This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A081591 #23 Jun 09 2025 10:39:14
%S A081591 1,15,78,190,351,561,820,1128,1485,1891,2346,2850,3403,4005,4656,5356,
%T A081591 6105,6903,7750,8646,9591,10585,11628,12720,13861,15051,16290,17578,
%U A081591 18915,20301,21736,23220,24753,26335,27966,29646,31375,33153,34980,36856,38781,40755
%N A081591 Third row of Pascal-(1,6,1) array A081581.
%C A081591 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
%H A081591 Vincenzo Librandi, <a href="/A081591/b081591.txt">Table of n, a(n) for n = 0..2000</a>
%H A081591 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A081591 a(n) = (2 - 21*n + 49*n^2)/2.
%F A081591 G.f.: (1+6*x)^2/(1-x)^3.
%F A081591 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
%F A081591 E.g.f.: exp(x)*(2 + 28*x + 49*x^2)/2. - _Elmo R. Oliveira_, Jun 09 2025
%t A081591 Table[(2-21n+49n^2)/2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,15,78},40] (* _Harvey P. Dale_, Aug 03 2012 *)
%o A081591 (Magma) [(2-21*n+49*n^2)/2: n in [0..50]]; // _Vincenzo Librandi_, Jun 18 2011
%o A081591 (PARI) a(n)=(2-21*n+49*n^2)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A081591 Cf. A016993, A081581, A081592.
%K A081591 easy,nonn
%O A081591 0,2
%A A081591 _Paul Barry_, Mar 23 2003