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.
%I A068009 #57 Mar 29 2024 08:47:27
%S A068009 1,2,1,4,1,1,8,2,1,1,16,4,2,1,1,32,8,4,1,1,1,64,16,6,2,1,1,1,128,32,
%T A068009 12,4,2,1,1,1,256,64,24,8,4,2,1,1,1,512,128,44,16,8,3,1,1,1,1,1024,
%U A068009 256,88,32,14,6,3,1,1,1,1,2048,512,176,64,26,12,5,2,1,1,1,1,4096,1024,344,128,52,22,10,4,2,1,1,1,1
%N A068009 Square array T(m,n) with m (row) >= 1 and n (column) >= 0 read by antidiagonals: number of subsets of {1,2,3,...n} that sum to 0 mod m (including the empty set, whose sum is 0).
%C A068009 When p is an odd prime, T(p,k+p) = 2*T(p,k) + (2^k * ((2^p) - 2)/p) for all k >= 0. [Sophie LeBlanc]
%C A068009 When m divides n (with n >= m), T(m,n) = (1/m) Sum_{d | m and d is odd} phi(d) * 2^(n/d). [N. Kitchloo and L. Pachter; D. Rusin]
%C A068009 A068009(C(i+1,2), i) = 2, A068009(C(i,2)+1, i) = A000009(i-1) + 1. [AK, cf. A068049]
%H A068009 Alois P. Heinz, <a href="/A068009/b068009.txt">Antidiagonals n = 0..140, flattened</a>
%H A068009 Antti Karttunen, <a href="/A068009/a068009s.txt">Scheme code for computing this table and its rows</a>.
%H A068009 N. Kitchloo and L. Pachter, <a href="https://web.archive.org/web/20150912170614/https://math.berkeley.edu/~lpachter/papers/modpaper2.pdf">An interesting result about subset sums</a>, Nov 27 1993.
%H A068009 William Kuszmaul, <a href="https://arxiv.org/abs/1402.3839">A new approach to enumerating statistics modulo n</a>, arXiv:1402.3839 [math.CO], 2014.
%H A068009 Lior Pachter, <a href="https://web.archive.org/web/20150913033707/https://liorpachter.wordpress.com/2015/04/09/subset-sums/">Subset sums</a>, 2015.
%H A068009 Bill Pet, Sophie LeBlanc, Will Self et al., <a href="https://groups.google.com/forum/#!topic/sci.math/glRjL-EuhJo/discussion">Subsets of {1,2,3,...,n}</a> (discussion in sci.math).
%H A068009 R. P. Stanley and M. F. Yoder, <a href="https://ipnpr.jpl.nasa.gov/progress_report2/XIV/XIVU.PDF">A study of Varshamov codes for asymmetric channels</a>, JPL Technical Report 32-1526, Vol. XIV (1972), 117-123.
%H A068009 <a href="/index/Su#subsetsums">Index entries for sequences related to subset sums mod m</a>
%e A068009 Table for T(m,n) (with rows m >= 1 and columns n >= 0) begins as follows:
%e A068009 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ...
%e A068009 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
%e A068009 1, 1, 2, 4, 6, 12, 24, 44, 88, 176, 344, ...
%e A068009 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, ...
%e A068009 1, 1, 1, 2, 4, 8, 14, 26, 52, ...
%e A068009 1, 1, 1, 2, 3, 6, 12, 22, ...
%e A068009 1, 1, 1, 1, 3, 5, 10, ...
%e A068009 1, 1, 1, 1, 2, 4, ...
%e A068009 1, 1, 1, 1, 2, ...
%e A068009 1, 1, 1, 1, ...
%e A068009 1, 1, 1, ...
%e A068009 1, 1, ...
%e A068009 1, ...
%e A068009 ...
%p A068009 b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
%p A068009 b(n-1, m, t)+ b(n-1, m, irem(t+n,m)))
%p A068009 end:
%p A068009 T:= (m, n)-> b(n, m, 0):
%p A068009 seq(seq(T(1+m, d-m), m=0..d), d=0..12); # _Alois P. Heinz_, Jan 18 2014
%t A068009 max = 13; row[m_] := (ClearAll[t]; im = IdentityMatrix[m]; v = Join[ {Last[im]}, Most[im] ]; t[0] = im[[1]]; t[k_] := t[k] = (im + MatrixPower[v, k]) . t[k-1]; Table[ t[k][[1]], {k, 0, max}]); rows = Table[ row[m], {m, 1, max}]; A068009 = Flatten[ Table[ rows[[m-n+1, n]], {m, 1, max, 1}, {n, m, 1, -1}]] (* _Jean-François Alcover_, Apr 02 2012, after Will Self *)
%t A068009 b[n_, m_, t_] := b[n, m, t] = If[n == 0, If[t == 0, 1, 0], b[n-1, m, t]+b[n-1, m, Mod[t+n, m]]]; T[m_, n_] := b[n, m, 0]; Table[Table[T[1+m, d-m], {m, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 13 2015, after _Alois P. Heinz_ *)
%Y A068009 Main diagonal: A000016, superdiagonal: A063776. The first term greater than one occurs on each row m in the position A002024(m) and these are given in A068049.
%Y A068009 Row 1: A000079, row 2: A011782, row 3: A068010, row 5: A068011, row 6: A068012, row 7: A068013, row 9: A068030, row 10: A068031, row 11: A068032, row 12: A068033, row 13: A068034, row 14: A068035, row 15: A068036, row 16: A068037, row 17: A068038, row 18: A068039, row 19: A068040, row 20: A068041, row 21: A068042, row 25: A068043, row 32: A068044, row 64: A068045.
%K A068009 nonn,nice,tabl
%O A068009 0,2
%A A068009 _Antti Karttunen_, Feb 11 2002