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A174067 Triangle, row sums = A000041 starting (1, 2, 3, 5, 7,...); derived from finite differences of p(x) = A(x)*A(x^2) = B(x)*B(x^3) = C(x)*C(x^4) = ... 5
1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 1, 1, 0, 1, 5, 2, 2, 1, 0, 1, 7, 2, 3, 1, 1, 0, 1, 9, 4, 3, 3, 1, 1, 0, 1, 12, 5, 5, 3, 3, 0, 1, 0, 1, 15, 8, 6, 5, 3, 2, 1, 1, 0, 1, 19, 10, 9, 6, 5, 2, 2, 1, 1, 0, 1, 25, 13, 12, 10, 5, 5, 2, 2, 1, 1, 0, 1, 31, 17, 16, 12, 9, 5, 4, 2, 2, 1, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums = A000041 starting with offset 1: (1, 2, 3, 5, 7, 11,...).

LINKS

Table of n, a(n) for n=1..91.

FORMULA

Given an array of rows satisfying p(x) = A(x)*A(x^2) = row 1 = A174065;

row = 2 A174068 satisfying p(x) = B(x)*B(x^3);, row 3 satisfies p(x) =

C(x)*C(x^4),...and so on; take finite differences from the top, becoming

rows of triangle A174067

EXAMPLE

First few rows of the array =

1, 1, 1, 2, 3, 4,. 5,..7,..9,..12,..15,..19,... = A174065

1, 1, 2, 2, 4, 5,. 7,..9,.13,..17,..23,..29,... = A174068

1, 1, 2, 3, 4, 6,. 9,.12,.16,..22,..29,..38,... satisfies p(x) = C(x)*C(x^4)

1, 1, 2, 3, 5, 6,.10,.13,.19,..25,..34,..44,... analogous for k=5

1, 1, 2, 3, 5, 7,.10,.14,.20,..28,..37,..49,..................k=6

1, 1, 2, 3, 5, 7,.11,.14,.21,..28,..39,..51,..................k=7

1, 1, 2, 3, 5, 7,.11,.15,.21,..29,..40,..53,..................k=8

1, 1, 2, 3, 5, 7,.11,.15,.22,..29,..41,..54,..................k=9

1, 1, 2, 3, 5, 7,.11,.15,.22,..30,..41,..55,..................k=10

...

Finally, take finite differences from the top, deleting the first 1, to

obtain triangle A174067

1;

1, 1;

2, 0, 1;

3, 1, 0, 1;

4, 1, 1, 0, 1;

5, 2, 2, 1, 0, 1;

7, 2, 3, 1, 1, 0, 1;

9, 4, 3, 3, 1, 1, 0, 1;

12, 5, 5, 3, 3, 0, 1, 0, 1;

15, 8, 6, 5, 3, 2, 1, 1, 0, 1;

19, 10, 9, 6, 5, 2, 2, 1, 1, 0, 1;

25, 13, 12, 10, 5, 5, 2, 2, 1, 1, 0, 1;

31, 17, 16, 12, 9, 5, 4, 2, 2, 1, 1, 0, 1;

38, 24, 20, 18, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1;

...

CROSSREFS

A000041, A174065, A174066, A174068

Sequence in context: A272472 A100260 A165317 * A124943 A169803 A099557

Adjacent sequences:  A174064 A174065 A174066 * A174068 A174069 A174070

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Mar 06 2010

STATUS

approved

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Last modified May 30 04:27 EDT 2017. Contains 287305 sequences.