%I A051336
%S A051336 1,3,7,13,22,33,48,65,86,110,138,168,204,242,284,330,381,434,493,554,
%T A051336 621,692,767,844,929,1017,1109,1205,1307,1411,1523,1637,1757,1881,2009,
%U A051336 2141,2282,2425,2572,2723,2882,3043,3212,3383,3560,3743,3930,4119
%N A051336 Number of arithmetic progressions in {1,2,3,...,n}, including trivial 
               arithmetic progressions of lengths 1 and 2.
%H A051336 T. D. Noe, <a href="b051336.txt">Table of n, a(n) for n=1..1000</a>
%F A051336 Theorem: the second differences give tau(n+1), the number of divisors 
               of n+1 (A000005).
%F A051336 The number of arithmetic subsequences of [1, ..., n] with successive-term 
               increment i and length k is (n-i*(k-1))(i > 0, k > 0, n > i*(k-1)). 
               - Robert E. Sawyer (rs.1(AT)mindspring.com)
%F A051336 a(n) = n + sum { i=1..n-1, j=1..floor(n/i) } (n - i*j) - Robert E. Sawyer 
               (rs.1(AT)mindspring.com)
%e A051336 a(1): [1]; a(2): [1],[2],[1,2]; a(3): [1],[2],[3],[1,2],[1,3],[2,3],[1,
               2,3]
%Y A051336 a(n) = n + A078567(n).
%Y A051336 Cf. A000005, A054519.
%Y A051336 Sequence in context: A155354 A136219 A078582 this_sequence A002623 A081662 
               A091652
%Y A051336 Adjacent sequences: A051333 A051334 A051335 this_sequence A051337 A051338 
               A051339
%K A051336 nonn,easy,nice
%O A051336 1,2
%A A051336 John W. Layman (layman(AT)math.vt.edu), Nov 02 1999