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%I #28 Jul 27 2022 02:19:05
%S 1,3,7,12,19,28,38,50,64,80,97,116,137,160,185,211,239,269,301,335,
%T 371,408,447,488,531,576,623,672,722,774,828,884,942,1002,1064,1128,
%U 1193,1260,1329,1400,1473,1548,1625,1704,1785,1867,1951,2037,2125,2215,2307,2401,2497,2595,2695,2796,2899,3004,3111,3220
%N Connell sum sequence (partial sums of the Connell sequence).
%C 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).
%C a(n) is a sharp upper bound for the value of a gamma-labeling of a graph with n edges (cf. Bullington).
%H Grady D. Bullington, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Bullington/bullington7.html">The Connell Sum Sequence</a>, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
%H Ian Connell, <a href="http://www.jstor.org/stable/2309358">Elementary Problem E1382</a>, American Mathematical Monthly, v. 66, no. 8 (October, 1959), p. 724.
%H Douglas E. Iannucci and Donna Mills-Taylor, <a href="http://www.cs.uwaterloo.ca/journals/JIS/IANN/iann1.html">On Generalizing the Connell Sequence</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
%F a(n) = (n-th triangular number) - n + (n-th partial sum of A122797).
%F a(n) = n^2 + n - R*((6*n+1)-R^2)/6, where R = round(sqrt(2*n)). - _Gerald Hillier_, Nov 29 2009
%o (Python)
%o from math import isqrt
%o 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
%Y Cf. A001614, A045928, A045929, A045930.
%Y Cf. A122794, A122795, A122796, A122797, A122798, A122799, A122800.
%Y Cf. A337300 (geometric Connell sums).
%K nonn,easy
%O 1,2
%A Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006
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