A104622 -id:A104622 - cp's OEIS frontend

A104621 Heptanacci-Lucas numbers.

7, 1, 3, 7, 15, 31, 63, 127, 247, 493, 983, 1959, 3903, 7775, 15487, 30847, 61447, 122401, 243819, 485679, 967455, 1927135, 3838783, 7646719, 15231991, 30341581, 60439343, 120393007, 239818559, 477709983, 951581183, 1895515647, 3775799303, 7521257025

Offset: 0

Jonathan Vos Post, Mar 17 2005

This 7th-order linear recurrence is a generalization of the Lucas sequence A000032. Mario Catalani would refer to this is a generalized heptanacci sequence, had he not stopped his series of sequences after A001644 "generalized tribonacci", A073817 "generalized tetranacci", A074048 "generalized pentanacci", A074584 "generalized hexanacci." T. D. Noe and I have noted that each of these has many more primes than the corresponding tribonacci A000073 (see A104576), tetranacci A000288 (see A104577), pentanacci, hexanacci and heptanacci (see A104414). For primes in Heptanacci-Lucas numbers, see A104622. For semiprimes in Heptanacci-Lucas numbers, see A104623.

G. C. Greubel, Table of n, a(n) for n = 0..3300 (terms 0..200 from T. D. Noe)
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
Spiros D. Dafnis, Andreas N. Philippou, Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Cf. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (-7+6*x+ 5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7) )); // G. C. Greubel, Apr 22 2019

A104621 := proc(n)
    option remember;
    if n <=6 then
        op(n+1,[7, 1, 3, 7, 15, 31, 63])
    else
        add(procname(n-i),i=1..7) ;
    end if;
end proc: # R. J. Mathar, Mar 26 2015

a[0]=7; a[1]=1; a[2]=3; a[3]=7; a[4]=15; a[5]=31; a[6]=63; a[n_]:= a[n]= a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]+a[n-7]; Table[a[n], {n,0,40}] (* Robert G. Wilson v, Mar 17 2005 *)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {7, 1, 3, 7, 15, 31, 63}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

my(x='x+O('x^40)); Vec((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7)) \\ G. C. Greubel, Dec 18 2017

polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6-x^7), 40) \\ G. C. Greubel, Apr 22 2019

((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6 +x^7)).series(x, 41).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7); a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63.
From R. J. Mathar, Nov 16 2007: (Start)
G.f.: (7 - 6*x - 5*x^2 - 4*x^3 - 3*x^4 - 2*x^5 - x^6)/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).
a(n) = 7*A066178(n) - 6*A066178(n-1) - 5*A066178(n-2) - ... - 2*A066178(n-5) - A066178(n-6) if n >= 6. (End)

A104623 Indices of semiprime (A001358) values of Heptanacci-Lucas numbers A104621.

Original entry on oeis.org

4, 8, 9, 11, 12, 14, 15, 16, 22, 23, 32, 34, 37, 41, 42, 50, 52, 57, 58, 66, 69, 76, 77, 81, 90, 120, 139

Offset: 0

Jonathan Vos Post, Mar 17 2005

nonn

The 7th-order linear recurrence A104622 (heptanacci-Lucas numbers) is a generalization of the Lucas sequence A000032. T. D. Noe and I have noted that the heptanacci-Lucas numbers have many more primes than the corresponding heptanacci (see A104414) which he found has only the first 3 primes that I identified through the first 5000 values, whereas these heptanacci-Lucas numbers have 17 primes among the first 100 values. For primes in Heptanacci-Lucas numbers, see A104622.

			A104621(4) = 15 = 3 * 5,
A104621(8) = 247 = 13 * 19,
A104621(9) = 493 = 17 * 29,
A104621(11) = 1959 = 3 * 653,
A104621(12) = 3903 = 3 * 1301,
A104621(14) = 15487 = 17 * 911,

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math.CO/0210201 v1, 2002
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358.

A105768 Prime Lucas 7-step numbers, A104621.

Original entry on oeis.org

3, 7, 31, 127, 983, 122401, 15231991, 30341581, 60439343, 239818559, 235883775871, 935968272887, 462162688688737, 3592979567873032703, 439785318101603999198591, 432569613524779275706080077

Offset: 1

T. D. Noe, Apr 22 2005

nonn

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Eric Weisstein's World of Mathematics, Lucas n-Step Number

Cf. A104622 (indices of prime Lucas 7-step numbers).

a={-1, -1, -1, -1, -1, -1, 7}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, s]], {n, 1000}]; lst

A105935 Indices of Lucas 7-step numbers A104621 which have a nontrivial divisor in common with index.

Original entry on oeis.org

6, 12, 18, 20, 21, 33, 36, 42, 54, 57, 60, 65, 87, 93, 99, 100, 104, 105, 108, 111, 141, 147, 152, 153, 155, 156, 162, 165, 171, 177, 180, 189, 192, 195, 210, 215, 220, 222, 230, 235, 238, 240, 249, 255, 261, 264, 273, 276, 279, 280, 286, 294, 295, 297, 300

Offset: 1

Jonathan Vos Post, Apr 26 2005

Wanted: closed-form formula for this as exists for Fibonacci and Lucas numbers. Lucas 7-step numbers also known as heptanacci-Lucas numbers. The prime Lucas 7-step numbers are A105768, their indices being A104622.

			gcd(a(n), A104621(6,12,18,33,36,57,60,87,93,108)) = 3,
gcd(a(n), A104621(20,65,100,105)) = 5,
gcd(a(n), A104621(21,42)) = 7,
gcd(a(n), A104621(54,99)) = 9,
gcd(a(n), A104621(104)) = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A104621, A105285, A105289.

m=300; s = LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {7, 1, 3, 7, 15, 31, 63}, m+1]; Select[Range[m], !CoprimeQ[#, s[[#+1]]] &] (* Amiram Eldar, Sep 05 2019 *)

gcd(a(n), A104621(a(n))) > 1.

Offset corrected and a(5)-a(6) and more terms added by Amiram Eldar, Sep 05 2019

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A104621 Heptanacci-Lucas numbers.

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A104623 Indices of semiprime (A001358) values of Heptanacci-Lucas numbers A104621.

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A105768 Prime Lucas 7-step numbers, A104621.

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A105935 Indices of Lucas 7-step numbers A104621 which have a nontrivial divisor in common with index.

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