

A134869


Row sums of triangle A134868.


4



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
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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+x2x^2+x^3)/(1x)^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



