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

Author

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

Keywords

• nonn
• easy

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

Examples

			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
		

Links

• Leo Tavares, Triangle/Square Pairs
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Cf. A000290.

Programs

Magma

[(3/2)*n^2 + (1/2)*n - 1 : n in [1..50]]; // Wesley Ivan Hurt, Dec 22 2015

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 *)

PARI

a(n)=(3/2)*n^2+(1/2)*n-1 \\ Charles R Greathouse IV, Sep 24 2015

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

Extensions

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