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A095794
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a(n) = A005449(n) - 1, where A005449 = second pentagonal numbers.
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13
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1, 6, 14, 25, 39, 56, 76, 99, 125, 154, 186, 221, 259, 300, 344, 391, 441, 494, 550, 609, 671, 736, 804, 875, 949, 1026, 1106, 1189, 1275, 1364, 1456, 1551, 1649, 1750, 1854, 1961, 2071, 2184, 2300, 2419, 2541, 2666, 2794, 2925, 3059, 3196, 3336, 3479, 3625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Row sums of triangle A131414.
Equals binomial transform of (1,5,3,0,0,0,..). Equals A051340 * (1,2,3,..).
a(n) is essentially the case -1 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n}((k-2)*i-(k-3). Thus P_{-1}(n) = n*(5-3*n)/2 and a(n) = -P_{-1}(n+2). [Peter Luschny, Jul 08 2011]
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| a(n) = (3/2)*n^2 + (1/2)*n - 1.
a(n) = A126890(n+1,n-2) for n>1. - Reinhard Zumkeller, Dec 30 2006
G.f.: x*(-1-3*x+x^2)/(-1+x)^3 = 1-3/(-1+x)^3-4/(-1+x)^2 . - R. J. Mathar, Nov 19 2007
a(n) = n*A016777(n-1)-sum(i=1..n-2, A016777(i))-(n-1) = (n+1)*(3*n-2)/2. - Bruno Berselli, May 04 2010
a(n) = 3*n+a(n-1)-1 (with a(1)=1). - Vincenzo Librandi, Nov 16 2010
a(n) = A115067(-n). - Bruno Berselli, Sep 02 2011
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EXAMPLE
| 1. a(4) = 25 = A005449(4) - 1
2. a(5) = 39 = (3/2)*5^2 + (1/2)*5 - 1.
3. a(7) = 76 = 3*56 - 3*39 + 25
4. a(5) = 39 = right term of M^4 * [1 1 1] = [1 5 39].
For n = 8, a(8) = 8*22-(1+4+7+10+13+16+19)-7 = 99. [From Bruno Berselli, May 04 2010]
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MAPLE
| A005449 := proc(n) RETURN(n*(3*n+1)/2) ; end: A095794 := proc(n) RETURN(A005449(n)-1) ; end: for n from 1 to 100 do printf("%a, ", A095794(n)) ; od: - R. J. Mathar, Jun 23 2006
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]-3 od: seq(-a[n], n=2..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 2008
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MATHEMATICA
| a[n_]:=Sum[i+n-3, {i, 1, n}]; [From Vladimir Orlovsky, Dec 04 2008]
s = 1; lst = {s}; Do[s += n + 4; AppendTo[lst, s], {n, 1, 200, 3}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
Table[Sum[i + n - 3, {i, 1, n}], {n, 2, 50}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
FoldList[## + 2 &, 1, 3 Range@ 45] (* Robert G. Wilson v, Feb 03 2011 *)
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CROSSREFS
| Cf. A005449, A051340, A131414.
A000217 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
Sequence in context: A083657 A010740 A185594 * A119867 A026055 A165986
Adjacent sequences: A095791 A095792 A095793 * A095795 A095796 A095797
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KEYWORD
| nonn,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2004, Jul 08 2007
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EXTENSIONS
| Corrected and extended by R. J. Mathar, Jun 23 2006
Comment corrected by Jason Bandlow (jbandlow(AT)math.upenn.edu), Feb 28 2009
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