A128862 Numbers simultaneously triangular and centered triangular.

1, 10, 136, 1891, 26335, 366796, 5108806, 71156485, 991081981, 13803991246, 192264795460, 2677903145191, 37298379237211, 519499406175760, 7235693307223426, 100780206894952201, 1403687203222107385, 19550840638214551186, 272308081731781609216

Offset

1,2

Comments

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..875

F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.

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) - 3/2 + (1/2)*sqrt(192*a(n)^2 - 96*a(n) - 15).

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;

Mathematica

Rest@ CoefficientList[Series[x (1 - 5 x + x^2)/((1 - x) (1 - 14 x + x^2)), {x, 0, 19}], x] (* Michael De Vlieger, Jul 19 2023 *)

Crossrefs

Intersection of A000217 and A005448.

Cf. A001570, 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