A081591 Third row of Pascal-(1,6,1) array A081581.

1, 15, 78, 190, 351, 561, 820, 1128, 1485, 1891, 2346, 2850, 3403, 4005, 4656, 5356, 6105, 6903, 7750, 8646, 9591, 10585, 11628, 12720, 13861, 15051, 16290, 17578, 18915, 20301, 21736, 23220, 24753, 26335, 27966, 29646, 31375, 33153, 34980

Offset

0,2

Comments

1. Smallest triangular number T(k) (other than the trivial adjacent ones) such that T(n) + T(k) is a square. ( T(n-1) and T(n+1) are trivial triangular numbers such that T(n) +T(n-1) and T(n) + T(n+1) both are squares.) 0+1 = 1, 1+15 = 16, 3+ 78= 81, 6 + 190 = 196 etc. 2. (7n+5)-th triangular number. - Amarnath Murthy, Jun 20 2003

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

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

Formula

a(n) = (2 - 21*n + 49*n^2)/2.

G.f.: (1+6*x)^2/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=15, a(2)=78. - Harvey P. Dale, Aug 03 2012

Mathematica

Table[(2-21n+49n^2)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 15, 78}, 40] (* Harvey P. Dale, Aug 03 2012 *)

Prog

(Magma) [(2-21*n+49*n^2)/2: n in [0..50]]; // Vincenzo Librandi, Jun 18 2011
(PARI) a(n)=(2-21*n+49*n^2)/2 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A016993, A081592.

Sequence in context: A303097 A128272 A180579 * A269620 A269436 A044202

Adjacent sequences: A081588 A081589 A081590 * A081592 A081593 A081594

Keyword

easy,nonn

Author

Paul Barry, Mar 23 2003

Status

approved