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%I #31 Oct 05 2022 04:55:02
%S 2,4,13,20,40,55,90,116,170,210,287,344,448,525,660,760,930,1056,1265,
%T 1420,1672,1859,2158,2380,2730,2990,3395,3696,4160,4505,5032,5424,
%U 6018,6460,7125,7620,8360,8911,9730,10340,11242,11914,12903,13640,14720,15525,16700
%N Sum of numbers in n-th upward diagonal of triangle the sum of {1; 2,3; 4,5,6; 7,8,9,10; ...} and {1; 2,3; 3,4,5; 4,5,6,7; ...}.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F a(n) = (n * ceiling(n/2)) + ((15 + 25*n + 15*n^2 + 14*n^3 - 3*(((-1)^n))*(5 + n*(3 + n))) / 96).
%F a(n) = A079824(n) + A093005(n).
%F G.f.: x*(2 + 2*x + 3*x^2 + x^3 - x^4)/((1 - x)^4*(1 + x)^3). - _Stefano Spezia_, Aug 19 2022
%e 2 = A079824(1) + A093005(1) = 1 + 1.
%e 4 = A079824(2) + A093005(2) = 2 + 2.
%e 13 = A079824(3) + A093005(3) = 7 + 6.
%e 20 = A079824(4) + A093005(4) = 12 + 8.
%o (Python)
%o def a(n): return (n * ((n + n % 2) // 2)) + (15 + 25*n + 15*(n**2) + 14*(n**3) - 3*(((-1)**n))*(5 + n*(3 + n))) // 96
%Y Cf. A079824, A093005.
%K nonn,easy
%O 1,1
%A _Torlach Rush_, Aug 02 2022
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