A134869 Row sums of triangle A134868.

1, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432

Offset

1,2

Comments

Where records occur in A182703. - Omar E. Pol, Feb 14 2012

Consider quadratic polynomials x^2+cx+d. Then a(n) is the number of these polynomials with 0 <= c < n, 0 <= d < n where no polynomial can be horizontally translated into another. For example, a(3) = 7, the coefficients are as follows: (c, d) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)}. Two polynomials are excluded, namely x^2+2x+1 = (x+1)^2+0(x+1)+0, and x^2+2x+2 = (x+1)^2+0(x+1)+1. - Griffin N. Macris, Jul 19 2016

a(n) gives the number of regions into which the square [0,1]x[0,1] is divided by the Bernstein polynomials of degree n. - Franck Maminirina Ramaharo, Feb 28 2018

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

Formula

a(n) = 1, then for n>1, a(n) = T(n) + 1, where A000217 = (1, 3, 6, 10, 15, ...).

Binomial transform of [1, 3, 0, 1, -1, 1, -1, 1, ...].

From R. J. Mathar, Oct 27 2008: (Start)

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

a(n) = 1 + A000217(n) = A000124(n), n > 1. (End)

Example

a(4) = 11 = sum of row 4 terms of triangle A134868: (2, + 2 + 3 + 4).
a(4) = 11 = 1 + 10, where 10 = T(4).
a(4) = 11 = (1, 3, 3, 1) dot (1, 3, 0, 1) = (1 + 9 + 0 + 1).

Maple

a:=n->sum((stirling2(j+1, n)), j=1..n):seq(a(n), n=1..50); # Zerinvary Lajos, Apr 12 2008

Mathematica

Table[(n^2 + n)/2 + Boole[n != 1], {n, 53}] (* or *)
Table[PolygonalNumber@ n + Boole[n != 1], {n, 53}] (* Version 10.4, or *)
Table[Sum[StirlingS2[k + 1, n], {k, n}], {n, 53}] (* or *)
Rest@ CoefficientList[Series[x (1 + x - 2 x^2 + x^3)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Jul 19 2016 *)

Prog

(PARI) a(n)=if(n>1, n*(n+1)/2+1, 1) \\ Charles R Greathouse IV, Aug 05 2016

Crossrefs

Cf. A000217, A134868.

Sequence in context: A269132 A310757 A231603 * A171377 A020683 A310758

Adjacent sequences: A134866 A134867 A134868 * A134870 A134871 A134872

Keyword

nonn,easy

Author

Gary W. Adamson, Nov 14 2007

Extensions

More terms from R. J. Mathar, Oct 27 2008

Status

approved