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A109900 The (n,r)-th term of the following triangle is T(n)-T(r) for r = 0 to n. The n-th row contains n+1 terms. T(n) = the n-th triangular number = n(n+1)/2. Sequence contains the sum of terms at a 45-degree angle.

%I A109900 #44 Sep 17 2022 03:44:57
%S A109900 0,1,3,8,15,27,42,64,90,125,165,216,273,343,420,512,612,729,855,1000,
%T A109900 1155,1331,1518,1728,1950,2197,2457,2744,3045,3375,3720,4096,4488,
%U A109900 4913,5355,5832,6327,6859,7410,8000,8610,9261,9933,10648,11385,12167,12972
%N A109900 The (n,r)-th term of the following triangle is T(n)-T(r) for r = 0 to n. The n-th row contains n+1 terms. T(n) = the n-th triangular number = n(n+1)/2. Sequence contains the sum of terms at a 45-degree angle.
%C A109900 Initial terms match those of A047866 with a difference of +1 or -1 in some cases. A047866: 0, 1, 3, 8, 15, 27, 42, 63, 90, 124, 165, 215, ...
%H A109900 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A109900 a(2n+1) = (n+1)^3; a(2n) = (2n+1)*T(n) = (2n+1)*(n+1)*n/2, where T=A000217. - _R. J. Mathar_, Feb 11 2008
%F A109900 a(n) = A034828(n+1). - _R. J. Mathar_, Aug 18 2008
%F A109900 G.f.: x*(1+x+x^2)/(1-2*x-x^2+4*x^3-x^4-2*x^5+x^6). - _Colin Barker_, Jan 04 2012
%F A109900 a(n) = (2*n^3+6*n^2+5*n+1-(n+1)*(-1)^n)/16. - _Luce ETIENNE_, May 12 2015
%F A109900 a(n) = Sum_{k=0..n} A001318(k). - _Jacob Szlachetka_, Dec 20 2021
%F A109900 Sum_{n>=1} 1/a(n) = 6 - 8*log(2) + zeta(3). - _Amiram Eldar_, Sep 17 2022
%e A109900 The (n,r)-th term of the following triangle is T(n)-T(r) for r = 0 to n. The n-th row contains n+1 terms.
%e A109900    0
%e A109900    1  0
%e A109900    3  2  0
%e A109900    6  5  3  0
%e A109900   10  9  7  4  0
%e A109900   15 14 12  9  5  0
%e A109900   21 20 18 15 11  6  0
%e A109900   28 27 ...
%e A109900   36 ...
%e A109900 Sequence contains the sum of terms at a 45-degree angle.
%e A109900 a(5) = 15 + 9 + 3 = 27.
%p A109900 A109900 := proc(n) if n mod 2 = 1 then ( (n+1)/2)^3 ; else (n+1)*(n/2+1)*(n/2)/2 ; fi ; end: seq(A109900(n),n=0..80) ; # _R. J. Mathar_, Feb 11 2008
%t A109900 LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 3, 8, 15, 27}, 50] (* _Amiram Eldar_, Sep 17 2022 *)
%Y A109900 Cf. A000217, A001318, A034828, A047866.
%K A109900 nonn,easy
%O A109900 0,3
%A A109900 _Amarnath Murthy_, Jul 13 2005
%E A109900 Corrected and extended by _R. J. Mathar_, Feb 11 2008