A133263 Binomial transform of (1, 2, 0, 1, -1, 1, -1, 1, ...).

1, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380, 1433

Offset

0,2

Comments

A007318 * [1, 2, 0, 1, -1, 1, -1, 1, ...]. Left column of A134249.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

From Emeric Deutsch, Nov 12 2007: (Start)

a(n) = (n^2 + n + 4)/2 for n > 0.

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

a(n) = A000124(n) + 1, n >= 1. - Zerinvary Lajos, Apr 12 2008

a(0)=1, a(1)=3; for n >= 2, a(n) = a(n-1) + n. - Philippe Lallouet (philip.lallouet(AT)orange.fr), May 27 2008; corrected by Michel Marcus, Nov 03 2018

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=3, a(2)=5, a(3)=8. - Harvey P. Dale, Feb 13 2012

a(n) = A238531(n+1) if n >= 0. - Michael Somos, Feb 28 2014

For n > 0: A228446(a(n)) = 5. - Reinhard Zumkeller, Mar 12 2014

a(n) = A022856(n+4) for n >= 1. - Georg Fischer, Nov 02 2018

Sum_{n>=0} 1/a(n) = 1/2 + 2*Pi*tanh(sqrt(15)*Pi/2)/sqrt(25). - Amiram Eldar, Jun 02 2025

From Elmo R. Oliveira, Nov 23 2025: (Start)

E.g.f.: exp(x)*(4 + 2*x + x^2)/2 - 1.

a(n) = A027689(n)/2 for n >= 1. (End)

Example

a(3) = 8 = (1, 3, 3, 1) dot (1, 2 0, 1) = (1 + 6 + 0 + 1).

Maple

1, seq((n^2+n+4)*1/2, n=1..50); # Emeric Deutsch, Nov 12 2007
# Alternative:
a:=n->add((Stirling2(j+1, n)), j=0..n): seq(a(n)+1, n=0..50); # Zerinvary Lajos, Apr 12 2008

Mathematica

Join[{1}, Table[(n^2+n+4)/2, {n, 50}]] (* Harvey P. Dale, Feb 13 2012 *)
(* Alternative: *)
Join[{1}, LinearRecurrence[ {3, -3, 1}, {3, 5, 8}, 50]] (* Harvey P. Dale, Feb 13 2012 *)

Prog

(PARI) a(n)=n*(n+1)/2+2 \\ Charles R Greathouse IV, Mar 26 2014

Crossrefs

Cf. A000124, A007318, A022856, A027689, A134249, A228446, A238531.

Essentially the same as A089071.

Sequence in context: A002579 A023544 A238531 * A038088 A018917 A167385

Adjacent sequences: A133260 A133261 A133262 * A133264 A133265 A133266

Keyword

nonn,easy,changed

Author

Gary W. Adamson, Oct 15 2007

Extensions

More terms from Emeric Deutsch, Nov 12 2007

Status

approved