A014979 Numbers that are both triangular and pentagonal.

0, 1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, 421418033734080886426, 81752926228785223683195, 15859646270350599313653420

Offset

1,3

References

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 210, p. 61, Ellipses, Paris 2008.

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.

Links

Table of n, a(n) for n=1..13.

C. Gill, solution to question no. 8, Mathematical Miscellany, 1 (1836), pp. 220-225, at p. 223.

J. C. Su, On some properties of two simultaneous polygonal sequences, JIS 10 (2007) 07.10.4, example 4.2.

Eric Weisstein's World of Mathematics, Pentagonal Triangular Number.

Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Formula

a(n) = 194 * a(n-1) - a(n-2) + 16.

G.f.: x^2 * (1 + 15*x) / ((1 - x) * (1 - 194*x + x^2)).

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

a(n+1) = 97*a(n)+8+7*(192*a(n)^2+32*a(n)+1)^(1/2) - Richard Choulet, Sep 19 2007

a(n) = A076139(2*n - 3) = A108281(2 - n). for all n in Z. - Michael Somos, Jun 16 2011

Example

G.f. = x^2 + 210*x^3 + 40755*x^4 + 7906276*x^5 + 1533776805*x^6 + ...
a(4) = 40755 which is 285*(285-1)/2 = 165*(3*165-1)/2.

Mathematica

a[ n_] := ChebyshevU[ 2 n - 3, 7] / 14 + ChebyshevT[ 2 n - 3, 7] / 84 - 1/12; (* Michael Somos, Feb 24 2015 *)
LinearRecurrence[{195, -195, 1}, {0, 1, 210}, 20] (* Harvey P. Dale, May 19 2017 *)

Prog

(PARI) {a(n) = polchebyshev( 2*n - 3, 2, 7) / 14 + polchebyshev( 2*n - 3, 1, 7) / 84 - 1 / 12}; /* Michael Somos, Jun 16 2011 */

Crossrefs

Cf. A046174, A046175, A076139, A108281.

Sequence in context: A229754 A089514 A289328 * A356690 A362945 A275458

Adjacent sequences: A014976 A014977 A014978 * A014980 A014981 A014982

Keyword

nonn,easy

Author

Glenn Johnston (glennj(AT)sonic.net)

Extensions

Corrected and extended by Warut Roonguthai

Edited by N. J. A. Sloane, Jul 24 2006

Status

approved