login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A140996 Triangle G(n, k) read by rows for 0 <= k <= n, where G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, G(n+4, n+1) = 8, and G(n+5, m) = G(n+1, m-1) + G(n+1, m) + G(n+2, m) + G(n+3, m) + G(n+4, m) for n >= 0 for m = 1..(n+1). 23
1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 16, 8, 4, 2, 1, 1, 31, 17, 8, 4, 2, 1, 1, 60, 35, 17, 8, 4, 2, 1, 1, 116, 72, 35, 17, 8, 4, 2, 1, 1, 224, 148, 72, 35, 17, 8, 4, 2, 1, 1, 432, 303, 149, 72, 35, 17, 8, 4, 2, 1, 1, 833, 618, 308, 149, 72, 35, 17, 8, 4, 2, 1, 1, 1606, 1257, 636, 308, 149, 72, 35, 17, 8, 4, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

From Petros Hadjicostas, Jun 12 2019: (Start)

This is a mirror of image of triangular array A140995. The current array has index of asymmetry s = 3 and index of obliqueness (obliquity) e = 0. Array A140995 has the same index of asymmetry, but has index of obliqueness e = 1. (In other related sequences, the author uses the letter y for the index of asymmetry and the letter z for the index of obliqueness, but the stone slab that appears over a tomb in a picture that he posted in those sequences, the letters s and e are used instead. See, for example, the documentation for sequences A140998, A141065, A141066, and A141067.)

In general, if the index of asymmetry (from the Pascal triangle A007318) is s, then the order of the recurrence is s + 2 (because the recurrence of the Pascal triangle has order 2). There are also s + 2 infinite sets of initial conditions (as opposed to the Pascal triangle that has only 2 infinite sets of initial conditions, namely, G(n, 0) = G(n+1, n+1) = 1 for n >= 0).

Pascal's triangle A007318 has s = 0 and is symmetric, arrays A140998 and A140993 have s = 1 (with e = 0 and e = 1, respectively), arrays A140997 and A140994 have s = 2 (with e = 0 and e = 1, respectively), and arrays A141020 and A141021 have s = 4 (with e = 0 and e = 1, respectively).

(End)

LINKS

Robert Price, Table of n, a(n) for n = 0..5150

Juri-Stepan Gerasimov, Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...

FORMULA

From Petros Hadjicostas, Jun 12 2019: (Start)

G(n, k) = A140995(n, n - k) for 0 <= k <= n.

Bivariate g.f.: Sum_{n,k >= 0} G(n, k)*x^n*y^k = (1 - x - x^2 - x^3 - x^4 + x^2*y + x^3*y + x^5*y)/((1 - x) * (1 - x*y) * (1 - x - x^2 - x^3 - x^4 - x^4*y)).

If we take the first derivative of the bivariate g.f. w.r.t. y and set y = 0, we get the g.f. of column k = 1: x/((1 - x) * (1 - x - x^2 - x^3 - x^4)). This is the g.f. of a shifted version of sequence A107066.

Substituting y = 1 in the above bivariate function and simplifying, we get the g.f. of row sums: 1/(1 - 2*x). Hence, the row sums are powers of 2; i.e., A000079.

(End)

EXAMPLE

Triangle (with rows n >= 0 and columns k >= 0) begins as follows:

  1

  1   1

  1   2   1

  1   4   2   1

  1   8   4   2   1

  1  16   8   4   2  1

  1  31  17   8   4  2  1

  1  60  35  17   8  4  2  1

  1 116  72  35  17  8  4  2 1

  1 224 148  72  35 17  8  4 2 1

  1 432 303 149  72 35 17  8 4 2 1

  1 833 618 308 149 72 35 17 8 4 2 1

  ...

MATHEMATICA

nlim = 100;

For[n = 0, n <= nlim, n++, G[n, 0] = 1];

For[n = 1, n <= nlim, n++, G[n, n] = 1];

For[n = 2, n <= nlim, n++, G[n, n-1] = 2];

For[n = 3, n <= nlim, n++, G[n, n-2] = 4];

For[n = 4, n <= nlim, n++, G[n, n-3] = 8];

For[n = 5, n <= nlim, n++, For[k = 1, k < n - 3, k++,

   G[n, k] = G[n-4, k-1] + G[n-4, k] + G[n-3, k] + G[n-2, k] + G[n-1, k]]];

A140996 = {}; For[n = 0, n <= nlim, n++,

For[k = 0, k <= n, k++, AppendTo[A140996, G[n, k]]]];

A140996 (* Robert Price, Jul 03 2019 *)

CROSSREFS

Cf. A007318, A107066, A140993, A140994, A140995, A140997, A140998, A141020, A141021, A141031, A141065, A141066, A141067, A141068, A141069, A141070, A141072, A141073.

Sequence in context: A114394 A059623 A140997 * A141020 A152568 A155038

Adjacent sequences:  A140993 A140994 A140995 * A140997 A140998 A140999

KEYWORD

nonn,tabl

AUTHOR

Juri-Stepan Gerasimov, Jul 08 2008

EXTENSIONS

Name edited by Petros Hadjicostas, Jun 12 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 July 23 11:44 EDT 2019. Contains 325254 sequences. (Running on oeis4.)