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
Examples
a(2)=10 because 10 is the third triangular number and the fourth centered triangular number.
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).
Programs
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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;
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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 *)
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)
Extensions
Offset set to 1 by R. J. Mathar, Apr 28 2020
More terms from Michel Marcus, Jan 20 2021
Comments