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A002939
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a(n) = 2*n*(2*n-1).
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89
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0, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8010, 8372
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OFFSET
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0,2
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COMMENTS
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Write 0,1,2,... in a spiral; sequence gives numbers on one of 4 diagonals (see Example section).
For n>2, the terms represent the sums of those primitive Pythagorean triples with hypotenuse (H) one unit longer than the longest side (L), or H = L + 1. - Richard R. Forberg, Jun 09 2015
For n>1, a(n) is the perimeter of a Pythagorean triangle with an odd leg 2*n-1. - Agola Kisira Odero, Apr 26 2016
A338109(n)/a(n+1) is the Kirchhoff index of the join of the disjoint union of two complete graphs on n vertices with the empty graph on n+1 vertices.
Equivalently, the graph can be described as the graph on 3*n + 1 vertices with labels 0..3*n and with i and j adjacent iff iff i+j> 0 mod 3.
A338588(n)/a(n+1) is the Kirchhoff index of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.
Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j> 0 mod 3.
These graphs are cographs. (End)
a(n), n>=1, is the number of paths of minimum length (length=2) from the origin to the cross polytope of size 2 in Z^n (column 2 in A371064). - Shel Kaphan, Mar 09 2024
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
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LINKS
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FORMULA
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Sum_{n >= 1} 1/a(n) = log(2) (cf. Tijdeman).
Log(2) = Sum_{n >= 1} ((1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ...) = Sum_{n >= 0} (-1)^n/(n+1). Log(2) = Integral_{x=0..1} 1/(1+x) dx. - Gary W. Adamson, Jun 22 2003
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*x*(1+3*x)/(1-x)^3. (End)
0 = 12 + a(n)*(-8 + a(n) - 2*a(n+1)) + a(n+1)*(-8 + a(n+1)) for all n in Z. - Michael Somos, Jan 28 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 - log(2)/2. - Amiram Eldar, Jul 31 2020
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EXAMPLE
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G.f. = 2*x + 12*x^2 + 30*x^3 + 56*x^4 + 90*x^5 + 132*x^6 + 182*x^7 + 240*x^8 + ...
On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step in any of the four cardinal directions and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie along one of the diagonals, as seen in the example below:
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99 64--65--66--67--68--69--70--71--72
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98 63 36--37--38--39--40--41--42 73
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97 62 35 16--17--18--19--20 43 74
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96 61 34 15 4---5---6 21 44 75
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95 60 33 14 3 *0* 7 22 45 76
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94 59 32 13 *2*--1 8 23 46 77
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93 58 31 *12*-11--10---9 24 47 78
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92 57 *30*-29--28--27--26--25 48 79
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91 *56*-55--54--53--52--51--50--49 80
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*90*-89--88--87--86--85--84--83--82--81
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MAPLE
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MATHEMATICA
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PROG
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(Haskell)
a002939 n = (* 2) . a000384
a002939_list = scanl1 (+) a017089_list
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CROSSREFS
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Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488 (this sequence is the case k=8). - Bruno Berselli, Jun 10 2013
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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