%I #26 Sep 08 2021 03:46:15

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,3,5,7,9,11,13,17,25,37,53,73,97,321,549,

%T 781,1017,1257,1501,1995,2985,4471,6453,8931,11905,39433,67457,95977,

%U 124993,154505,184513,245273,367041,549817,793601,1098393,1464193,4849891

%N a(n)*a(n-13) = a(n-1)*a(n-12)+a(n-6)+a(n-7) with initial terms a(1)=...=a(13)=1.

%H Seiichi Manyama, <a href="/A133854/b133854.txt">Table of n, a(n) for n = 1..5754</a>

%H P. Heideman and E. Hogan, <a href="http://arXiv.org/abs/0709.2529">A New Family of Somos-Like Recurrences</a>, arXiv:0709.2529 [math.CO], 2007-2009.

%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,124,0,0,0,0,0,0,0,0,0,0,0,-124,0,0,0,0,0,0,0,0,0,0,0,1).

%F Sequence also generated by the linear recurrence 124*(u(n-12)-u(n-24))+u(n-36) with the initial 36 terms given by the quadratic recurrence.

%F G.f.: x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 +x^8 +x^9 +x^10 +x^11 -123*x^12 -121*x^13 -119*x^14 -117*x^15 -115*x^16 -113*x^17 -111*x^18 -107*x^19 -99*x^20 -87*x^21 -71*x^22 -51*x^23 +97*x^24 +73*x^25 +53*x^26 +37*x^27 +25*x^28 +17*x^29 +13*x^30 +11*x^31 +9*x^32 +7*x^33 +5*x^34 +3*x^35) / ((1 -x)*(1 +x)*(1 -x +x^2)*(1 +x^2)*(1 +x +x^2)*(1 -x^2 +x^4)*(1 +11*x^6 -x^12)*(1 -11*x^6 -x^12)). - _Colin Barker_, Jul 18 2016

%p a := proc(n) option remember; if n<=13 then RETURN(1); else RETURN((a(n-1)*a(n-12)+a(n-6)+a(n-7))/a(n-13)); fi; end;

%t RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]== a[8]== a[9]== a[10]== a[11]== a[12]==a[13]==1,a[n]==(a[n-1]a[n-12]+a[n-6]+ a[n-7])/ a[n-13]},a,{n,50}] (* _Harvey P. Dale_, Nov 24 2015 *)

%o (PARI) a(k=13, n) = {K = (k-1)/2; vds = vector(n); for (i=1, 2*K+1, vds[i] = 1;); for (i=2*K+2, n, vds[i] = (vds[i-1]*vds[i-2*K]+vds[i-K]+vds[i-K-1])/vds[i-2*K-1];); for (i=1, n, print1(vds[i], ","););} \\ _Michel Marcus_, Nov 01 2012

%o (PARI) Vec(x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 +x^8 +x^9 +x^10 +x^11 -123*x^12 -121*x^13 -119*x^14 -117*x^15 -115*x^16 -113*x^17 -111*x^18 -107*x^19 -99*x^20 -87*x^21 -71*x^22 -51*x^23 +97*x^24 +73*x^25 +53*x^26 +37*x^27 +25*x^28 +17*x^29 +13*x^30 +11*x^31 +9*x^32 +7*x^33 +5*x^34 +3*x^35) / ((1 -x)*(1 +x)*(1 -x +x^2)*(1 +x^2)*(1 +x +x^2)*(1 -x^2 +x^4)*(1 +11*x^6 -x^12)*(1 -11*x^6 -x^12)) + O(x^60)) \\ _Colin Barker_, Jul 18 2016

%Y Cf. A072881, A092264, A133846, A133847, A133848.

%K easy,nonn

%O 1,14

%A _Emilie Hogan_, Sep 26 2007