login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A327889 Triangle read by rows T(n, k) = (-1)^k * A000217(k) * F(binomial(n,k)), where F(x) = 1 if the largest decimal digit of x is 1, and 0 otherwise. 2
0, 0, -1, 0, 0, 3, 0, 0, 0, -6, 0, 0, 0, 0, 10, 0, 0, 3, -6, 0, -15, 0, 0, 3, 0, 10, 0, 21, 0, 0, 0, 0, 0, 0, 0, -28, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 10, -15, 0, 0, 0, -45, 0, -1, 0, -6, 0, 0, 0, -28, 0, -45, 55, 0, -1, 0, -6, 0, 0, 0, 0, 36, 0, 55, -66, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -66, 78 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Represents alternating, normalized (linear) modification of A327853, transformed by first (decimal) digit indicator function F(x).

The scatterplot of the sequence represents a fractal-like structure, made out of arc-like structures. For comparison, the A327853 represents a Sierpinski's gasket (triangle), bounded by a function of A003056 (the positive inverse of triangular numbers).

If a number base other than decimal is used, then for a larger value of the base, the scatterplot structure will appear to be "zoomed in". The smallest base that will still represent the structure is ternary, since in binary we have F(x)=1 for all x, and the scatterplot will degrade to a simple triangle structure.

If we modify F(x) to look at other digits than the largest digit, then the structure appears to lose "density".

Why does Pascal's triangle (Sierpinski's gasket) converge to such arc-like structure when the digit indicator function F(x) is applied (in some number base)? Are there sequences other than those related to binomial coefficients, that can replicate this structure?

LINKS

Matej Veselovac, Table of n, a(n) for n = 1..100000

Math StackExchange, Pattern in Pascal's triangle .

Matej Veselovac, Scatterplot of the sequence, for terms a(n), n=1...10^5..

FORMULA

The entries of the triangle are given by T(n, k) = (-1)^k * A000217(k) * F(binomial(n,k)), then it is read by rows, where F(x) = 1 if the largest decimal digit of x is 1, and 0 otherwise.

EXAMPLE

First 16 rows of the T(n, k):

0;

0, -1;

0,  0, 3;

0,  0, 0, -6;

0,  0, 0,  0, 10;

0,  0, 3, -6,  0, -15;

0,  0, 3,  0, 10,   0, 21;

0,  0, 0,  0,  0,   0,  0, -28;

0,  0, 0,  0,  0,   0,  0,   0, 36;

0,  0, 0,  0, 10, -15,  0,   0,  0,-45;

0, -1, 0, -6,  0,   0,  0, -28,  0,-45, 55;

0, -1, 0, -6,  0,   0,  0,   0, 36,  0, 55, -66;

0, -1, 0,  0,  0,   0,  0,   0,  0,  0,  0, -66, 78;

0, -1, 0,  0,  0, -15, 21, -28, 36,  0,  0,   0, 78, -91;

0, -1, 0,  0, 10,   0,  0,   0,  0,  0, 55,   0,  0, -91, 105;

0, -1, 3,  0, 10,   0,  0,   0,  0,  0,  0, -66,  0, -91, 105, -120;

MATHEMATICA

d[n_, b_: 10] := Mod[Floor[n/b^(Floor[Log[b, n]])], b]; t[n_] := n (n + 1)/2; f[x_] := x ; r[n0_, b_: 10] := Flatten[Table[(-1)^k  Floor[f[t[k]]]*If[d[Binomial[n, k], b] == 1, 1, 0], {n, 0, n0}, {k, 0, n}]]; r[20] (* Matej Veselovac, Sep 29 2019 *)

CROSSREFS

Cf. A001317, A007318, A003056, A000217.

Cf. A327853 (original sequence, before applying the transformation).

Sequence in context: A330734 A081805 A181189 * A221702 A084681 A261038

Adjacent sequences:  A327886 A327887 A327888 * A327890 A327891 A327892

KEYWORD

sign,base,tabl,look

AUTHOR

Matej Veselovac, Sep 29 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 16 21:37 EST 2020. Contains 331975 sequences. (Running on oeis4.)