A122797 A P_3-stuttered arithmetic progression with a(n+1) = a(n) if n is a triangular number, a(n+1) = a(n) + 1 otherwise.

1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Author

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

Keywords

• nonn
• easy

Comments

P_3(i) = the i-th triangular number.

As a triangle [1; 1,2; 2,3,4; ...], row sums = A064808: (1, 3, 9, 22, 45, 81, ...). - Gary W. Adamson, Aug 10 2007

a(n) = n - A003056(n-1). - Reinhard Zumkeller, Feb 12 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
• Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
• J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187. [From _Johannes W. Meijer_, Jun 21 2010]

Crossrefs

Cf. A001614, A122793, A122794, A122795, A122796, A122798, A122799, A122800.

Cf. A064808.

Programs

Haskell

a122797 n = a122797_list !! (n-1)
a122797_list  = 1 : zipWith (+) a122797_list (map ((1 -) . a010054) [1..])
-- Reinhard Zumkeller, Feb 12 2012

Mathematica

nxt[{n_,a_}]:={n+1,If[OddQ[Sqrt[8n+1]],a,a+1]}; NestList[nxt,{1,1},100][[All,2]] (* Harvey P. Dale, Oct 10 2018 *)

PARI

isTriang(n) = {if (! issquare(8*n+1), return (0)); return (1);}
lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! isTriang(i), aa = aa + 1););} \\ Michel Marcus, Apr 01 2013

Python

from math import isqrt
def A122797(n): return n+1-(k:=isqrt(m:=n<<1))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

Formula

a(n) = A001614(n) - n + 1.

Extensions

Definition corrected by Michel Marcus, Apr 01 2013