A134249 -id:A134249 - cp's OEIS frontend

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

Gary W. Adamson, Oct 15 2007

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

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

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

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

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 *)

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

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

More terms from Emeric Deutsch, Nov 12 2007

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

Cf. A058331, A134249.

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

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

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

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

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

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

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

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

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

R

Formula

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