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
Examples
a(3) = 8 = (1, 3, 3, 1) dot (1, 2 0, 1) = (1 + 6 + 0 + 1).
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).
Programs
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Maple
1, seq((n^2+n+4)*1/2,n=1..50); # Emeric Deutsch, Nov 12 2007 a:=n->add((Stirling2(j+1,n)), j=0..n): seq(a(n)+1, n=0..50); # Zerinvary Lajos, Apr 12 2008
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Mathematica
Join[{1},Table[(n^2+n+4)/2,{n,50}]] (* or *) Join[{1}, LinearRecurrence[ {3,-3,1},{3,5,8},50]] (* Harvey P. Dale, Feb 13 2012 *) -
PARI
a(n)=n*(n+1)/2+2 \\ Charles R Greathouse IV, Mar 26 2014
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
Extensions
More terms from Emeric Deutsch, Nov 12 2007
Comments