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, 36856, 38781, 40755

Offset: 0

Author

Paul Barry, Mar 23 2003

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A016993, A081581, A081592.

Programs

Magma

[(2-21*n+49*n^2)/2: n in [0..50]]; // Vincenzo Librandi, Jun 18 2011

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 *)

PARI

a(n)=(2-21*n+49*n^2)/2 \\ Charles R Greathouse IV, Jun 17 2017

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

E.g.f.: exp(x)*(2 + 28*x + 49*x^2)/2. - Elmo R. Oliveira, Jun 09 2025