A073817 -id:A073817 - cp's OEIS frontend

A001648 Tetranacci numbers A073817 without the leading term 4.

1, 3, 7, 15, 26, 51, 99, 191, 367, 708, 1365, 2631, 5071, 9775, 18842, 36319, 70007, 134943, 260111, 501380, 966441, 1862875, 3590807, 6921503, 13341626, 25716811, 49570747, 95550687, 184179871, 355018116, 684319421, 1319068095, 2542585503, 4900991135

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 1..3501 (terms 1..200 from T. D. Noe)
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Shingo Saito, Tatsushi Tanaka, and Noriko Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq. 14 (2011) # 11.2.4, Table 3.
Eric Weisstein's World of Mathematics, Lucas n-Step Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Cf. A000288, A073817.

I:=[1,3,7,15]; [n le 4 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 18 2017

A001648:=-(1+2*z+3*z**2+4*z**3)/(-1+z+z**2+z**3+z**4); # conjectured by Simon Plouffe in his 1992 dissertation

Rest@ CoefficientList[ Series[(4 - 3 x - 2 x^2 - x^3)/(1 - x - x^2 - x^3 - x^4), {x, 0, 40}], x] (* Or *)
a[0] = 4; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[n_] := 2*a[n - 1] - a[n - 5]; Array[a, 33] (* Robert G. Wilson v *)
LinearRecurrence[{1, 1, 1, 1}, {1, 3, 7, 15}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

a(n):=n*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1),i,0,((n-k)/4))/k,k,ceiling(n/5),n); /* Vladimir Kruchinin, Jan 20 2012 */

a(n)=if(n<0,0,polcoeff(x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4)+x*O(x^n),n))

G.f.: x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4).
a(n) = trace of M^n, where M = the 4 X 4 matrix [ 0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 1 1 1]. E.g., the trace (sum of diagonal terms) of M^12 = a(12) = 2631 = (108 + 316 + 717 + 1490). - Gary W. Adamson, Feb 22 2004
a(n) = n*Sum_{k=ceiling(n/5)..n} Sum_{i=0..(n-k)/4} (-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1)/k, n>0. - Vladimir Kruchinin, Jan 20 2012

A074058 Reflected tetranacci numbers A073817.

Original entry on oeis.org

4, -1, -1, -1, 7, -6, -1, -1, 15, -19, 4, -1, 31, -53, 27, -6, 63, -137, 107, -39, 132, -337, 351, -185, 303, -806, 1039, -721, 791, -1915, 2884, -2481, 2303, -4621, 7683, -7846, 7087, -11545, 19987, -23375, 22020, -30177, 51519, -66737, 67415, -82374, 133215, -184993, 201567, -232163, 348804

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 16 2002

Also a(n) is the trace of A^(-n), where A is the 4 X 4 matrix ((1,1,0,0), (1,0,1,0), (1,0,0,1), (1,0,0,0)).

R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete Mathematics", Addison-Wesley, Reading, MA, 1998.

Indranil Ghosh, Table of n, a(n) for n = 0..8998
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,1).

Cf. A073817, A073937.

CoefficientList[Series[(4+3*x+2*x^2+x^3)/(1+x+x^2+x^3-x^4), {x, 0, 1}], x]

polsym(polrecip(1+x+x^2+x^3-x^4), 55) \\ Joerg Arndt, Jan 21 2023

a(n) = -a(n-1)-a(n-2)-a(n-3)+a(n-4), a(0)=4, a(1)=-1, a(2)=-1, a(3)=-1.
G.f.: (4+3x+2x^2+x^3)/(1+x+x^2+x^3-x^4).
From Peter Bala, Jan 19 2023: (Start)
a(n) = (-1)^n*A073937(n).
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. (End)

A106295 Period of the Lucas 4-step sequence A073817 mod n.

Original entry on oeis.org

1, 5, 26, 10, 312, 130, 342, 20, 78, 1560, 120, 130, 84, 1710, 312, 40, 4912, 390, 6858, 1560, 4446, 120, 12166, 260, 1560, 420, 234, 1710, 280, 1560, 61568, 80, 1560, 24560, 17784, 390, 1368, 34290, 1092, 1560, 240, 22230, 162800, 120, 312, 60830, 103822

Offset: 1

T. D. Noe, May 02 2005

nonn

This sequence is the same as the period of Fibonacci 4-step sequence (A000078) mod n for n<563 because the discriminant of the characteristic polynomial x^4-x^3-x^2-x-1 is -563. The two sequences differ only at n that are multiples of 563.

Chai Wah Wu, Table of n, a(n) for n = 1..1486
Marcellus E. Waddill, Some Properties of the Tetranacci Sequence Modulo m, The Fibonacci Quarterly, vol. 30, no. 3, 232-238 (1992).
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A000078, A073817, A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1).

n=4; Table[p=i; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 60}]

from itertools import count
def A106295(n):
    a = b = (4%n,1%n,3%n,7%n)
    s = sum(b) % n
    for m in count(1):
        b, s = b[1:] + (s,), (s+s-b[0]) % n
        if a == b:
            return m # Chai Wah Wu, Feb 22-27 2022

Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)).

a(2^k) = 5*2^(k-1) for k > 0. If a(p) != a(p^2) for p prime, then a(p^k) = p^(k-1)*a(p) for k > 0 [Waddill, 1992]. - Chai Wah Wu, Feb 25 2022

A075116 Binomial transform of A073817: a(n)=Sum(Binomial(n,k)*A073817(k),(k=0,..,n)).

Original entry on oeis.org

4, 5, 9, 23, 69, 210, 627, 1846, 5405, 15809, 46254, 135382, 396327, 1160294, 3396892, 9944688, 29113741, 85232259, 249522603, 730492701, 2138562494, 6260774221, 18328804756, 53658712275, 157089206159, 459888386910

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 02 2002

N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-1).

Cf. A073817.

CoefficientList[Series[(4-15*z+16*z^2-6*z^3)/(1-5*z+8*z^2-6*z^3+z^4), {z, 0, 30}], z]
LinearRecurrence[{5,-8,6,-1},{4,5,9,23},30] (* Harvey P. Dale, Jun 24 2017 *)

a(n)=5a(n-1)-8a(n-2)+6a(n-3)-a(n-4), a(0)=4, a(1)=5, a(2)=9, a(3)=23. G.f.: (4-15*z+16*z^2-6*z^3)/(1-5*z+8*z^2-6*z^3+z^4).

A104577 Indices of prime generalized tetranacci numbers, A073817.

Original entry on oeis.org

2, 3, 8, 9, 16, 19, 24, 27, 46, 68, 71, 78, 107, 198, 309, 377, 477, 1057, 1631, 2419, 3974, 4293, 8247, 10513, 10709, 12011, 15042, 30543, 31607, 39664, 47552, 145858

Offset: 1

T. D. Noe, Mar 16 2005

nonn

The sequence of generalized tetranacci numbers is defined as beginning with 1, 3, 7, 15. Subsequent terms are the sum of the previous four terms. Note that the sequence of these generalized tetranacci numbers has many more primes than the tetranacci sequence A000078 (whose prime indices are in A104534).

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

Cf. A104576 (indices of prime generalized tribonacci numbers).

a={-1, -1, -1, 4}; Do[s=Plus@@a; a=RotateLeft[a]; a[[4]]=s; If[PrimeQ[s], Print[n]], {n, 30000}]

A106296 Period of the Lucas 4-step sequence A073817 mod prime(n).

Original entry on oeis.org

5, 26, 312, 342, 120, 84, 4912, 6858, 12166, 280, 61568, 1368, 240, 162800, 103822, 303480, 205378, 226980, 100254, 357910, 2664, 998720, 1157520, 9320, 368872, 1030300, 10608, 1225042, 2614040, 13874, 2048382, 4530768, 136, 772880, 3307948

Offset: 1

T. D. Noe, May 02 2005

nonn

This sequence is the same as the period of Fibonacci 4-step sequence (A000078) mod prime(n) except for n=103, which corresponds to the prime 563 because the discriminant of the characteristic polynomial x^4-x^3-x^2-x-1 is -563. We have a(n) < prime(n) for primes 563 and A106280.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106280 (primes p such that x^4-x^3-x^2-x-1 mod p has 4 distinct zeros), A106295.

n=4; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 60}]

a(n) = A106295(prime(n)).

A075128 Binomial transform of generalized tetranacci numbers A073817: a(n)=Sum((-1)^k Binomial(n,k)*A073817(k),(k=0,..,n)).

Original entry on oeis.org

4, 3, 5, 3, 5, 8, 23, 52, 109, 201, 350, 586, 983, 1680, 2952, 5288, 9549, 17207, 30803, 54761, 96910, 171223, 302736, 536225, 951487, 1690208, 3003408, 5335509, 9473756, 16814058, 29833868, 52932503, 93922925, 166678207, 295825369

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 03 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (3, -2, -2, 3).

Cf. A073817.

CoefficientList[Series[(4-9*z+4*z^2+2*z^3)/(1-3*z+2*z^2+2*z^3-3*z^4), {z, 0, 40}], z]
LinearRecurrence[{3,-2,-2,3},{4,3,5,3},40] (* Harvey P. Dale, Jul 13 2023 *)

a(n)=3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4), a(0)=4, a(1)=3, a(2)=5, a(3)=3. G.f.: (4 - 9*z + 4*z^2 + 2*z^3)/(1 - 3*z + 2*z^2 + 2*z^3 - 3*z^4).

A106300 Primes that do not divide any term of the Lucas 4-step sequence A073817.

Original entry on oeis.org

2789, 3847, 4451, 4751, 5431, 6203, 8317, 9533, 9629, 9907, 10093, 11839, 13903, 13907, 14207, 15823, 16319, 16759, 19543, 20939, 21379, 21859, 25303, 26683, 29483, 30871, 31267, 31699, 32003, 32771, 33967, 34963, 36229, 37061, 39983

Offset: 1

T. D. Noe, May 02 2005

nonn

If a prime p divides a term a(k) of this sequence, then k must be less than the period of the sequence mod p. Hence these primes are found by computing A073817(k) mod p for increasing k and stopping when either A073817(k) mod p = 0 or the end of the period is reached. Interestingly, for all of these primes, the period of the sequence A073817(k) mod p appears to be (p-1)/d, where d is a small integer.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Cf. A053028 (primes not dividing any Lucas number), A106299 (primes not dividing any Lucas 3-step number), A106301 (primes not dividing any Lucas 5-step number).

n=4; lst={}; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; While[s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; !(a==a0 || s==0)]; If[s>0, AppendTo[lst, p]], {i, 10000}]; lst

A106791 Sum of two consecutive squares of Lucas 4-step numbers (A073817).

Original entry on oeis.org

17, 10, 58, 274, 901, 3277, 12402, 46282, 171170, 635953, 2364489, 8785386, 32637202, 121265666, 450571589, 1674090725, 6220049810, 23110593298, 85867345570, 319039636721, 1185390110881, 4404311472106, 16364198176874

Offset: 0

Jonathan Vos Post, May 16 2005

A106729 is sum of two consecutive squares of Lucas numbers (A001254), for which L(n)^2 + L(n+1)^2 = 5*{F(n)^2 + F(n+1)^2} = 5*A001519(n). A106789 is sum of two consecutive squares of Lucas 3-step numbers (A001644). Sum of two consecutive squares of Lucas 4-step numbers can be expressed in terms of tetranacci numbers, but not quite as neatly.

			a(0) = A073817(0)^2 + A073817(1)^2 = 4^2 + 1^2 = 16 + 1 = 17.
a(1) = A073817(1)^2 + A073817(2)^2 = 1^2 + 3^2 = 1 + 9 = 10.
a(2) = A073817(2)^2 + A073817(3)^2 = 3^2 + 7^2 = 9 + 49 = 58.
a(3) = A073817(3)^2 + A073817(4)^2 = 7^2 + 15^2 = 49 + 225 = 274.
a(4) = A073817(4)^2 + A073817(5)^2 = 15^2 + 26^2 = 225 + 676 = 901 = 30^2 + 1.
a(5) = A073817(5)^2 + A073817(6)^2 = 26^2 + 51^2 = 676 + 2601 = 3277.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,6,12,-4,-6,0,-2,0,1).

Cf. A001254, A001519, A001644, A073817, A106729.

a:=[17,10,58,274,901,3277,12402, 46282,171170,635953];; for n in [11..40] do a[n]:=2*a[n-1]+4*a[n-2]+6*a[n-3]+12*a[n-4]-4*a[n-5] -6*a[n-6]-2*a[n-8]+a[n-10]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+17*x^9)/(1-2*x-4*x^2 -6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{2,4,6,12,-4,-6,0,-2,0,1}, {17,10,58,274,901,3277,12402, 46282,171170,635953}, 40] (* G. C. Greubel, Apr 23 2019 *)

my(x='x+O('x^40)); Vec((17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+17*x^9)/(1-2*x-4*x^2-6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10)) \\ G. C. Greubel, Apr 23 2019

((17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+ 17*x^9)/(1-2*x-4*x^2-6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

a(n) = A073817(n)^2 + A073817(n+1)^2.
a(n) = 5*A073817(n)^2 + 4*A073817(n)*A073817(n-4) + A073817(n-4)^2.
G.f.: (17-24*x-30*x^2+16*x^3-143*x^4-21*x^5+46*x^6-32*x^7+2*x^8+17*x^9)/( (1- 3*x-3*x^2+x^3+x^4)*(1+x+2*x^2+2*x^3-2*x^4+x^5-x^6)). - Colin Barker, Dec 17 2012

A075112 a(n)=Sum((-1)^(i+Floor(n/2))S(2i+e),(i=0,..,Floor(n/2))), where S(n) are generalized Tetranacci numbers (A073817) and e=(1/2)(1-(-1)^n).

Original entry on oeis.org

4, 1, -1, 6, 16, 20, 35, 79, 156, 288, 552, 1077, 2079, 3994, 7696, 14848, 28623, 55159, 106320, 204952, 395060, 761489, 1467815, 2829318, 5453688, 10512308, 20263123, 39058439, 75287564, 145121432, 279730552, 539197989, 1039337543, 2003387514, 3861653592, 7443576640, 14347955295

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 01 2002

a(n) is the convolution of S(n) with the sequence (1,0,-1,0,1,0,-1,0,....) A056594.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 2, 2, 1, 1).

Cf. A073817, A056594, A074662, A074677, A074678, A075111.

CoefficientList[Series[(4 - 3*x - 2*x^2 - x^3)/(1 - x - 2*x^3 - 2*x^4 - x^5 - x^6), {x, 0, 40}], x]
LinearRecurrence[{1,0,2,2,1,1},{4,1,-1,6,16,20},40] (* Harvey P. Dale, Mar 09 2013 *)

a(n)=a(n-1)+2a(n-3)+2a(n-4)+a(n-5)+a(n-6), a(0)=4, a(1)=1, a(2)=-1, a(3)=6, a(4)=16, a(5)=20. G.f.: (4 - 3*x - 2*x^2 - x^3)/(1 - x - 2*x^3 - 2*x^4 - x^5 - x^6).

cp's OEIS Frontend

A001648 Tetranacci numbers A073817 without the leading term 4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Formula

A074058 Reflected tetranacci numbers A073817.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A106295 Period of the Lucas 4-step sequence A073817 mod n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A075116 Binomial transform of A073817: a(n)=Sum(Binomial(n,k)*A073817(k),(k=0,..,n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A104577 Indices of prime generalized tetranacci numbers, A073817.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A106296 Period of the Lucas 4-step sequence A073817 mod prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A075128 Binomial transform of generalized tetranacci numbers A073817: a(n)=Sum((-1)^k Binomial(n,k)*A073817(k),(k=0,..,n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A106300 Primes that do not divide any term of the Lucas 4-step sequence A073817.

Views

Author

Keywords

Comments