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A014979 Numbers that are both triangular and pentagonal. 4
0, 1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, 421418033734080886426, 81752926228785223683195, 15859646270350599313653420 (list; graph; refs; listen; history; text; internal format)
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.

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

Index to sequences with 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). - Michael Somos, Jun 16 2011

EXAMPLE

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.

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: A187308 A229754 A089514 * A229671 A134236 A136350

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

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Last modified November 26 20:33 EST 2014. Contains 250119 sequences.