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.
0, 1, 3, 8, 15, 27, 42, 64, 90, 125, 165, 216, 273, 343, 420, 512, 612, 729, 855, 1000, 1155, 1331, 1518, 1728, 1950, 2197, 2457, 2744, 3045, 3375, 3720, 4096, 4488, 4913, 5355, 5832, 6327, 6859, 7410, 8000, 8610, 9261, 9933, 10648, 11385, 12167, 12972
Offset: 0
Examples
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. 0 1 0 3 2 0 6 5 3 0 10 9 7 4 0 15 14 12 9 5 0 21 20 18 15 11 6 0 28 27 ... 36 ... Sequence contains the sum of terms at a 45-degree angle. a(5) = 15 + 9 + 3 = 27.
Links
- Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
Programs
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Maple
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
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Mathematica
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 3, 8, 15, 27}, 50] (* Amiram Eldar, Sep 17 2022 *)
Formula
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
a(n) = A034828(n+1). - R. J. Mathar, Aug 18 2008
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
a(n) = (2*n^3+6*n^2+5*n+1-(n+1)*(-1)^n)/16. - Luce ETIENNE, May 12 2015
a(n) = Sum_{k=0..n} A001318(k). - Jacob Szlachetka, Dec 20 2021
Sum_{n>=1} 1/a(n) = 6 - 8*log(2) + zeta(3). - Amiram Eldar, Sep 17 2022
Extensions
Corrected and extended by R. J. Mathar, Feb 11 2008
Comments