Emilie Hogan - cp's OEIS frontend

A162672 Lunar product 19*n.

0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150

Offset: 0

Emilie Hogan, Dennis Hou, Kellen Myers and N. J. A. Sloane, Apr 09 2010

Since 19 is the smallest lunar prime, this is a kind of lunar analog of the even numbers.

As the b-file shows, this sequence is not monotonic and contains repetitions.

			19 * 3 = 13, so 13 is a member. 1109 has just two divisors, 9 and 109, so 1109 is not a member.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Cf. A087062, A078645.

For a two-digit number n, the lunar product 19*n is obtained by putting a 1 in front of n.

Entry revised by N. J. A. Sloane, May 28 2011, to correct errors in some of the comments

A133848 a(n)a(n-11) = a(n-1)a(n-10)+a(n-5)+a(n-6) with initial terms a(1)=...=a(11)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 9, 11, 15, 23, 35, 51, 71, 239, 411, 587, 767, 951, 1325, 2075, 3201, 4703, 6581, 22185, 38165, 54521, 71253, 88361, 123141, 192889, 297605, 437289, 611941, 2062927, 3548881, 5069803, 6625693, 8216551, 11450719

Offset: 1

Emilie Hogan, Sep 26 2007

Seiichi Manyama, Table of n, a(n) for n = 1..5091
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 (0,0,0,0,0,0,0,0,0,94,0,0,0,0,0,0,0,0,0,-94,0,0,0,0,0,0,0,0,0,1).

Cf. A072881, A092264, A133846, A133847, A133854.

a := proc(n) option remember; if n<=11 then RETURN(1); else RETURN((a(n-1)*a(n-10)+a(n-5)+a(n-6))/a(n-11)); fi; end;

Rest@ CoefficientList[Series[x (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 - 93 x^10 - 91 x^11 - 89 x^12 - 87 x^13 - 85 x^14 - 83 x^15 - 79 x^16 - 71 x^17 - 59 x^18 - 43 x^19 + 71 x^20 + 51 x^21 + 35 x^22 + 23 x^23 + 15 x^24 + 11 x^25 + 9 x^26 + 7 x^27 + 5 x^28 + 3 x^29)/((1 - x) (1 + x) (1 - x + x^2 - x^3 + x^4) (1 + x + x^2 + x^3 + x^4) (1 - 93 x^10 + x^20)), {x, 0, 47}], x] (* Michael De Vlieger, Jul 18 2016 *)

a(k=11, 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 -93*x^10 -91*x^11 -89*x^12 -87*x^13 -85*x^14 -83*x^15 -79*x^16 -71*x^17 -59*x^18 -43*x^19 +71*x^20 +51*x^21 +35*x^22 +23*x^23 +15*x^24 +11*x^25 +9*x^26 +7*x^27 +5*x^28 +3*x^29) / ((1 -x)*(1 +x)*(1 -x +x^2 -x^3 +x^4)*(1 +x +x^2 +x^3 +x^4)*(1 -93*x^10 +x^20)) + O(x^60)) \\ Colin Barker, Jul 18 2016

Sequence also generated by the linear recurrence 94*(u(n-10)-u(n-20))+u(n-30) with the initial 30 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 -93*x^10 -91*x^11 -89*x^12 -87*x^13 -85*x^14 -83*x^15 -79*x^16 -71*x^17 -59*x^18 -43*x^19 +71*x^20 +51*x^21 +35*x^22 +23*x^23 +15*x^24 +11*x^25 +9*x^26 +7*x^27 +5*x^28 +3*x^29) / ((1 -x)*(1 +x)*(1 -x +x^2 -x^3 +x^4)*(1 +x +x^2 +x^3 +x^4)*(1 -93*x^10 +x^20)). - 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

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

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

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User: Emilie Hogan

A162672 Lunar product 19*n.

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A133848 a(n)*a(n-11) = a(n-1)*a(n-10)+a(n-5)+a(n-6) with initial terms a(1)=...=a(11)=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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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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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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A133848 a(n)a(n-11) = a(n-1)a(n-10)+a(n-5)+a(n-6) with initial terms a(1)=...=a(11)=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.

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.