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A128862 Numbers simultaneously triangular and centered triangular.

%I A128862 #43 Jul 19 2023 15:15:58
%S A128862 1,10,136,1891,26335,366796,5108806,71156485,991081981,13803991246,
%T A128862 192264795460,2677903145191,37298379237211,519499406175760,
%U A128862 7235693307223426,100780206894952201,1403687203222107385,19550840638214551186,272308081731781609216
%N A128862 Numbers simultaneously triangular and centered triangular.
%C A128862 A129803 is an essentially identical sequence. - _R. J. Mathar_, Jun 13 2008
%H A128862 Michael De Vlieger, <a href="/A128862/b128862.txt">Table of n, a(n) for n = 1..875</a>
%H A128862 F. Javier de Vega, <a href="https://doi.org/10.17654/0972555523015">On the parabolic partitions of a number</a>, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
%H A128862 J. Sadek and R. Euler, <a href="http://arxiv.org/abs/1003.2375">A Formula For All K-Gonal Numbers that Are Centered</a>, 2010, arXiv:1003.2375 [math.NT], 2010.
%H A128862 S. C. Schlicker, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.5.339">Numbers Simultaneously Polygonal and Centered Polygonal</a>, Mathematics Magazine,  Vol. 84, No. 5, December 2011, pp. 339-350.
%H A128862 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%F A128862 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))).
%F A128862 a(n) = (3*A001570(n) + 1)/4. - _Ralf Stephan_, May 20 2007
%F A128862 From _Richard Choulet_, Oct 01 2007: (Start)
%F A128862 a(n+2) = 14*a(n+1) - a(n) - 3.
%F A128862 a(n+1) = 7*a(n) - 3/2 + (1/2)*sqrt(192*a(n)^2 - 96*a(n) - 15).
%F A128862 G.f.: x*(1-5*x+x^2)/((1-x)*(1-14*x+x^2)). (End)
%e A128862 a(2)=10 because 10 is the third triangular number and the fourth centered triangular number.
%p A128862 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;
%t A128862 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 *)
%Y A128862 Intersection of A000217 and A005448.
%Y A128862 Cf. A001570, A129803.
%K A128862 easy,nonn
%O A128862 1,2
%A A128862 _Steven Schlicker_, Apr 24 2007
%E A128862 Offset set to 1 by _R. J. Mathar_, Apr 28 2020
%E A128862 More terms from _Michel Marcus_, Jan 20 2021