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A109900
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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.
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3
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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
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OFFSET
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0,3
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COMMENTS
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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, ...
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LINKS
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FORMULA
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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
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
Sum_{n>=1} 1/a(n) = 6 - 8*log(2) + zeta(3). - Amiram Eldar, Sep 17 2022
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EXAMPLE
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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.
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MAPLE
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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
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LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 3, 8, 15, 27}, 50] (* Amiram Eldar, Sep 17 2022 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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