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A081498 Consider the triangle in which the n-th row starts with n, contains n terms and the difference of successive terms is 1,2,3,... up to n-1. Sequence gives row sums. 2
1, 3, 5, 6, 5, 1, -7, -20, -39, -65, -99, -142, -195, -259, -335, -424, -527, -645, -779, -930, -1099, -1287, -1495, -1724, -1975, -2249, -2547, -2870, -3219, -3595, -3999, -4432, -4895, -5389, -5915, -6474, -7067, -7695, -8359, -9060, -9799, -10577, -11395, -12254, -13155, -14099, -15087, -16120 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The triangle whose row sums are being considered is:

  1;

  2,   1;

  3,   2,   0;

  4,   3,   1,  -2;

  5,   4,   2,  -1,  -5;

  6,   5,   3,   0,  -4,  -9;

  7,   6,   4,   1,  -3,  -8, -14;

The leading diagonal is given by A080956(n-1) = n*(3-n)/2.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = n^2 - binomial(n+1, n-2). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

a(n) = binomial(n,2)+binomial(n,1)-binomial(n,3). - Zerinvary Lajos, Jul 23 2006

a(n) = n*(1+6*n-n^2)/6. - Karen Yeats (kayeats(AT)bu.edu), Nov 20 2006

From Michael Somos, Jul 04 2012: (Start)

G.f.: x * (1 - x - x^2) / (1 - x)^4.

a(-1 - n) = A008778(n). (End)

E.g.f.: x*(6 +3*x -x^2)*exp(x)/6. - G. C. Greubel, Mar 06 2019

EXAMPLE

G.f. = x * (1 + 3*x + 5*x^2 + 6*x^3 + 5*x^4 + x^5 - 7*x^6 - 20*x^7 - 39*x^8 - 65*x^9 + ...).

MAPLE

seq(n^2-binomial(n+1, n-2), n=1..50); # C. Ronaldo

[seq(binomial(n, 2)+binomial(n, 1)-binomial(n, 3), n=1..49)]; # Zerinvary Lajos, Jul 23 2006

MATHEMATICA

LinearRecurrence[{4, -6, 4, -1}, {1, 3, 5, 6}, 50] (* G. C. Greubel, Mar 06 2019 *)

PROG

(PARI) {a(n) = if( n< 0, n = -2 - n; polcoeff( (1 + x - x^2) / (1 - x)^4 + x * O(x^n), n), polcoeff( (1 - x - x^2) / (1 - x)^4 + x * O(x^n), n))} /* Michael Somos, Jul 04 2012 */

(PARI) vector(50, n, n*(1+6*n-n^2)/6) \\ G. C. Greubel, Mar 06 2019

(GAP) List([1..50], n->n^2-Binomial(n+1, n-2)); # Muniru A Asiru, Mar 05 2019

(MAGMA) [n*(1+6*n-n^2)/6: n in [1..50]]; // G. C. Greubel, Mar 06 2019

(Sage) [n*(1+6*n-n^2)/6 for n in (1..50)] # G. C. Greubel, Mar 06 2019

CROSSREFS

Cf. A008778, A080956, A081499.

Sequence in context: A078064 A091517 A106117 * A110279 A161435 A224831

Adjacent sequences:  A081495 A081496 A081497 * A081499 A081500 A081501

KEYWORD

sign,easy

AUTHOR

Amarnath Murthy, Mar 25 2003

EXTENSIONS

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

Offset changed to 1 at the suggestion of Michel Marcus, Mar 05 2019

Formulas and programs addapted for offset 1 by Michel Marcus, Mar 05 2019

STATUS

approved

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Last modified March 25 08:15 EDT 2019. Contains 321469 sequences. (Running on oeis4.)