th row of A068009.

a(n) = 2^(-4+n) for n>7. G.f.: (2*x^8 -x^7 -2*x^6 +x^5 +x^4 +x^3 +x^2 +x -1) / (2*x -1). - Colin Barker, Dec 22 2012

A068039 Number of subsets of {1,2,3,...,n} that sum to 0 mod 18.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 8, 16, 30, 58, 115, 229, 456, 912, 1824, 3643, 7286, 14572, 29132, 58262, 116522, 233024, 466048, 932096, 1864150, 3728300, 7456600, 14913112, 29826220, 59652436, 119304704, 238609408, 477218816, 954437292, 1908874584

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

18th row of A068009.

Empirical G.f.: -(12*x^35 +2*x^34 -16*x^33 +8*x^32 +16*x^31 -6*x^30 +2*x^29 +12*x^28 +16*x^27 -8*x^26 -32*x^25 -18*x^24 +6*x^23 +16*x^22 +6*x^21 +2*x^20 +12*x^19 -12*x^18 -6*x^17 -x^16 +8*x^15 -4*x^14 -8*x^13 +3*x^12 -x^11 -6*x^10 +4*x^8 +6*x^7 +x^6 +x^5 +x^4 -3*x^3 -x^2 -x +1) / ((2*x -1)*(2*x^3 -1)*(2*x^6 -1)*(2*x^9 -1)*(2*x^18 -1)). [Colin Barker, Dec 23 2012]

A068040 Number of subsets of {1,2,3,...,n} that sum to 0 mod 19.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 7, 14, 27, 54, 108, 216, 431, 862, 1724, 3450, 6899, 13798, 27596, 55190, 110378, 220754, 441506, 883010, 1766020, 3532046, 7064092, 14128182, 28256364, 56512728, 113025456, 226050910, 452101820, 904203640, 1808407284

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

19th row of A068009.

Empirical G.f.: -(2*x^19+x^17-2*x^16+x^13+x^9-3*x^7+x^5+x^4+x^3+x^2+x-1) / ((2*x-1)*(2*x^19-1)). - Colin Barker, Dec 22 2012

cp's OEIS Frontend

A068012 Number of subsets of {1,2,3,...,n} that sum to 0 mod 6.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A068013 Number of subsets of {1,2,3,...,n} that sum to 0 mod 7.

Views

Author

Keywords

Crossrefs

Formula

A068032 Number of subsets of {1,2,3,...,n} that sum to 0 mod 11.

Views

Author

Keywords

Crossrefs

Formula

A068033 Number of subsets of {1,2,3,...,n} that sum to 0 mod 12.

Views

Author

Keywords

Crossrefs

Formula

A068034 Number of subsets of {1,2,3,...,n} that sum to 0 mod 13.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A068035 Number of subsets of {1,2,3,...,n} that sum to 0 mod 14.

Views

Author

Keywords

Crossrefs

Formula

A068036 Number of subsets of {1,2,3,...,n} that sum to 0 mod 15.

Views

Author

Keywords

Crossrefs

Formula

A068037 Number of subsets of {1,2,3,...,n} that sum to 0 mod 16.

Views

Author

Keywords

Links

Crossrefs

Formula

A068039 Number of subsets of {1,2,3,...,n} that sum to 0 mod 18.

Views

Author

Keywords

Crossrefs

Formula

A068040 Number of subsets of {1,2,3,...,n} that sum to 0 mod 19.

Views

Author

Keywords

Crossrefs

Formula