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

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

Offset

1,2

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

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

Links

Table of n, a(n) for n=1..49.

Leo Tavares, Triangle/Square Pairs

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

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

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

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, for n>1, a(1)=1. - Vincenzo Librandi, Nov 16 2010

a(n) = A115067(-n). - Bruno Berselli, Sep 02 2011

From Wesley Ivan Hurt, Dec 22 2015: (Start)

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

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

E.g.f.: 1 + exp(x)*(3*x^2 + 4*x - 2)/2. - Stefano Spezia, Jun 04 2021

From Amiram Eldar, Feb 22 2022: (Start)

Sum_{n>=1} 1/a(n) = Pi/(5*sqrt(3)) + 3*log(3)/5 + 2/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(5*sqrt(3)) + 4*log(2)/5 - 2/5. (End)

a(n) = A000217(n) + A000290(n) - 1. - Leo Tavares, Jun 02 2023

Example

a(4) = 25 = A005449(4) - 1.
a(5) = 39 = (3/2)*5^2 + (1/2)*5 - 1.
a(7) = 76 = 3*56 - 3*39 + 25.
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. - Bruno Berselli, May 04 2010

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, Feb 18 2008

Mathematica

FoldList[## + 2 &, 1, 3 Range@ 45] (* Robert G. Wilson v, Feb 03 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 6, 14}, 50] (* Harvey P. Dale, Dec 09 2013 *)

Prog

(PARI) a(n)=(3/2)*n^2+(1/2)*n-1 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [(3/2)*n^2 + (1/2)*n - 1 : n in [1..50]]; // Wesley Ivan Hurt, Dec 22 2015

Crossrefs

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

Cf. A000290.

Sequence in context: A317333 A010740 A185594 * A119867 A293400 A026055

Adjacent sequences: A095791 A095792 A095793 * A095795 A095796 A095797

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 06 2004, Jul 08 2007

Extensions

Corrected and extended by R. J. Mathar, Jun 23 2006

Status

approved