A122793 Connell sum sequence (partial sums of the Connell sequence).

1, 3, 7, 12, 19, 28, 38, 50, 64, 80, 97, 116, 137, 160, 185, 211, 239, 269, 301, 335, 371, 408, 447, 488, 531, 576, 623, 672, 722, 774, 828, 884, 942, 1002, 1064, 1128, 1193, 1260, 1329, 1400, 1473, 1548, 1625, 1704, 1785, 1867, 1951, 2037, 2125, 2215, 2307, 2401, 2497, 2595, 2695, 2796, 2899, 3004, 3111, 3220

Offset

1,2

Comments

a(n) is the sum of the n highest entries in the projection of the n-th tetrahedron or tetrahedral number (e.g., a(7) = 7+6+6+5+5+5+4+4).

a(n) is a sharp upper bound for the value of a gamma-labeling of a graph with n edges (cf. Bullington).

Links

Table of n, a(n) for n=1..60.

Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))

Ian Connell, Elementary Problem E1382, American Mathematical Monthly, v. 66, no. 8 (October, 1959), p. 724.

Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Formula

a(n) = (n-th triangular number) - n + (n-th partial sum of A122797).

a(n) = n^2 + n - R*((6*n+1)-R^2)/6, where R = round(sqrt(2*n)). - Gerald Hillier, Nov 29 2009

Prog

(Python)
from math import isqrt
def A122793(n): return n*(n+1)-(r:=(k:=isqrt(m:=n<<1))+int((m<<2)>(k<<2)*(k+1)+1))*((6*n+1)-r**2)//6 # Chai Wah Wu, Jul 26 2022

Crossrefs

Cf. A001614, A045928, A045929, A045930.

Cf. A122794, A122795, A122796, A122797, A122798, A122799, A122800.

Cf. A337300 (geometric Connell sums).

Sequence in context: A025713 A022791 A025742 * A062714 A337300 A039677

Adjacent sequences: A122790 A122791 A122792 * A122794 A122795 A122796

Keyword

nonn,easy

Author

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

Status

approved