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A081591 Third row of Pascal-(1,6,1) array A081581. 3
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 (list; graph; refs; listen; history; text; internal format)
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(0)=1, a(1)=15, a(2)=78, a(n)=3*a(n-1)-3*a(n-2)+a(n-3) -- From 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

CROSSREFS

Cf. A016993, A081592.

Sequence in context: A205433 A128272 A180579 * A269620 A269436 A044202

Adjacent sequences:  A081588 A081589 A081590 * A081592 A081593 A081594

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 23 2003

STATUS

approved

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Last modified December 3 20:40 EST 2016. Contains 278745 sequences.