A037235 a(n) = n*(2*n^2 - 3*n + 4)/3.

0, 1, 4, 13, 32, 65, 116, 189, 288, 417, 580, 781, 1024, 1313, 1652, 2045, 2496, 3009, 3588, 4237, 4960, 5761, 6644, 7613, 8672, 9825, 11076, 12429, 13888, 15457, 17140, 18941, 20864, 22913, 25092, 27405, 29856, 32449, 35188, 38077, 41120, 44321, 47684, 51213

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Row sums of triangle A134249. Also, binomial transform of (1, 3, 6, 4, 0, 0, 0, ...). - Gary W. Adamson, Oct 15 2007

Binomial transform of a(n) starts: 0, 1, 6, 28, 112, 400, 1312, 4032, ... . - Wesley Ivan Hurt, Oct 21 2014

Number of equivalence classes of n-tuples from the set {1,0,-1} where at the number of nonzero elements is 1,2, or 3 and two n-tuples are equivalent if they are negatives of each other. - Michael Somos, Oct 19 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A058331, A134249.

Programs

Magma

[n*(2*n^2-3*n+4)/3: n in [0..40]]; // Vincenzo Librandi, Jun 15 2011

Maple

A037235:=n->n*(2*n^2-3*n+4)/3: seq(A037235(n), n=0..50); # Wesley Ivan Hurt, Oct 21 2014

Mathematica

Table[n (2 n^2 - 3 n + 4)/3, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 21 2014 *)

PARI

A037235(n) = n*(2*n^2-3*n+4)/3 \\ Michael B. Porter, Dec 07 2009

R

a <- c(0, 1, 4, 13)
for(n in (length(a)+1):30) a[n] <- 4*a[n-1] -6*a[n-2] +4*a[n-3] -a[n-4]
a
# Yosu Yurramendi, Sep 03 2013

Formula

G.f.: x*(1+3*x^2)/(1-x)^4.

a(n) = Sum_{k=0..n-1} (2*k^2 + 1). - Mike Warburton, Sep 08 2007

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with n>3, a(0)=0, a(1)=1, a(2)=4, a(3)=13. - Yosu Yurramendi, Sep 03 2013

a(n+1) = a(n) + A058331(n). - Michael Somos, Oct 19 2022