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A056000
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a(n) = n*(n+9)/2.
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29
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0, 5, 11, 18, 26, 35, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 200, 221, 243, 266, 290, 315, 341, 368, 396, 425, 455, 486, 518, 551, 585, 620, 656, 693, 731, 770, 810, 851, 893, 936, 980, 1025, 1071, 1118, 1166, 1215, 1265, 1316, 1368, 1421, 1475
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OFFSET
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0,2
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COMMENTS
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Numbers m >= 0 such that 8m+81 is a square. - Bruce J. Nicholson, Jul 29 2017
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..5000
Leo Tavares, Illustration: Triangular Pairs
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = A000217(n+4) - 10.
G.f.: x(5-4x)/(1-x)^3.
From Zerinvary Lajos, Oct 01 2006: (Start)
a(n) = A000096(n) + 3*n.
a(n) = A055999(n) + n.
a(n) = A056115(n) - n.
(End)
a(n) = binomial(n,2) - 4*n, n >= 9. - Zerinvary Lajos, Nov 25 2006
a(n) = A126890(n,4) for n > 3. - Reinhard Zumkeller, Dec 30 2006
a(n) = A028569(n)/2. - Zerinvary Lajos, Feb 12 2007
If we define f(n,i,a) = Sum_{k=0..(n-i)} binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,5), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = n + a(n-1) + 4. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+4). - Gary Detlefs, Aug 10 2010
Sum_{n>=1} 1/a(n) = 7129/11340. - R. J. Mathar, Jul 14 2012
a(n) = 5n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: (1/2)*(x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 1879/11340. - Amiram Eldar, Jul 03 2020
a(n) = A000217(n+1) + A008585(n) - 1. - Leo Tavares, Sep 22 2022
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MATHEMATICA
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Table[n (n + 9)/2, {n, 0, 50}] (* or *)
FoldList[#1 + #2 + 4 &, Range[0, 50]] (* or *)
Table[PolygonalNumber[n + 4] - 10, {n, 0, 50}] (* or *)
CoefficientList[Series[x (5 - 4 x)/(1 - x)^3, {x, 0, 50}], x] (* Michael De Vlieger, Jul 30 2017 *)
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PROG
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(PARI) a(n)=n*(n+9)/2 \\ Charles R Greathouse IV, Sep 24 2015
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CROSSREFS
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Cf. A000096, A055998, A055999, A001477.
Column m=2 of (1, 5)-Pascal triangle A096940.
Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
Cf. A000217, A008585.
Sequence in context: A145005 A004083 A190365 * A080566 A094684 A240438
Adjacent sequences: A055997 A055998 A055999 * A056001 A056002 A056003
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KEYWORD
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easy,nonn,changed
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AUTHOR
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Barry E. Williams, Jun 16 2000
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EXTENSIONS
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More terms from James A. Sellers, Jul 04 2000
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STATUS
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approved
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