A159920 Sums of the antidiagonals of Sundaram's sieve (A159919).

4, 14, 32, 60, 100, 154, 224, 312, 420, 550, 704, 884, 1092, 1330, 1600, 1904, 2244, 2622, 3040, 3500, 4004, 4554, 5152, 5800, 6500, 7254, 8064, 8932, 9860, 10850, 11904, 13024, 14212, 15470, 16800, 18204, 19684, 21242, 22880, 24600, 26404

Offset

2,1

Comments

For every n >= 2, a(n) is the sum of numbers in the (n-1)-th antidiagonal of the Sundaram sieve. (It is not clear why the offset was set to 2 rather than 1.) Thus, if T(j, k) is the element in row j and column k of the Sundaram sieve, we have a(n) = Sum_{i = 1..n-1} T(i, n-i) = Sum_{i = 1..n-1} (2*i*(n-i) + i + (n-i)) = (n - 1)*n*(n + 4)/3 for the sum of the numbers in the (n-1)-th antidiagonal. - Petros Hadjicostas, Jun 19 2019

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000

Andrew Baxter, Sundaram's Sieve [broken link].

Julian Havil, Sundaram's Sieve, Plus Magazine, March 2009.

New Zealand Maths, Newletter 18, October 2002.

H. Sharp, Jr., Enumeration of vacuously transitive relations, Discrete Math. 4 (1973), 185-196.

Wikipedia, Sundaram's Sieve.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = (n - 1)*n*(n + 4)/3.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = 2*A005581(n), n > 1.

a(n) = Sum_{i=1..n-1} i*(i + 3). - Wesley Ivan Hurt, Oct 19 2013

From G. C. Greubel, Oct 03 2022: (Start)

G.f.: 2*x^2*(2 - x)/(1-x)^4.

E.g.f.: (1/3)*x^2*(6 + x)*exp(x). (End)

a(n) = 2*A097900(n)/(n-2)! for n >= 2. - Cullen M. Vaney, Jul 14 2025

Example

For n = 5, (4*5*9)/3 = 60. Indeed, T(1, 4) + T(2, 3) + T(3, 2) + T(4, 1) = 13 + 17 + 17 + 13 = 60 for the sum of the terms in the 4th antidiagonal of the Sundaram sieve.

Maple

A159920:=n->n*(n-1)*(n+4)/3; seq(A159920(k), k=2..100); # Wesley Ivan Hurt, Oct 19 2013

Mathematica

Table[(n-1)*n*(n+4)/3, {n, 2, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{4, -6, 4, -1}, {4, 14, 32, 60}, 61]  (* Harvey P. Dale, Apr 23 2011 *)

Prog

(Magma) [n*(n-1)*(n+4)/3: n in [2..60]]; // G. C. Greubel, Oct 03 2022
(SageMath) [n*(n-1)*(n+4)/3 for n in range(2, 60)] # G. C. Greubel, Oct 03 2022

Crossrefs

Cf. A005581, A097900, A159919, A347701 (interpretation as transitive relations).

Sequence in context: A001740 A129395 A023539 * A036486 A023627 A023649

Adjacent sequences: A159917 A159918 A159919 * A159921 A159922 A159923

Keyword

nonn,easy,changed

Author

Russell Walsmith, Apr 26 2009

Status

approved