

A104249


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


14



1, 3, 8, 16, 27, 41, 58, 78, 101, 127, 156, 188, 223, 261, 302, 346, 393, 443, 496, 552, 611, 673, 738, 806, 877, 951, 1028, 1108, 1191, 1277, 1366, 1458, 1553, 1651, 1752, 1856, 1963, 2073, 2186, 2302, 2421, 2543, 2668, 2796, 2927, 3061, 3198, 3338, 3481
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OFFSET

0,2


COMMENTS

Second differences are all 3.
Related to the sequence of odd numbers A005408 since for these numbers the first differences are all 2.
Column 2 of A114202.  Paul Barry, Nov 17 2005
Equals third row of A167560 divided by 2.  Johannes W. Meijer, Nov 12 2009
A242357(a(n)) = n + 1.  Reinhard Zumkeller, May 11 2014
Also, this sequence is related to A011379, for n>0, by A011379(n) = n*a(n)  Sum_{i=0..n1} a(i).  Bruno Berselli, Jul 08 2015


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
GuoNiu Han, Enumeration of Standard Puzzles
GuoNiu Han, Enumeration of Standard Puzzles [Cached copy]
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

G.f.: (1+2*x^2)/(1x)^3.
Recurrence: {u(1) = 3, u(2) = 8, (n+3)*u(n+3)+(5n)*u(n+2)*(2+2*n)*u(n+1) +(22*n)*u(n), u(0) = 1}.
a(0)=1, a(n) = a(n1)+3n1, n>0; a(n) = Sum_{k=0..n} C(n, k)C(2, k)J(k+1), J(n) = A001045(n).  Paul Barry, Nov 17 2005
Binomial transform of [1,2,3,0,...].  Gary W. Adamson, Apr 23 2008


EXAMPLE

The sequence of first differences delta_a(n) = a(n+1)  a(n) is: 2,5,8,11,14,17,20,23,26,...
The sequence of second differences delta_delta_a(n) = a(n+2)  2*a(n+1) + a(n) is: 3,3,3,3,3,3,3,3,3,... E.g. 78  2*58 + 41 = 3.


MAPLE

a := proc (n) local i, u; option remember; u[0] := 1; u[1] := 3; u[2] := 8; for i from 3 to n do u[i] := (4*u[i3]8*u[i2]2*u[i1]+(2*u[i3]+2*u[i2]u[i1])*i)/i end do; [seq(u[i], i = 0 .. n)] end proc;


MATHEMATICA

A104249[n_] := (3*n^2 + n + 2)/2; Table[A104249[n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)


PROG

(MAGMA) [(3*n^2+n+2)/2: n in [0..50]]; // Vincenzo Librandi, May 09 2011
(Haskell)
a104249 n = n*(3*n+1) `div` 2 + 1  Reinhard Zumkeller, May 11 2014
(PARI) a(n)=n*(3*n+1)/2+1 \\ Charles R Greathouse IV, Oct 07 2015


CROSSREFS

Cf. A001399, A002597, A005408, A011379, A016777, A143689.
Sequence in context: A115006 A211480 A122796 * A225253 A254875 A025202
Adjacent sequences: A104246 A104247 A104248 * A104250 A104251 A104252


KEYWORD

nonn,easy


AUTHOR

Thomas Wieder, Feb 26 2005


STATUS

approved



