A133848 -id:A133848 - cp's OEIS frontend

A072881 a(1)=a(2)=a(3)=1; for n>3, a(n)=(a(n-1)*a(n-2)+a(n-1)+a(n-2))/a(n-3).

1, 1, 1, 3, 7, 31, 85, 393, 1093, 5071, 14119, 65523, 182449, 846721, 2357713, 10941843, 30467815, 141397231, 393723877, 1827222153, 5087942581, 23612490751, 65749529671, 305135157603, 849655943137, 3943144558081

Offset: 1

Benoit Cloitre, Jul 28 2002, revised Feb 03 2005

What accounts for the high proportion of semiprimes in this sequence? Primes: 3, 7, 31, 1093, 846721, 393723877, ... Semiprimes: 85 = 5 * 17 393 = 3 * 131 5071 = 11 * 461 14119 = 7 * 2017 65523 = 3 * 21841 182449 = 43 * 4243 5087942581 = 11113 * 457837 849655943137 = 17 * 49979761361 3943144558081 = 31 * 127198211551 - Jonathan Vos Post, Feb 04 2005

Seiichi Manyama, Table of n, a(n) for n = 1..1803
P. Heideman and E. Hogan, A new family of Somos-like recurrences.
P. Heideman and E. Hogan, A new family of Somos-like recurrences, El. J. Combin. 15 (2008) #R54. [From _R. J. Mathar_, Dec 04 2008]
Index entries for linear recurrences with constant coefficients, signature (0, 14, 0, -14, 0, 1).

A048736 [From Jaume Oliver Lafont, Sep 25 2009]

Cf. A092264, A133846, A133847, A133848, A133854.

LinearRecurrence[{0, 14, 0, -14, 0, 1},{1, 1, 1, 3, 7, 31},26] (* Ray Chandler, Jul 24 2016 *)
nxt[{a_,b_,c_}]:={b,c,(c*b+c+b)/a}; NestList[nxt,{1,1,1},30][[All,1]] (* Harvey P. Dale, Mar 11 2019 *)

a(k=3, 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, Oct 28 2012

Both sequences u=(a(2n-1)){n>0} and u=(a(2n)){n>0} satisfy the order 3 linear recursion : u(n)=14u(n-1)-14u(n-2)+u(n-3).
a(2*n-1) = ceiling((1/11)*sqrt(1002/5-78*sqrt(33/5))*(sqrt(15)/2+sqrt(11)/ 2)^(2*n-1)).
a(2*n) = ceiling((1/11)*(13-sqrt(165))*(sqrt(15)/2+sqrt(11)/2)^(2*n)).
G.f.: x*(1+x-13*x^2-11*x^3+7*x^4+3*x^5)/(1-14*x^2+14*x^4-x^6). - Jaume Oliver Lafont, Sep 25 2009
a(n) = (4-(-1)^n)*a(n-1)-a(n-2)-1. - Bruno Langlois, Aug 21 2016
Sequences u=(a(2n)) and v=(a(2n-1)) satisfy order 2 linear recursions : u(n)=13*u(n-1)-u(n-2)-5 and v(n)=13*v(n-1)-v(n-2)-7. - Bruno Langlois, Aug 21 2016

A092264 a(n)a(n-5) = a(n-1)a(n-4)+a(n-2)+a(n-3), with initial terms a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 9, 17, 65, 117, 227, 449, 1737, 3137, 6105, 12097, 46819, 84565, 164593, 326161, 1262361, 2280101, 4437891, 8794241, 34036913, 61478145, 119658449, 237118337, 917734275, 1657629797, 3226340217, 6393400849

Offset: 1

Paul Heideman (ppheideman(AT)wisc.edu), Feb 19 2004

Seiichi Manyama, Table of n, a(n) for n = 1..2801
P. Heideman and E. Hogan, A new family of Somos-like recurrences, The Electronic Journal of Combinatorics, Volume 15 (2008), #R54.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 28, 0, 0, 0, -28, 0, 0, 0, 1).

Cf. A072881, A133846, A133847, A133848, A133854.

R := proc(n) option remember; if n<5 then 1 else RETURN((R(n-1)*R(n-4)+R(n-2)+R(n-3))/R(n-5)); fi; end;

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==1,a[n]==(a[n-1]a[n-4]+a[n-2]+a[n-3])/a[n-5]},a,{n,40}] (* or *) LinearRecurrence[ {0,0,0,28,0,0,0,-28,0,0,0,1},{1,1,1,1,1,3,5,9,17,65,117,227},40] (* Harvey P. Dale, Aug 08 2013 *)

a(k=5, 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

Vec(x*(1 +x +x^2 +x^3 -27*x^4 -25*x^5 -23*x^6 -19*x^7 +17*x^8 +9*x^9 +5*x^10 +3*x^11) / ((1 -x)*(1 +x)*(1 +x^2)*(1 +5*x^2 -x^4)*(1 -5*x^2 -x^4)) + O(x^50)) \\ Colin Barker, Jul 18 2016

a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=3, a(7)=5, a(8)=9, a(9)=17, a(10)=65, a(11)=117, a(12)=227, a(n)=28*a(n-4)-28*a(n-8)+a(n-12). - Harvey P. Dale, Aug 08 2013

G.f.: x*(1 +x +x^2 +x^3 -27*x^4 -25*x^5 -23*x^6 -19*x^7 +17*x^8 +9*x^9 +5*x^10 +3*x^11) / ((1 -x)*(1 +x)*(1 +x^2)*(1 +5*x^2 -x^4)*(1 -5*x^2 -x^4)). - Colin Barker, Jul 18 2016

A133846 a(n)*a(n-7) = a(n-1)a(n-6)+a(n-3)+a(n-4) with initial terms a(1)=...=a(7)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 11, 19, 31, 111, 195, 283, 465, 831, 1381, 4969, 8741, 12697, 20885, 37353, 62101, 223471, 393121, 571051, 939331, 1680031, 2793151, 10051203, 17681675, 25684567, 42248981, 75564019, 125629681, 452080641, 795282225

Offset: 1

Emilie Hogan, Sep 26 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000
P. Heideman and E. Hogan, A New Family of Somos-Like Recurrences, arXiv:0709.2529 [math.CO], 2007-2009.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,46,0,0,0,0,0,-46,0,0,0,0,0,1).

Cf. A072881, A092264, A133847, A133848, A133854.

a := proc(n) option remember; if n<=7 then RETURN(1); else RETURN((a(n-1)*a(n-6)+a(n-3)+a(n-4))/a(n-7)); fi; end;

nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,(g*b+e+d)/a}; Transpose[ NestList[ nxt,{1,1,1,1,1,1,1},40]][[1]] (* or *) LinearRecurrence[ {0,0,0,0,0,46,0,0,0,0,0,-46,0,0,0,0,0,1},{1,1,1,1,1,1,1,3,5,7,11,19,31,111,195,283,465,831},40] (* Harvey P. Dale, Aug 21 2014 *)

a(k=7, 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

Vec(x*(1 +x +x^2 +x^3 +x^4 +x^5 -45*x^6 -43*x^7 -41*x^8 -39*x^9 -35*x^10 -27*x^11 +31*x^12 +19*x^13 +11*x^14 +7*x^15 +5*x^16 +3*x^17) / ((1 -x)*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -45*x^6 +x^12)) + O(x^50)) \\ Colin Barker, Jul 18 2016

G.f.: x*(1 +x +x^2 +x^3 +x^4 +x^5 -45*x^6 -43*x^7 -41*x^8 -39*x^9 -35*x^10 -27*x^11 +31*x^12 +19*x^13 +11*x^14 +7*x^15 +5*x^16 +3*x^17) / ((1 -x)*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -45*x^6 +x^12)). - Colin Barker, Jul 18 2016

A133847 a(n)a(n-9) = a(n-1)a(n-8)+a(n-4)+a(n-5) with initial terms a(1)=...=a(9)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 9, 13, 21, 33, 49, 169, 293, 421, 553, 823, 1365, 2179, 3265, 11289, 19585, 28153, 36993, 55081, 91393, 145929, 218689, 756163, 1311861, 1885783, 2477929, 3689557, 6121925, 9775033, 14648881, 50651601, 87875061

Offset: 1

Emilie Hogan, Sep 26 2007

Seiichi Manyama, Table of n, a(n) for n = 1..4390
P. Heideman and E. Hogan, A New Family of Somos-Like Recurrences, arXiv:0709.2529 [math.CO], 2007-2009.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,68,0,0,0,0,0,0,0,-68,0,0,0,0,0,0,0,1).

Cf. A072881, A092264, A133846, A133848, A133854.

a := proc(n) option remember; if n<=9 then RETURN(1); else RETURN((a(n-1)*a(n-8)+a(n-4)+a(n-5))/a(n-9)); fi; end;

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]==a[8]==a[9]==1,a[n]==(a[n-1]a[n-8]+a[n-4]+a[n-5])/a[n-9]},a,{n,50}] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,68,0,0,0,0,0,0,0,-68,0,0,0,0,0,0,0,1},{1,1,1,1,1,1,1,1,1,3,5,7,9,13,21,33,49,169,293,421,553,823,1365,2179},50] (* Harvey P. Dale, Jan 14 2016 *)

a(k=9, 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

Vec(x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 -67*x^8 -65*x^9 -63*x^10 -61*x^11 -59*x^12 -55*x^13 -47*x^14 -35*x^15 +49*x^16 +33*x^17 +21*x^18 +13*x^19 +9*x^20 +7*x^21 +5*x^22 +3*x^23) / ((1 -x)*(1 +x)*(1 +x^2)*(1 +x^4)*(1 -67*x^8 +x^16)) + O(x^50)) \\ Colin Barker, Jul 18 2016

Sequence also generated by the linear recurrence 68*(u(n-8)-u(n-16))+u(n-24) with the initial 24 terms given by the quadratic recurrence.

G.f.: x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 -67*x^8 -65*x^9 -63*x^10 -61*x^11 -59*x^12 -55*x^13 -47*x^14 -35*x^15 +49*x^16 +33*x^17 +21*x^18 +13*x^19 +9*x^20 +7*x^21 +5*x^22 +3*x^23) / ((1 -x)*(1 +x)*(1 +x^2)*(1 +x^4)*(1 -67*x^8 +x^16)). - Colin Barker, Jul 18 2016

A133854 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.

Original entry on oeis.org

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, 781, 1017, 1257, 1501, 1995, 2985, 4471, 6453, 8931, 11905, 39433, 67457, 95977, 124993, 154505, 184513, 245273, 367041, 549817, 793601, 1098393, 1464193, 4849891

Offset: 1

Emilie Hogan, Sep 26 2007

Seiichi Manyama, Table of n, a(n) for n = 1..5754
P. Heideman and E. Hogan, A New Family of Somos-Like Recurrences, arXiv:0709.2529 [math.CO], 2007-2009.
Index entries for linear recurrences with constant coefficients, 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).

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

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;

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 *)

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

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

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.

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

A243709 Coxeter numbers of the D-type Niemeier root systems.

Original entry on oeis.org

6, 10, 14, 18, 22, 30, 46

Offset: 1

N. J. A. Sloane, Jun 19 2014

These are exactly the even integers 2n such that the group Gamma_0(2n)+n has genus zero.

Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A. Harvey, Umbral moonshine and the Niemeier lattices, Research in the Mathematical Sciences, 2014, 1:3. See Eq. (23).

Cf. A091401.

a(n) = 2*A133848(n+11). - Omar E. Pol, Dec 20 2014

cp's OEIS Frontend

A072881 a(1)=a(2)=a(3)=1; for n>3, a(n)=(a(n-1)*a(n-2)+a(n-1)+a(n-2))/a(n-3).

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A092264 a(n)*a(n-5) = a(n-1)*a(n-4)+a(n-2)+a(n-3), with initial terms a(1) = ... = a(5) = 1.

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A133846 a(n)*a(n-7) = a(n-1)a(n-6)+a(n-3)+a(n-4) with initial terms a(1)=...=a(7)=1.

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A133847 a(n)*a(n-9) = a(n-1)*a(n-8)+a(n-4)+a(n-5) with initial terms a(1)=...=a(9)=1.

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A133854 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.

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A243709 Coxeter numbers of the D-type Niemeier root systems.

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A092264 a(n)a(n-5) = a(n-1)a(n-4)+a(n-2)+a(n-3), with initial terms a(1) = ... = a(5) = 1.

A133847 a(n)a(n-9) = a(n-1)a(n-8)+a(n-4)+a(n-5) with initial terms a(1)=...=a(9)=1.

A133854 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.