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A014370 If n = binomial(b,2) + binomial(c,1), b > c >= 0 then a(n) = binomial(b+1,3) + binomial(c+1,2).

%I A014370 #26 Jul 02 2025 16:01:55
%S A014370 1,2,4,5,7,10,11,13,16,20,21,23,26,30,35,36,38,41,45,50,56,57,59,62,
%T A014370 66,71,77,84,85,87,90,94,99,105,112,120,121,123,126,130,135,141,148,
%U A014370 156,165,166,168,171,175,180,186,193,201,210,220,221,223,226,230,235,241,248,256,265,275,286
%N A014370 If n = binomial(b,2) + binomial(c,1), b > c >= 0 then a(n) = binomial(b+1,3) + binomial(c+1,2).
%D A014370 W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge, 1993, p. 159.
%H A014370 Paolo Xausa, <a href="/A014370/b014370.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened).
%F A014370 a(n) = Sum_{m = 1..n} b(m), b(m) = 1,1,2,1,2,3,1,2,3,4,... = A002260. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%F A014370 a(n*(n+1)/2+m) = n*(n+1)*(n+2)/6 + m*(m+1)/2 = A000292(n)+ A000217(m), m = 0...n+1, n = 1, 2, 3.. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%F A014370 a(n) = a(n-1) + A002260(n). As a triangle with n >= k >= 1: a(n, k) = a(n-1, k) + (n-1)*n/2 = a(n, k-1) + k = (n^3-n+3k^2+3k)/6. - _Henry Bottomley_, Nov 14 2001
%F A014370 a(n) = b(n) * (b(n) + 1) * (b(n) + 2) / 6 + c(n) * (c(n) + 1) / 2, where b(n) = [sqrt(2 * n) - 1/2] and c(n) = n - b(n) * (b(n) + 1) / 2. - Robert A. Stump (bee_ess107(AT)msn.com), Sep 20 2002
%F A014370 As a triangle, T(n,k) = binomial(n+1, 3) + binomial(k+1,2). - _Franklin T. Adams-Watters_, Jan 27 2014
%e A014370 The triangle starts:
%e A014370   1
%e A014370   2 4
%e A014370   5 7 10
%e A014370   11 13 16 20
%e A014370   21 23 26 30 35
%p A014370 a := 0: for i from 1 to 15 do for j from 1 to i do a := a+j: printf(`%d,`,a); od:od:
%t A014370 A014370[n_, k_] := Binomial[n + 1, 3] + Binomial[k + 1, 2];
%t A014370 Table[A014370[n, k], {n, 12}, {k, n}] (* _Paolo Xausa_, Mar 11 2025 *)
%Y A014370 Cf. A002260, A000292 (main diagonal), A000217, A014368, A014369, A006046, A050407 (1st column), A005581 (subdiagonal), A071239 (row sums), A212013.
%K A014370 nonn,easy,tabl
%O A014370 1,2
%A A014370 _N. J. A. Sloane_
%E A014370 More terms from _James Sellers_, Feb 05 2000