The On-Line Encyclopedia of Integer Sequences (OEIS)

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A165997

Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.
(history; published version)

#10 by Peter Luschny at Wed Mar 29 08:58:29 EDT 2023

STATUS

reviewed

approved

#9 by Michel Marcus at Wed Mar 29 07:33:28 EDT 2023

STATUS

proposed

reviewed

#8 by Joerg Arndt at Wed Mar 29 05:08:57 EDT 2023

STATUS

editing

proposed

#7 by Joerg Arndt at Wed Mar 29 05:08:51 EDT 2023

CROSSREFS

Cf. A060941 (Duchon's numbers, is diagonal T(3*n, 2*n)).

STATUS

proposed

editing

#6 by Jon E. Schoenfield at Tue Mar 28 22:18:16 EDT 2023

STATUS

editing

proposed

#5 by Jon E. Schoenfield at Tue Mar 28 22:16:14 EDT 2023

CROSSREFS

Cf. A060941 (Duchon's numbers, is diagonal T(3*n, 2*n)).

STATUS

proposed

editing

Discussion

Tue Mar 28

22:18

Jon E. Schoenfield: “numbers, is diagonal” looks odd to me. Maybe it’s fine.

#4 by Kevin Ryde at Tue Mar 28 19:34:39 EDT 2023

STATUS

editing

proposed

#3 by Kevin Ryde at Tue Mar 28 19:33:05 EDT 2023

NAME

Triangle Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

COMMENTS

There are f(n) = floor((2*n+3)/3) = A004396(n+1) terms in row n.

EXAMPLE

Triangle begins: [1] [1] [1, 1] [1, 2, 2] [1, 3, 5] [1, 4, 9, 9] [1, 5, 14, 23, 23] [1, 6, 20, 43, 66] [1, 7, 27, 70, 136, 136] [1, 8, 35, 105, 241, 377, 377] [1, 9, 44, 149, 390, 767, 1144] [1, 10, 54, 203, 593, 1360, 2504, 2504] [1, 11, 65, 268, 861, 2221, 4725, 7229, 7229] [1, 12, 77, 345, 1206, 3427, 8152, 15381, 22610] ...

Triangle begins:

k=0 1 2 3 4 5 6 7 8

n=0: 1

n=1: 1

n=2: 1, 1

n=3: 1, 2, 2

n=4: 1, 3, 5

n=5: 1, 4, 9, 9

n=6: 1, 5, 14, 23, 23

n=7: 1, 6, 20, 43, 66

n=8: 1, 7, 27, 70, 136, 136

n=9: 1, 8, 35, 105, 241, 377, 377

n=10: 1, 9, 44, 149, 390, 767, 1144

n=11: 1, 10, 54, 203, 593, 1360, 2504, 2504

n=12: 1, 11, 65, 268, 861, 2221, 4725, 7229, 7229

n=13: 1, 12, 77, 345, 1206, 3427, 8152, 15381, 22610

...

CROSSREFS

Cf. A004396 (row lengths).

Cf. A060941 (Duchon's numbers, is diagonal T(3*n, 2*n))

STATUS

approved

editing

Discussion

Tue Mar 28

19:34

Kevin Ryde: Some layout for the example triangle.  Could think about shortening to stop at n=10 ...

#2 by Russ Cox at Fri Mar 30 17:28:33 EDT 2012

AUTHOR

_Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), _, Oct 03 2009

Discussion

Fri Mar 30

17:28

OEIS Server: https://oeis.org/edit/global/148

#1 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010

NAME

Triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

DATA

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 1, 4, 9, 9, 1, 5, 14, 23, 23, 1, 6, 20, 43, 66, 1, 7, 27, 70, 136, 136, 1, 8, 35, 105, 241, 377, 377, 1, 9, 44, 149, 390, 767, 1144, 1, 10, 54, 203, 593, 1360, 2504, 2504, 1, 11, 65, 268, 861, 2221, 4725, 7229, 7229, 1, 12, 77, 345, 1206

OFFSET

0,6

COMMENTS

There are f(n) = floor((2*n+3)/3) terms in row n.

EXAMPLE

PROG

(PARI) f(n) = floor((2*(n-1)+3)/3); s=14; M=matrix(s, s); for(n=1, s, M[n, 1]=1); for(n=2, s, for(k=2, f(n), M[n, k]=M[n, k-1]+M[n-1, k])); for(n=1, s, for(k=1, f(n), print1(M[n, k], ", ")))

CROSSREFS

A060941 (Duchon's numbers, is diagonal T(3*n, 2*n))

KEYWORD

nonn,tabf

AUTHOR

Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 03 2009

STATUS

approved