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A051336 Number of arithmetic progressions in {1,2,3,...,n}, including trivial arithmetic progressions of lengths 1 and 2. 4
1, 3, 7, 13, 22, 33, 48, 65, 86, 110, 138, 168, 204, 242, 284, 330, 381, 434, 493, 554, 621, 692, 767, 844, 929, 1017, 1109, 1205, 1307, 1411, 1523, 1637, 1757, 1881, 2009, 2141, 2282, 2425, 2572, 2723, 2882, 3043, 3212, 3383, 3560, 3743, 3930, 4119 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

FORMULA

Theorem: the second differences give tau(n+1), the number of divisors of n+1 (A000005).

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)

a(n) = n + sum { i=1..n-1, j=1..floor(n/i) } (n - i*j). - Robert E. Sawyer (rs.1(AT)mindspring.com)

EXAMPLE

a(1): [1];

a(2): [1],[2],[1,2];

a(3): [1],[2],[3],[1,2],[1,3],[2,3],[1,2,3].

MATHEMATICA

nmax = 48; t = Table[ DivisorSigma[0, n], {n, 1, nmax}]; Accumulate[ Accumulate[t]+1] - Accumulate[t] (* Jean-Fran├žois Alcover, Nov 08 2011 *)

With[{c=Accumulate[DivisorSigma[0, Range[50]]]}, Accumulate[c+1]-c] (* Harvey P. Dale, Dec 23 2015 *)

CROSSREFS

a(n) = n + A078567(n).

Cf. A000005, A054519.

Sequence in context: A155354 A136219 A078582 * A253896 A002623 A173196

Adjacent sequences:  A051333 A051334 A051335 * A051337 A051338 A051339

KEYWORD

nonn,easy,nice

AUTHOR

John W. Layman, Nov 02 1999

STATUS

approved

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Last modified November 19 01:27 EST 2017. Contains 294912 sequences.