A140995 Triangle G(n, k) read by rows, for 0 <= k <= n, where G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, 2) = 4, G(n+4, 3) = 8, and G(n+5, m) = G(n+1, m-3) + G(n+1, m-4) + G(n+2, m-3) + G(n+3, m-2) + G(n+4, m-1) for n >= 0 and m = 4..(n+4).

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 8, 1, 1, 2, 4, 8, 16, 1, 1, 2, 4, 8, 17, 31, 1, 1, 2, 4, 8, 17, 35, 60, 1, 1, 2, 4, 8, 17, 35, 72, 116, 1, 1, 2, 4, 8, 17, 35, 72, 148, 224, 1, 1, 2, 4, 8, 17, 35, 72, 149, 303, 432, 1, 1, 2, 4, 8, 17, 35, 72, 149, 308, 618, 833, 1, 1, 2, 4, 8, 17, 35, 72, 149, 308, 636, 1257, 1606, 1

Offset

0,5

Comments

From Petros Hadjicostas, Jun 13 2019: (Start)

This is a mirror image of the triangular array A140996. The current array has index of asymmetry s = 3 and index of obliqueness (obliquity) e = 1. Array A140996 has the same index of asymmetry, but has index of obliqueness e = 0. (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 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.)

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

If A(x,y) = Sum_{n,k >= 0} G(n, k)*x^n*y^k is the bivariate g.f. for this array (with G(n, k) = 0 for 0 <= n < k) and B(x, y) = Sum_{n, k} A140996(n, k)*x^n*y^k, then A(x, y) = B(x*y, y^(-1)). This can be proved using formal manipulation of double series expansions and the fact G(n, k) = A140996(n, n-k) for 0 <= k <= n.

If we let b(k) = lim_{n -> infinity} G(n, k) for k >= 0, then b(0) = 1, b(1) = 2, b(2) = 4, b(3) = 8, and b(k) = b(k-1) + b(k-2) + 2*b(k-3) + b(k-4) for k >= 4. (The existence of the limit can be proved by induction on k.) Thus, the limiting sequence is 1, 2, 4, 8, 17, 35, 72, 149, 308, 636, 1314, 2715, 5609, 11588, 23941, 49462, 102188, 211120, 436173, ... (sequence A309462). (End)

Links

Table of n, a(n) for n=0..90.

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

Formula

From Petros Hadjicostas, Jun 13 2019: (Start)

G(n, k) = A140996(n, n-k) for 0 <= k <= n.

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

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 begins:
  1
  1 1
  1 2 1
  1 2 4 1
  1 2 4 8  1
  1 2 4 8 16  1
  1 2 4 8 17 31  1
  1 2 4 8 17 35 60   1
  1 2 4 8 17 35 72 116   1
  1 2 4 8 17 35 72 148 224   1
  1 2 4 8 17 35 72 149 303 432   1
  1 2 4 8 17 35 72 149 308 618 833 1
  ...

Crossrefs

Cf. A000079, A007318, A140993, A140994, A140996, A140997, A140998, A141020, A141021, A141031, A141065, A141066, A141067, A141068, A141069, A141070, A141072, A141073, A309462.

Sequence in context: A023506 A232089 A141021 * A140994 A245163 A140993

Adjacent sequences: A140992 A140993 A140994 * A140996 A140997 A140998

Keyword

nonn,tabl

Author

Juri-Stepan Gerasimov, Jul 08 2008

Extensions

Entries checked by R. J. Mathar, Apr 14 2010

Name edited by and more terms from Petros Hadjicostas, Jun 13 2019

Status

approved