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A178239 Triangle read by rows, antidiagonals of an array generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1). 5
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 5, 2, 1, 1, 1, 6, 1, 7, 3, 3, 1, 1, 1, 7, 1, 9, 4, 7, 1, 1, 1, 1, 8, 1, 11, 5, 13, 1, 4, 1, 1, 1, 9, 1, 13, 6, 21, 1, 7, 3, 1, 1, 1, 10, 1, 15, 7, 31, 1, 10, 5, 5, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Partial sums of array terms in groups of 1, next 2, next 4,...8 = powers of (r+2).

Row sums = A178240: (1, 2, 3, 5, 7, 11, 16, 23,...)

Row 1 of the array = A002487.

Row 2 = .............A116528.

Row 3 = .............A178241.

Row 4 = .............A178242.

...

Row 10 = ............A178243.

Polcoeff row r of the array as f(x) satisfies f(x)/f(x^2) = (1 + x + r*x^2).

Let q(x) = (1 + x + r*x^2). Then polcoeff row 4 = q(x) * q(x^2) * q(x^4) * q(x^8) * ...

LINKS

Table of n, a(n) for n=0..77.

FORMULA

Antidiagonals of an array generated from a(n) = a(2n); a(2n+1) = r*a(n) + a(n+1).

Given a triangle M with columns stepped down twice from the previous column,

for columns >0, with (1, 1, r, 0, 0, 0,...) in each column, r-th row of the

array = Lim_{n->inf} M^n.

EXAMPLE

First few rows of the array =

1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...

1,...1,...2,...1,...3,...2,...3,...1,...4,...3,...5,...2,...5,...3,...4,...

1,...1,...3,...1,...5,...3,...7,...1,...7,...5,..13,...3,..13,...7,..15,...

1,...1,...4,...1,...7,...4,..13,...1,..10,...7,..25,...4,..25,..13,..40,...

1,...1,...5,...1,...9,...5,..21,...1,..13,...9,..41,...5,..41,..21,..85,...

1,...1,...6,...1,..11,...6,..31,...1,..16,..11,..61,...6,..61,..31,.156,...

...

Example: In row 3: (1, 1, 4, 1, 7, 4, 13,...) = A178241, r = 3.

A178241(7) = 13 = 3*4 + 1. In blocks of 1, 2, 4, 8,...terms, partial sums are

powers of (r+2) = 5: (1, 5, 25,...).

First few rows of the triangle =

1;

1, 1;

1, 1, 1;

1, 1, 2, 1;

1, 1, 3, 1, 1;

1, 1, 4, 1, 3, 1;

1, 1, 5, 1, 5, 2, 1;

1, 1, 6, 1, 7, 3, 3, 1;

1, 1, 7, 1, 9, 4, 7, 1, 1;

1, 1, 8, 1, 11, 5, 13, 1, 4, 1;

1, 1, 9, 1, 13, 6, 21, 1, 7, 3, 1;

1, 1, 10, 1, 15, 7, 31, 1, 10, 5, 5, 1;

1, 1, 11, 1, 17, 8, 43, 1, 13, 7, 13, 2, 1;

1, 1, 12, 1, 19, 9, 57, 1, 16, 9, 21, 3, 5, 1;

1, 1, 13, 1, 21, 11, 73, 1, 19, 11, 31, 4, 13, 2, 1;

...

CROSSREFS

Cf. A002487, A116528, A178240, A178241, A178241, A178243

Sequence in context: A294306 A181348 A107682 * A260534 A085476 A124944

Adjacent sequences:  A178236 A178237 A178238 * A178240 A178241 A178242

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, May 23 2010

STATUS

approved

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Last modified August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)