OFFSET
1,3
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 210, p. 61, Ellipses, Paris 2008.
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 35.
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
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-5)-(1-sqrt(3))^(4*n-5))/(2^(2*n-1)*sqrt(3)))^2)/2-1/8. - John Sillcox (johnsillcox(AT)hotmail.com), Sep 01 2003 [adapted to offset by Stefano Spezia, Sep 20 2025]
a(n+1) = 97*a(n)+8+7*(192*a(n)^2+32*a(n)+1)^(1/2) - Richard Choulet, Sep 19 2007
E.g.f.: (exp(97*x)*(362*cosh(56*sqrt(3)*x) - 209*sqrt(3)*sinh(56*sqrt(3)*x)) - 360 - 2*exp(x))/24. - Stefano Spezia, Sep 20 2025
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
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
a(14)-a(16) from Stefano Spezia, Sep 20 2025
STATUS
approved