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A095794 a(n) = A005449(n) - 1, where A005449 = second pentagonal numbers. 23

%I

%S 1,6,14,25,39,56,76,99,125,154,186,221,259,300,344,391,441,494,550,

%T 609,671,736,804,875,949,1026,1106,1189,1275,1364,1456,1551,1649,1750,

%U 1854,1961,2071,2184,2300,2419,2541,2666,2794,2925,3059,3196,3336,3479,3625

%N a(n) = A005449(n) - 1, where A005449 = second pentagonal numbers.

%C Row sums of triangle A131414.

%C Equals binomial transform of (1,5,3,0,0,0,..). Equals A051340 * (1,2,3,..).

%C 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

%C Beginning with n=2, a(n) is the falling diagonal starting with T(1,3) in A049777 (as a square array). - _Bob Selcoe_, Oct 27 2014

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = (3/2)*n^2 + (1/2)*n - 1.

%F a(n) = A126890(n+1,n-2) for n>1. - _Reinhard Zumkeller_, Dec 30 2006, corrected by Jason Bandlow (jbandlow(AT)math.upenn.edu), Feb 28 2009

%F 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

%F 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

%F a(n) = 3*n + a(n-1)-1, for n>1, a(1)=1. - _Vincenzo Librandi_, Nov 16 2010

%F a(n) = A115067(-n). - _Bruno Berselli_, Sep 02 2011

%F From _Wesley Ivan Hurt_, Dec 22 2015: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.

%F a(n) = Sum_{i=n..2n} (i-1). (End)

%F E.g.f.: 1 + exp(x)*(3*x^2 + 4*x - 2)/2. - _Stefano Spezia_, Jun 04 2021

%e 1. a(4) = 25 = A005449(4) - 1.

%e 2. a(5) = 39 = (3/2)*5^2 + (1/2)*5 - 1.

%e 3. a(7) = 76 = 3*56 - 3*39 + 25.

%e 4. a(5) = 39 = right term of M^4 * [1 1 1] = [1 5 39].

%e For n = 8, a(8) = 8*22 - (1+4+7+10+13+16+19) - 7 = 99. - _Bruno Berselli_, May 04 2010

%p 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

%p 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_, Feb 18 2008

%t a[n_] := Sum[i+n-2, {i, n+1}]; Table[a[n], {n, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 04 2008 *)

%t s = 1; lst = {s}; Do[s += n + 4; AppendTo[lst, s], {n, 1, 200, 3}]; lst (* _Zerinvary Lajos_, Jul 11 2009 *)

%t Table[Sum[i+n-3, {i, n}], {n, 2, 50}] (* _Zerinvary Lajos_, Jul 11 2009 *)

%t FoldList[## + 2 &, 1, 3 Range@ 45] (* _Robert G. Wilson v_, Feb 03 2011 *)

%t LinearRecurrence[{3,-3,1},{1,6,14},50] (* _Harvey P. Dale_, Dec 09 2013 *)

%o (PARI) a(n)=(3/2)*n^2+(1/2)*n-1 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (MAGMA) [(3/2)*n^2 + (1/2)*n - 1 : n in [1..50]]; // _Wesley Ivan Hurt_, Dec 22 2015

%Y Cf. A000217, A005449, A016777, A049777, A051340, A115067, A126890, A131414.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Jun 06 2004, Jul 08 2007

%E Corrected and extended by _R. J. Mathar_, Jun 23 2006

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Last modified August 6 00:46 EDT 2021. Contains 346493 sequences. (Running on oeis4.)