A068009 -id:A068009 - cp's OEIS frontend

A068049 The first term greater than one on each row of A068009. a(n) = A068009[n, A002024[n]].

2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 4, 3, 3, 2, 2, 2, 5, 4, 3, 3, 2, 2, 2, 6, 5, 4, 3, 3, 2, 2, 2, 7, 6, 5, 4, 3, 3, 2, 2, 2, 9, 7, 6, 5, 4, 3, 3, 2, 2, 2, 11, 9, 7, 6, 5, 4, 3, 3, 2, 2, 2, 13, 11, 9, 7, 6, 5, 4, 3, 3, 2, 2, 2, 16, 13, 11, 9, 7, 6, 5, 4, 3, 3, 2, 2, 2, 19, 16, 13, 11, 9, 7, 6, 5

Offset: 1

Antti Karttunen, Feb 11 2002

In row 1 of A068009 the first term > 1 is found at position 1, for rows 2 & 3 at position 2, for rows 4,5,6 at position 3, for rows 7,8,9,10 at position 4 etc., thus it is natural to view this also as a triangular table.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

a(n) = A000009(A025581(n-1))+1. Specifically, the left edge is equal to A000009[n]+1 (i.e. apart from the first term = A052839) and the right edge is all-2 sequence A007395.

[seq(A000009(A025581(j-1))+1,j=1..99)];
A025581 := n-> binomial(1+floor(1/2+sqrt(2+2*n)),2)-(n+1);
N := 100; t1 := series(mul(1+x^k,k=1..N),x,N); A000009 := proc(n) coeff(t1,x,n); end;

a[n_] := PartitionsQ[(1/2)(Floor[Sqrt[2n]+1/2]^2 + Floor[Sqrt[2n]+1/2] - 2n)] + 1; Array[a, 100] (* Jean-François Alcover, Mar 02 2016 *)

A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 7, 7, 6, 5, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4

Offset: 0

N. J. A. Sloane, Mar 22 2000

Or, triangle T(n,k) read by rows, giving number of subsets of {1,2,...,n} with sum k. - Roger CUCULIERE (cuculier(AT)imaginet.fr), Nov 19 2000
Row n consists of A000124(n) terms. These are also the successive vectors (their nonzero elements) when one starts with the infinite vector (of zeros) with 1 inserted somewhere and then shifts it one step (right or left) and adds to the original, then shifts the result two steps and adds, three steps and adds, etc. - Antti Karttunen, Feb 13 2002
T(n,k) = number of partitions of k into distinct parts <= n. Triangle of distribution of Wilcoxon's signed rank statistic. - Mitch Harris, Mar 23 2006
T(n,k) = number of binary words of length n in which the sum of the positions of the 0's is k. Example: T(4,5)=2 because we have 0110 (sum of the positions of the 0's is 1+4=5) and 1001 (sum of the positions of the 0's is 2+3=5). - Emeric Deutsch, Jul 23 2006
A fair coin is flipped n times. You receive i dollars for a "success" on the i-th flip, 1<=i<=n. T(n,k)/2^n is the probability that you will receive exactly k dollars. Your expectation is n(n+1)/4 dollars. - Geoffrey Critzer, May 16 2010
From Gus Wiseman, Jan 02 2023: (Start)
With offset 1, also the number of integer compositions of n whose partial sums add up to k for k = n..n(n+1)/2. For example, row n = 6 counts the following compositions:
  6 15 24 33  42  51  141  231  321  411  1311 2211  3111  12111 21111 111111
          114 123 132 222  312  1131 1221 2121 11121 11211
                  213 1113 1122 1212 2112 1111
(End)

			Triangle begins:
  1;
  1, 1;
  1, 1, 1, 1;
  1, 1, 1, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1;
  ...
Row n = 4 counts the following binary words, where k = sum of positions of zeros:
  1111  0111  1011  0011  0101  0110  0001  0010  0100  1000  0000
                    1101  1110  1001  1010  1100
Row n = 5 counts the following strict partitions of k with all parts <= n (0 is the empty partition):
  0  1  2  3  4  5  42  43  53  54  532  542  543  5431 5432 54321
           21 31 32 51  52  431 432 541  5321 5421
                 41 321 421 521 531 4321

A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.

Alois P. Heinz, Rows n = 0..40, flattened
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
FindStat - Combinatorial Statistic Finder, The major index of an integer composition
Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences. (Russian), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy] See Table 1.
F. Wilcoxon, Individual Comparisons by Ranking Methods, Biometrics Bulletin, v. 1, no. 6 (1945), pp. 80-83.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
A. V. Yurkin, About the evident description of distribution of beams and "wavy geometrical trajectories" in long thin pipes, 2014 (original in Russian).
A. V. Yurkin, Symmetric triangle of Pascal and non-linear arithmetic parallelepiped, Book Manuscript, Research Gate 2015.

Cf. A053633, A068009.
Rows reduced modulo 2 and interpreted as binary numbers: A068052, A068053. Rows converge towards A000009.
Row sums give A000079.
Cf. A001788, A028362.
Cf. A285101 (multiplicative encoding of each row), A285103 (number of odd terms on row n), A285105 (number of even terms).
Row lengths are A000124.
A reciprocal version is (A033999, A219977, A291983, A291984, A291985, ...).
A negative version is A231599.
A version for partitions is A358194, reversed partitions A264034.
Cf. A025591, A029931, A063865, A152947, A318283, A359042.

with(gfun,seriestolist); map(op,[seq(seriestolist(series(mul(1+(z^i), i=1..n),z,binomial(n+1,2)+1)), n=0..10)]); # Antti Karttunen, Feb 13 2002
# second Maple program:
g:= proc(n) g(n):= `if`(n=0, 1, expand(g(n-1)*(1+x^n))) end:
T:= n-> seq(coeff(g(n), x, k), k=0..degree(g(n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 19 2012

Table[CoefficientList[ Series[Product[(1 + t^i), {i, 1, n}], {t, 0, 100}], t], {n, 0, 8}] // Grid (* Geoffrey Critzer, May 16 2010 *)

From Mitch Harris, Mar 23 2006: (Start)
T(n,k) = T(n-1, k) + T(n-1, k-n), T(0,0)=1, T(0,k) = 0, T(n,k) = 0 if k < 0 or k > (n+1 choose 2).
G.f.: (1+x)*(1+x^2)*...*(1+x^n). (End)
Sum_{k>=0} k * T(n,k) = A001788(n). - Alois P. Heinz, Feb 09 2017
max_{k>=0} T(n,k) = A025591(n). - Alois P. Heinz, Jan 20 2023

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94, 172, 316, 586, 1096, 2048, 3856, 7286, 13798, 26216, 49940, 95326, 182362, 349536, 671092, 1290556, 2485534, 4793492, 9256396, 17895736, 34636834, 67108864, 130150588, 252645136, 490853416

Offset: 0

N. J. A. Sloane

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.
Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).
Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015
Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016
From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:
  1  2  3    4    5      6      7        8
        123  234  135    246    147      258
                  345    456    357      468
                  12345  1236   567      678
                         1456   2347     1348
                         23456  2567     1568
                                12467    3458
                                13457    3678
                                34567    12458
                                1234567  14578
                                         23578
                                         24568
                                         45678
                                         123468
                                         135678
                                         2345678
(End)
Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

			For n=3 the 2 output sequences are 000111000111... and 010101...
For n=5 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
For n=6 there are 6 such sequences.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
J. Hedetniemi and K. R. Hutson, Equilibrium of shortest path load in ring network, Congressus Numerant., 203 (2010), 75-95. See p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Diff. Eqs. 20 (1) (2008) 201, eq. (39)

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
A. E. Brouwer, The Enumeration of Locally Transitive Tournaments, Math. Centr. Report ZW138, Amsterdam, 1980.
S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, Estimating the size of correcting codes using extremal graph problems, Optimization, 227-243, Springer Optim. Appl., 32, Springer, New York, 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Sébastien Designolle, Tamás Vértesi, and Sebastian Pokutta, Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms, arXiv:2310.20677 [quant-ph], 2023.
T. M. A. Fink, Exact dynamics of the critical Kauffman model with connectivity one, arXiv:2302.05314 [cond-mat.stat-mech], 2023.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, Challenge Problems: Independent Sets in Graphs
Yan Bo Ti, Gabriel Verret, and Lukas Zobernig, Abelian Varieties with p-rank Zero, arXiv:2203.08401 [math.NT], 2022.
Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra Appl. 530, 414-444 (2017).
Index entries for sequences related to tournaments
Index entries for sequences related to necklaces
Index entries for sequences related to subset sums modulo m

The main diagonal of table A068009, the left edge of triangle A053633.
Cf. A000048, A000031, A000013, A053634, A182469.
Subsets whose mean is an element are A065795.
Dominated by A082550.
Partitions containing their mean are A237984.
Subsets containing n but not their mean are A327477.
Cf. A051293, A135342, A240850, A327471, A327478.

a000016 0 = 1
a000016 n = (`div` (2 * n)) $ sum $
   zipWith (*) (map a000010 oddDivs) (map ((2 ^) . (div n)) $ oddDivs)
   where oddDivs = a182469_row n
-- Reinhard Zumkeller, May 01 2012

A000016 := proc(n) local d, t; if n = 0 then return 1 else t := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t := t + NumberTheory:-Totient(d)* 2^(n/d)/(2*n) fi od; return t fi end:

a[0] = 1; a[n_] := Sum[Mod[k, 2] EulerPhi[k]*2^(n/k)/(2*n), {k, Divisors[n]}]; Table[a[n], {n, 0, 35}](* Jean-François Alcover, Feb 17 2012, after Pari *)

a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n));

from sympy import totient, divisors
def A000016(n): return sum(totient(d)<>(~n&n-1).bit_length(),generator=True))//n if n else 1 # Chai Wah Wu, Feb 21 2023

a(n) = Sum_{odd d divides n} (phi(d)*2^(n/d))/(2*n), n>0.
a(n) = A063776(n)/2.
a(n) = 2^(n-1) - A327477(n). - Gus Wiseman, Sep 14 2019

More terms from Michael Somos, Dec 11 1999

A063776 Number of subsets of {1,2,...,n} which sum to 0 modulo n.

Original entry on oeis.org

2, 2, 4, 4, 8, 12, 20, 32, 60, 104, 188, 344, 632, 1172, 2192, 4096, 7712, 14572, 27596, 52432, 99880, 190652, 364724, 699072, 1342184, 2581112, 4971068, 9586984, 18512792, 35791472, 69273668, 134217728, 260301176, 505290272, 981706832

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 16 2001

From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} that are empty or contain n and have integer mean. If the subsets are not required to contain n, we get A327475. For example, the a(1) = 2 through a(6) = 12 subsets are:
  {}   {}   {}       {}       {}           {}
  {1}  {2}  {3}      {4}      {5}          {6}
            {1,3}    {2,4}    {1,5}        {2,6}
            {1,2,3}  {2,3,4}  {3,5}        {4,6}
                              {1,3,5}      {1,2,6}
                              {3,4,5}      {1,5,6}
                              {1,2,4,5}    {2,4,6}
                              {1,2,3,4,5}  {4,5,6}
                                           {1,2,3,6}
                                           {1,4,5,6}
                                           {2,3,5,6}
                                           {2,3,4,5,6}
(End)

			G.f. = 2*x + 2*x^2 + 4*x^3 + 4*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + 32*x^8 + 60*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..3333 (first 200 terms from T. D. Noe)
T. Amdeberhan, M. Can and V. Moll, Broken bracelets, Molien series, paraffin wax and the elliptic curve 48a4, SIAM Journal of Discrete Math., v.25, 2011, p. 1843. See equation (1.2).
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
N. Kitchloo and L. Pachter, An interesting result about subset sums, preprint (pdf file), 1993.
N. Kitchloo and L. Pachter, An interesting result about subset sums, preprint (pdf file), 1993.

The superdiagonal of A068009.
Cf. A000010, A000013, A051293, A053633, A053634, A053636, A054539, A082550.
Cf. A000016, A065795, A327475, A327481.

a063776 n = a053636 n `div` n  -- Reinhard Zumkeller, Sep 13 2013

Table[a = Select[ Divisors[n], OddQ[ # ] &]; Apply[Plus, 2^(n/a)*EulerPhi[a]]/n, {n, 1, 35}]
a[ n_] := If[ n < 1, 0, 1/n Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]]; (* Michael Somos, May 09 2013 *)
Table[Length[Select[Subsets[Range[n]],#=={}||MemberQ[#,n]&&IntegerQ[Mean[#]]&]],{n,0,10}] (* Gus Wiseman, Sep 14 2019 *)

{a(n) = if( n<1, 0, 1 / n * sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))}; /* Michael Somos, May 09 2013 */

a(n) = sumdiv(n, d, (d%2)* 2^(n/d)*eulerphi(d))/n; \\ Michel Marcus, Feb 10 2016

from sympy import totient, divisors
def A063776(n): return (sum(totient(d)<>(~n&n-1).bit_length(),generator=True))<<1)//n # Chai Wah Wu, Feb 21 2023

a(n) = (1/n) * Sum_{d divides n and d is odd} 2^(n/d) * phi(d).
a(n) = (1/n) * A053636(n). - Michael Somos, May 09 2013
a(n) = 2 * A000016(n).
For odd n, a(n) = A000031(n).
G.f.: -Sum_{m >= 0} (phi(2*m + 1)/(2*m + 1)) * log(1 - 2*x^(2*m + 1)). - Petros Hadjicostas, Jul 13 2019
a(n) = A082550(n) + 1. - Gus Wiseman, Sep 14 2019

More terms from Vladeta Jovovic, Aug 20 2001

A068010 Number of subsets of {1,2,3,...,n} that sum to 0 mod 3.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 24, 44, 88, 176, 344, 688, 1376, 2736, 5472, 10944, 21856, 43712, 87424, 174784, 349568, 699136, 1398144, 2796288, 5592576, 11184896, 22369792, 44739584, 89478656, 178957312, 357914624, 715828224, 1431656448, 2863312896

Offset: 0

Antti Karttunen, Feb 11 2002

Third row of A068009.

			a(4)=6 because we have: {}, {3}, {1,2}, {2,4}, {1,2,3}, {2,3,4}. - _Geoffrey Critzer_, Jan 18 2014

Charles R Greathouse IV, Table of n, a(n) for n = 0..3323
Sophie LeBlanc, Jan 20 2002, sci.math posting
Index entries for linear recurrences with constant coefficients, signature (2,0,2,-4).

A068010 := n -> (2^n + 2^((n + 1 + (4/sqrt(3))*cos(((4*n)+1)*Pi/6))/3))/3;

Table[nn=(n^2+n)/2;Total[Table[Coefficient[Series[Product[1+x^i,{i,1,n}],{x,0,nn}],x^(3k)],{k,1,nn}]]+1,{n,1,33}] (* Geoffrey Critzer, Jan 18 2014 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -4,2,0,2]^n*[1;1;2;4])[1,1] \\ Charles R Greathouse IV, Mar 27 2017

a(0)=1, a(1)=1, a(n) = 2*a(n-1) if 3 does not divide n-1 and a(n) = 2*a(n-1)-(2^((n-1)/3)) if 3 divides n-1.
a(n) = (2^n + 2^((n + 1 + (4/sqrt(3))*cos(((4*n)+1)*Pi/6))/3))/3. - Fred Galvin
G.f.: (1-x-2*x^3)/(1-2*x-2*x^3+4*x^4). - Colin Barker, Feb 03 2012
a(0)=1, a(1)=1, a(2)=2, a(n) = 2*a(n-3) + 2^(n - 2), n>=3. - Baris Arslan, Mar 27 2017

A068030 Number of subsets of {1,2,3,...,n} that sum to 0 mod 9.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 15, 30, 60, 116, 230, 458, 912, 1824, 3648, 7286, 14572, 29144, 58264, 116524, 233044, 466048, 932096, 1864192, 3728300, 7456600, 14913200, 29826224, 59652440, 119304872, 238609408, 477218816, 954437632, 1908874584

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

Robert Israel, Table of n, a(n) for n = 0..3321
Mathematics StackExchange, Let S = 1,2,3,4,...,2070 find the number of subsets of S whose sum of elements in S is divisible by 9
Robert Israel, Proof of conjectured g.f.

9th row of A068009.

G:= Array(0..100, 0..8):
G[0,0]:= 1; for j from 1 to 8 do G[0,j]:= 0 od:
for n from 1 to 100 do
  for j from 0 to 8 do
     k:= j - n mod 9;
     G[n,j]:= G[n-1,j] + G[n-1,k];
od od:
seq(G[n,0],n=0..100); # Robert Israel, Apr 30 2019

Empirical G.f.: -(4*x^12-2*x^9-x^7+2*x^6+2*x^5+2*x^4-3*x^3-x^2-x+1) / ((2*x-1)*(2*x^3-1)*(2*x^9-1)). [Colin Barker, Dec 22 2012]

Empirical G.f. verified: see link. - Robert Israel, Apr 30 2019

A068031 Number of subsets of {1,2,3,...,n} that sum to 0 mod 10.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 13, 26, 52, 104, 206, 410, 820, 1640, 3280, 6556, 13108, 26216, 52432, 104864, 209720, 419432, 838864, 1677728, 3355456, 6710896, 13421776, 26843552, 53687104, 107374208, 214748384, 429496736, 858993472, 1717986944, 3435973888, 6871947712

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

10th 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, 10)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..36);  # Alois P. Heinz, May 02 2025

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

A068038 Number of subsets of {1,2,3,...,n} that sum to 0 mod 17.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 8, 15, 30, 60, 120, 241, 482, 964, 1928, 3856, 7712, 15422, 30842, 61682, 123362, 246722, 493446, 986896, 1973790, 3947580, 7895160, 15790320, 31580642, 63161284, 126322568, 252645136, 505290272, 1010580544, 2021161084

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

17th row of A068009.

{A068038(n)=local(v,v1);v=vector(17);v[1]=1;for(i=1,n,v1=vector(17);for(j=0,16,v1[j+1]=v[j+1]+v[(j-i)%17+1]);v=v1);v[1]} \\ Max Alekseyev, Jul 23 2005

Empirical G.f.: -(2*x^17-x^12+x^8-2*x^7-x^6+x^5+x^4+x^3+x^2+x-1) / ((2*x-1)*(2*x^17-1)). - Colin Barker, Dec 22 2012

Rechecked by Max Alekseyev, Jul 23 2005

A068045 Number of subsets of {1,2,3,...,n} that sum to 0 mod 64.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 21, 105, 316, 705, 1297, 2181, 3837, 7596, 15962, 32998, 66173, 131379, 261910, 524012, 1048568, 2097228, 4194318, 8388600, 16777216, 33554434, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296

Offset: 0

Antti Karttunen, Feb 11 2002

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

64th row of A068009.

G.f.: (4*x^32 -2*x^31 -16*x^30+36*x^29 +138*x^28 -92*x^27 -544*x^26 -192*x^25 +848*x^24 +967*x^23 -177*x^22 -1074*x^21 -770*x^20 +78*x^19 +525*x^18 +413*x^17 +113*x^16 -73*x^15 -106*x^14 -63*x^13 -17*x^12 +x^10 +x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x-1) / (2*x-1). - Alois P. Heinz, Jan 18 2014

A068011 Number of subsets of {1,2,3,...,n} that sum to 0 mod 5.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 14, 26, 52, 104, 208, 412, 820, 1640, 3280, 6560, 13112, 26216, 52432, 104864, 209728, 419440, 838864, 1677728, 3355456, 6710912, 13421792, 26843552, 53687104, 107374208, 214748416, 429496768, 858993472, 1717986944, 3435973888, 6871947776

Offset: 0

Antti Karttunen, Feb 11 2002

nonn

For n>2, a(n) = 2 * A068031(n).

Sophie LeBlanc, Jan 20 2002, sci.math posting
Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,2,-4).

5th row of A068009.

A068011_rec := proc(n); if(0 = n) then RETURN(1); fi; if(1 = (n mod 5)) then RETURN(2*A068011_rec(n-1)-2^((n-1)/5)); fi; if(2 = (n mod 5)) then RETURN(2*A068011_rec(n-1)-2^((n-2)/5)); fi; RETURN(2*A068011_rec(n-1)); end;
# second Maple program:
b:= proc(n, s) option remember; `if`(n=0, `if`(s=0, 1, 0),
      b(n-1, s)+b(n-1, irem(s+n, 5)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, May 02 2025

LinearRecurrence[{2, 0, 0, 0, 2, -4}, {1, 1, 1, 2, 4, 8}, 40] (* Jean-François Alcover, Mar 06 2016 *)

a(k+1) = 2*a(k) if k == 2, 3, or 4 mod 5, 2*a(k)-2^(k/5) if k == 0 mod 5, 2*a(k)-2^((k-1)/5) if k == 1 mod 5.
G.f.: -(x^2-x+1)*(2*x^3+2*x^2-1) / ((2*x-1)*(2*x^5-1)). - Colin Barker, Dec 22 2012
If n == 0 mod 5, then a(n) = (2^n + 4*2^(n/5))/5. - Giorgos Kalogeropoulos, May 02 2025
a(n) ~ 2^n/5. - Stefano Spezia, May 02 2025

cp's OEIS Frontend

A068049 The first term greater than one on each row of A068009. a(n) = A068009[n, A002024[n]].

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A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).

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A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

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A063776 Number of subsets of {1,2,...,n} which sum to 0 modulo n.

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A068010 Number of subsets of {1,2,3,...,n} that sum to 0 mod 3.

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A068030 Number of subsets of {1,2,3,...,n} that sum to 0 mod 9.

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A068031 Number of subsets of {1,2,3,...,n} that sum to 0 mod 10.

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