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A042948
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Numbers congruent to 0 or 1 (mod 4).
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74
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0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108
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OFFSET
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0,3
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COMMENTS
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Maximum number of squares attacked by a bishop on an (n + 1) X (n + 1) chessboard. - Stewart Gordon, Mar 23 2001
Maximum vertex degree of the (n + 1) X (n + 1) bishop graph and black bishop graph. - Eric W. Weisstein, Jun 26 2017
Also number of squares attacked by a bishop on a toroidal chessboard. - Diego Torres (torresvillarroel(AT)hotmail.com), May 30 2001
Numbers n such that {1, 2, 3, ..., n-1, n} is a perfect Skolem set. - Emeric Deutsch, Nov 24 2006
The number of terms which lie on the principal diagonals of an n X n square spiral. - William A. Tedeschi, Mar 02 2008
Possible nonnegative discriminants of quadratic equation a*x^2 + b*x + c or discriminants of binary quadratic forms a*x^2 + b*x*y + c^y^2. - Artur Jasinski, Apr 28 2008
Nonnegative m for which floor(k*m/4) = k*floor(m/4), where k = 2 or 3. Example: 13 is in the sequence because floor(2*13/4) = 2*floor(13/4), and also floor(3*13/4) = 3*floor(13/4). - Bruno Berselli, Dec 09 2015
Also number of maximal cliques in the n X n white bishop graph. - Eric W. Weisstein, Dec 01 2017
Numbers k for which the binomial coefficient C(k,2) is even. - Tanya Khovanova, Oct 20 2018
Numbers m such that there exists a permutation (x(1), x(2), ..., x(m)) with all absolute differences |x(k) - k| distinct. - Jukka Kohonen, Oct 02 2021
Numbers m such that there exists a multiset of integers whose size is m, and sum and product are both -m. - Yifan Xie, Mar 25 2024
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LINKS
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Harry Tamvakis and O. P. Lossers, Amenable Numbers: 10454, The American Mathematical Monthly, Vol. 105, No. 4 (Apr., 1998), p. 368.
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FORMULA
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G.f.: x*(1 + 3*x)/((1 + x)*(1 - x)^2).
a(n) = a(n-1) + 2 + (-1)^n. (End)
a(n) = n + 2*floor(n/2) = 2*n - (n mod 2). - Bruno Berselli, Apr 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + 3*log(2)/4. - Amiram Eldar, Dec 05 2021
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MAPLE
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a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+4 od: seq(a[n], n=0..54); # Zerinvary Lajos, Mar 16 2008
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MATHEMATICA
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Select[Range[0, 150], Or[Mod[#, 4] == 0, Mod[#, 4] == 1] &] (* Vincenzo Librandi, Dec 09 2015 *)
LinearRecurrence[{1, 1, -1}, {1, 4, 5}, {0, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x (1 + 3 x)/((-1 + x)^2 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
{#, # + 1} & /@ (4 Range[0, 40]) // Flatten (* Harvey P. Dale, Jan 15 2024 *)
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PROG
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(PARI) a(n)=2*n-n%2;
(PARI) concat(0, Vec(x*(1+3*x)/((1+x)*(1-x)^2) + O(x^100))) \\ Altug Alkan, Dec 09 2015
(Maxima) makelist(-1/2+1/2*(-1)^n+2*n, n, 0, 60); /* Martin Ettl, Nov 05 2012 */
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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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STATUS
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approved
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