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A006451
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Numbers n such that n*(n+1)/2+1 is a square.
(Formerly M1472)
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28
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0, 2, 5, 15, 32, 90, 189, 527, 1104, 3074, 6437, 17919, 37520, 104442, 218685, 608735, 1274592, 3547970, 7428869, 20679087, 43298624, 120526554, 252362877, 702480239, 1470878640, 4094354882, 8572908965, 23863649055, 49966575152
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Or triangular numbers such that distance to the nearest square is 1. [From Ctibor O. ZIZKA (c.zizka(AT)email.cz), Oct 07 2009]
Also numbers n such that (ceiling(sqrt(n*(n+1)/2)))^2 - n*(n+1)/2 = 1. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Nov 10 2009]
For n>0, with T(n) the nth triangular number,
a(2n-1)=A002315(n-1)+A001108(n) and T(a(2n-1)+1=(A001653(n)+A001109(n))^2;
e.g., a(5)=41+49=90 and T(90)+1=(29+35)^2;
a(2n)=A002315(n)-A001108(n)-1 and T(a(2n))+1=(A001653(n+1)-A001109(n))^2;
e.g., a(6)=239-49-1=189 and T(189)+1=(169-35)^2.
In general, if b(n)=the numbers such that T(b(n))+T(k) is a square, then
b(2n-1)=k*A002315(n-1)+A001108(n) and T(b(2n-1)+T(k))=(k*A001653(n)+A001109(n))^2;
e.g., for k=3, b(5)=3*41+49=172 and T(172)+6=(3*29+35)^2;
b(2n)=k*A002315(n)-A001108(n)-1 and T(b(2n))+1=(k*A001653(n+1)-A001109(n))^2;
e.g., for k=4, b(6)=4*239-49-1=906 and T(906)+10=(4*169-35)^2.
See A154138, A154140, A154143, A154145, A154148, A154153.
- Charlie Marion (charliemath(AT)optonline.net), Dec 05 2010
a(2n-1)=A001652(n)-A001653(n); a(2n)=A001652(n-1)+A001653(n+1).
In general, indices k such that A001109(2j) plus the k-th triangular
number is a perfect square may be found as follows:
b(2n-1)=A001652(n+j-1)-A001653(n-j);
b(2n)=A001652(n-j-1)+A001653(n+j);
indices k such that A001109(2j-1) plus the k-th triangular number is
a perfect square may be found as follows:
b(2n-1)=A001652(n+j-1)-A001653(n-j+1);
b(2n)=A001652(n-j)+A001653(n+j). See also A154140.
- Charlie Marion, Mar 11 2011
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REFERENCES
| A. J. Gottlieb, How four dogs meet in a field, etc., Technology Review, Problem J/A2, Jul/August 1973 pp. 73-74; solution Jan 1974 (see link).
J. O. Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. J. Gottlieb, How four dogs meet in a field, etc. (scanned copy)
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n)=6a(n-2)-a(n-4)+2 with a(0)=0, a(1)=2, a(2)=5, a(3)=15. - Zak Seidov, Apr 15 2008
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MAPLE
| A006451:=z*(-2-3*z+2*z**2+z**3)/(z-1)/(z**2+2*z-1)/(z**2-2*z-1); [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| a[0] = a[1] = 1; a[2] = 2; a[3] = 4; a[n_] := 6 a[n - 2] - a[n - 4]; t = Array[a, 30, 0]; Join[{0}, Drop[ Floor[ Sqrt[2 #^2 - 1]] & /@ t, 2]] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 11 2010]
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PROG
| for(n=1, 10000, t=n*(n+1)/2+1; if(issquare(t), print1(n, ", "))) - Joerg Arndt, Oct 10 2009
(Haskell)
a006451 n = a006451_list !! n
a006451_list = 0 : 2 : 5 : 15 : map (+ 2)
(zipWith (-) (map (* 6) (drop 2 a006451_list)) a006451_list)
-- Reinhard Zumkeller, 10 Jan 2012
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CROSSREFS
| Cf. A000124, A006452, A006454.
Sequence in context: A078528 A077686 A034499 * A000962 A118387 A034522
Adjacent sequences: A006448 A006449 A006450 * A006452 A006453 A006454
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and Jeffrey Shallit
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Feb 07 2001
Edited by N. J. A. Sloane, Oct 24 2009, following discussions by several correspondents in the Sequence Fans Mailing List, Oct 10 2009
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