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A122793
Connell sum sequence (partial sums of the Connell sequence).
9
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
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. A337300 (geometric Connell sums).
Sequence in context: A025713 A022791 A025742 * A062714 A337300 A039677
KEYWORD
nonn,easy
AUTHOR
Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006
STATUS
approved