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 A014979 #31 Feb 16 2025 08:32:33
%S A014979 0,1,210,40755,7906276,1533776805,297544793910,57722156241751,
%T A014979 11197800766105800,2172315626468283465,421418033734080886426,
%U A014979 81752926228785223683195,15859646270350599313653420
%N A014979 Numbers that are both triangular and pentagonal.
%D A014979 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 210, p. 61, Ellipses, Paris 2008.
%D A014979 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 22.
%H A014979 C. Gill, solution to question no. 8, <a href="https://archive.org/details/mathematicalmis00unkngoog">Mathematical Miscellany</a>, 1 (1836), pp. 220-225, at p. 223.
%H A014979 J. C. Su, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Su/su.html">On some properties of two simultaneous polygonal sequences</a>, JIS 10 (2007) 07.10.4, example 4.2.
%H A014979 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalTriangularNumber.html">Pentagonal Triangular Number.</a>
%H A014979 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195,-195,1).
%F A014979 a(n) = 194 * a(n-1) - a(n-2) + 16.
%F A014979 G.f.: x^2 * (1 + 15*x) / ((1 - x) * (1 - 194*x + x^2)).
%F A014979 a(n)=((((1+sqrt(3))^(4*n-1)-(1-sqrt(3))^(4*n-1))/(2^(2*n+1)*sqrt(3)))^2)/2-1/8. - John Sillcox (johnsillcox(AT)hotmail.com), Sep 01 2003
%F A014979 a(n+1) = 97*a(n)+8+7*(192*a(n)^2+32*a(n)+1)^(1/2) - _Richard Choulet_, Sep 19 2007
%F A014979 a(n) = A076139(2*n - 3) = A108281(2 - n). for all n in Z. - _Michael Somos_, Jun 16 2011
%e A014979 G.f. = x^2 + 210*x^3 + 40755*x^4 + 7906276*x^5 + 1533776805*x^6 + ...
%e A014979 a(4) = 40755 which is 285*(285-1)/2 = 165*(3*165-1)/2.
%t A014979 a[ n_] := ChebyshevU[ 2 n - 3, 7] / 14 + ChebyshevT[ 2 n - 3, 7] / 84 - 1/12; (* _Michael Somos_, Feb 24 2015 *)
%t A014979 LinearRecurrence[{195,-195,1},{0,1,210},20] (* _Harvey P. Dale_, May 19 2017 *)
%o A014979 (PARI) {a(n) = polchebyshev( 2*n - 3, 2, 7) / 14 + polchebyshev( 2*n - 3, 1, 7) / 84 - 1 / 12}; /* _Michael Somos_, Jun 16 2011 */
%Y A014979 Cf. A046174, A046175, A076139, A108281.
%K A014979 nonn,easy
%O A014979 1,3
%A A014979 Glenn Johnston (glennj(AT)sonic.net)
%E A014979 Corrected and extended by _Warut Roonguthai_
%E A014979 Edited by _N. J. A. Sloane_, Jul 24 2006