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!)
A249223 Triangle read by rows: row n gives partial alternating sums of row n of A237048. 33
1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

All entries in the triangle are nonnegative since the number of 1's in odd-numbered columns of A237048 prior to column j, 1 <= j <= row(n), is at least as large as the number of 1's in even-numbered columns through column j. As a consequence:

(a)  The two adjacent symmetric Dyck paths whose legs are defined by adjacent rows of triangle A237593 never cross each other (see also A235791 and A237591) and the rows in this triangle describe the widths between the legs.

(b)  Let legs(n) denote the n-th row of triangle A237591, widths(n) the n-th row of this triangle, and c(n) the rightmost entry in the n-th row of this triangle (center of the Dyck path). Then area(n) = 2 * legs(n) . width(n) - c(n), where "." is the inner product, is the area between the two adjacent symmetric Dyck paths.

(c)  For certain sequences of integers, it is known that area(n) = sigma(n); see A238443, A245685, A246955, A246956 and A247687.

Right border gives A067742. - Omar E. Pol, Jan 21 2017

LINKS

G. C. Greubel, Table of n, a(n) for the first 150 rows, flattened

FORMULA

T(n, k) = Sum_{j=1..k} (-1)^(j+1)*A237048(n, j), for n>=1 and 1 <= k <= floor((sqrt(8*n + 1) - 1)/2). - corrected by Hartmut F. W. Hoft, Jan 25 2018

EXAMPLE

The sequence written as a triangle:

  n\k  1    2    3    4    5    6

   1   1

   2   1

   3   1    0

   4   1    1

   5   1    0

   6   1    1    2

   7   1    0    0

   8   1    1    1

   9   1    0    1

  10   1    1    1    0

  11   1    0    0    0

  12   1    1    2    2

  13   1    0    0    0

  14   1    1    1    0

  15   1    0    1    1    2

  16   1    1    1    1    1

  17   1    0    0    0    0

  18   1    1    2    1    1

  19   1    0    0    0    0

  20   1    1    1    1    2

  21   1    0    1    1    1    0

  22   1    1    1    0    0    0

  23   1    0    0    0    0    0

  24   1    1    2    2    2    2

The triangle shows that area(n) has width 1 for powers of 2 and that area(p) for primes p consists of only 1 horizontal leg of width 1 (and its symmetric vertical leg in the mirror symmetric duplicate of this triangle).

MAPLE

r := proc(n) floor((sqrt(1+8*n)-1)/2) ; end proc: # R. J. Mathar 2015 A003056

A237048:=proc(n, k) local i; global r;

if n<(k-1)*k/2 or k>r(n) then return(0); fi;

if (k mod 2)=1 and (n mod k)=0 then return(1); fi;

if (k mod 2)=0 and ((n-k/2) mod k) = 0 then return(1); fi;

return(0);

end;

A249223:=proc(n, k) local i; global r, A237048;

if n<(k-1)*k/2 or k>r(n) then return(0); fi;

add( (-1)^(i+1)*A237048(n, i), i=1..k);

end;

for n from 1 to 12 do lprint([seq(A249223(n, k), k=1..r(n))]); od; # N. J. A. Sloane, Jan 15 2021

MATHEMATICA

cd[n_, k_] := If[Divisible[n, k], 1, 0]; row[n_] := Floor[(Sqrt[8 n + 1] - 1)/2]; a237048[n_, k_] := If[OddQ[k], cd[n, k], cd[n - k/2, k]];

a1[n_, k_] := Sum[(-1)^(j + 1)*a237048[n, j], {j, 1, k}];

a2[n_] := Drop[FoldList[Plus, 0, Map[(-1)^(# + 1) &, Range[row[n]]] a237048[n]], 1]; Flatten[Map[a2, Range[24]]] (* data *) (* Corrected by G. C. Greubel, Apr 16 2017 *)

PROG

(PARI) t237048(n, k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0);

kmax(n) = (sqrt(1+8*n)-1)/2;

t(n, k) = sum(j=1, k, (-1)^(j+1)*t237048(n, j));

tabf(nn) = {for (n=1, nn, for (k=1, kmax(n), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Sep 20 2015

CROSSREFS

Cf. A000203, A237048, A237270, A237271, A237593, A238443, A245685, A246955, A246956, A247687.

Sequence in context: A339367 A082858 A255318 * A115953 A225245 A204770

Adjacent sequences:  A249220 A249221 A249222 * A249224 A249225 A249226

KEYWORD

nonn,tabf

AUTHOR

Hartmut F. W. Hoft, Oct 23 2014

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 April 17 11:26 EDT 2021. Contains 343064 sequences. (Running on oeis4.)