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A128862 Numbers simultaneously triangular and centered triangular. 3
1, 10, 136, 1891, 26335, 366796, 5108806, 71156485, 991081981, 13803991246, 192264795460, 2677903145191, 37298379237211, 519499406175760, 7235693307223426, 100780206894952201, 1403687203222107385, 19550840638214551186, 272308081731781609216 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A129803 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008

LINKS

Table of n, a(n) for n=1..19.

J. Sadek and R. Euler, A Formula For All K-Gonal Numbers that Are Centered, 2010, arXiv:1003.2375 [math.NT], 2010.

S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine,  Vol. 84, No. 5, December 2011, pp. 339-350.

Index entries for linear recurrences with constant coefficients, signature (15, -15, 1).

FORMULA

Define x(n) and y(n) by (3+sqrt(3))*(2+sqrt(3))^n = x(n) + y(n)*sqrt(3); let s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+3*(s(n)^2-s(n))).

a(n) = (3*A001570(n) + 1)/4. - Ralf Stephan, May 20 2007

From Richard Choulet, Oct 01 2007: (Start)

a(n+2) = 14*a(n+1)-a(n)-3.

a(n+1) = 7*a(n)-1.5+0.5*(192*a(n)^2-96*a(n)-15)^0.5.

G.f.: x*(1-5*x+x^2)/((1-x)*(1-14*x+x^2)). (End)

EXAMPLE

a(2)=10 because 10 is the third triangular number and the fourth centered triangular number.

MAPLE

CP := n -> 1+1/2*3*(n^2-n): N:=10: u:=2: v:=1: x:=3: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+3*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp), CP(s)]: end do: k_pcp;

CROSSREFS

Cf. A000217, A001570, A005448, A129803.

Sequence in context: A287473 A240917 A240654 * A129803 A065024 A026244

Adjacent sequences:  A128859 A128860 A128861 * A128863 A128864 A128865

KEYWORD

easy,nonn

AUTHOR

Steven Schlicker, Apr 24 2007

EXTENSIONS

Offset set to 1 by R. J. Mathar, Apr 28 2020

More terms from Michel Marcus, Jan 20 2021

STATUS

approved

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Last modified March 1 03:32 EST 2021. Contains 341732 sequences. (Running on oeis4.)