A000078 -id:A000078 - cp's OEIS frontend

A086088 Decimal expansion of the limit of the ratio of consecutive terms in the tetranacci sequence A000078.

1, 9, 2, 7, 5, 6, 1, 9, 7, 5, 4, 8, 2, 9, 2, 5, 3, 0, 4, 2, 6, 1, 9, 0, 5, 8, 6, 1, 7, 3, 6, 6, 2, 2, 1, 6, 8, 6, 9, 8, 5, 5, 4, 2, 5, 5, 1, 6, 3, 3, 8, 4, 7, 2, 7, 1, 4, 6, 6, 4, 7, 0, 3, 8, 0, 0, 9, 6, 6, 6, 0, 6, 2, 2, 9, 7, 8, 1, 5, 5, 5, 9, 1, 4, 9, 8, 1, 8, 2, 5, 3, 4, 6, 1, 8, 9, 0, 6, 5, 3, 2, 5

Offset: 1

Eric W. Weisstein, Jul 08 2003

The tetranacci constant corresponds to the Golden Section in a quadripartite division 1 = u_1 + u_2 + u_3 + u_4 of a unit line segment, i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = u_3/u_4 = c, c is the tetranacci constant. - Seppo Mustonen, Apr 19 2005
The other 3 polynomial roots of 1+x+x^2+x^3-x^4 are -0.77480411321543385... and the complex-conjugated pair -0.07637893113374572508475 +- i * 0.814703647170386526841... - R. J. Mathar, Oct 25 2008
The continued fraction expansion starts 1, 1, 12, 1, 4, 7, 1, 21, 1, 2, 1, 4, 6, 1, 10, 1, 2, 2, 1, 7, 1, 1,... - R. J. Mathar, Mar 09 2012
For n>=4, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - Vladimir Shevelev, Mar 21 2014
Note that we have: c + c^(-4) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 09 2022

			1.927561975...

Martin Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.

Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.
Gültekin, İnci; Deveci, Ömür, On the arrowhead-Fibonacci numbers. Open Math. 14, 1104-1113 (2016).
S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Tetranacci Number
Eric Weisstein's World of Mathematics, Disk Covering Problem
Eric Weisstein's World of Mathematics, Tetranacci Constant
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Index entries for algebraic numbers, degree 4

Cf. A000078.

k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), this sequence (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).

RealDigits[Root[ -1-#1-#1^2-#1^3+#1^4&, 2], 10, 110][[1]]

real(polroots(1+x+x^2+x^3-x^4)[2]) \\ Charles R Greathouse IV, Jul 19 2012

polrootsreal(1+x+x^2+x^3-x^4)[2] \\ Charles R Greathouse IV, Apr 14 2014

Equals 1/4 + sqrt(11/48 - s/72 + 7/s) + sqrt(11/24 + s/72 - 7/s + 1 / sqrt(704/507 - 128 * s/1521 + 7168 / (169 * s))) where s = (sqrt(177304464) + 7020)^(1/3). - Michal Paulovic, Oct 08 2022

A104411 Number of prime factors, with multiplicity, of the tetranacci numbers A000078.

Original entry on oeis.org

0, 0, 1, 2, 3, 2, 1, 4, 5, 5, 1, 1, 3, 4, 6, 2, 2, 8, 5, 9, 2, 2, 4, 5, 6, 4, 2, 7, 5, 8, 2, 4, 3, 6, 12, 1, 3, 9, 7, 8, 2, 3, 4, 7, 4, 6, 4, 7, 3, 8, 6, 6, 6, 6, 7, 1, 3, 11, 5, 8, 5, 5, 5, 4, 7, 2, 3, 9, 5, 9, 4, 6, 6, 8, 12, 7, 4, 25, 8, 10, 4, 4, 4, 7, 6, 4, 11, 5, 6, 7, 3, 4, 3, 8, 8, 5, 6, 13, 8, 7, 5, 5, 5

Offset: 3

Jonathan Vos Post, Mar 05 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 3..244
Marcellus E. Waddill, The Tetranacci Sequence and Generalizations, The Fibonacci Quarterly, 30.1 (1992) 9.
Marcellus E. Waddill, Some Properties of the Tetranacci Sequence Modulo m, The Fibonacci Quarterly, 30.3 (1992) 232.
Eric Weisstein's World of Mathematics, Tetranacci Number.

Cf. A000078, A001222.

PrimeOmega[LinearRecurrence[{1, 1, 1, 1}, {1, 1, 2, 4}, 100]] (* Amiram Eldar, May 16 2021 *)

a(n) = A001222(A000078(n)). a(n) = bigomega(A000078(n)).

More terms from R. J. Mathar, Dec 14 2009

Offset changed to 3 by Joerg Arndt, Dec 19 2020

A287656 Number of partitions of n into distinct tetranacci numbers (with a single type of 1) (A000078).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Ilya Gutkovskiy, Jun 05 2017

nonn

			a(15) = 2 because we have [15] and [8, 4, 2, 1].

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Tetranacci Number
Index entries for related partition-counting sequences

Cf. A000078, A000119, A000121, A003263, A117546, A288120.

G.f.: Product_{k>=4} (1 + x^A000078(k)).

A359849 a(n) is the smallest tetranacci number (A000078) with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 15, 1490, 39648, 28074040, 100808458960497, 9966792788887776, 4997150614173857218560, 1835682610171974487231869, 889487735339682550112673527109223032, 52499930084496170026238596234557616056408988199026780675759699719704592

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

nonn

			a(4) = 39648, because 39648 is a tetranacci number with 4 distinct prime factors {2, 3, 7, 59} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tetranacci Number

Cf. A000078, A001221, A060319, A359848, A359851.

a(n) = A000078(A359851(n)). - Daniel Suteu, Jan 18 2023

a(11) from Daniel Suteu, Jan 18 2023

A359851 a(n) is the index of the smallest tetranacci number (A000078) with exactly n distinct prime factors.

Original entry on oeis.org

3, 5, 8, 15, 20, 30, 53, 60, 80, 89, 130, 252

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tetranacci Number

Cf. A000078, A001221, A060320, A359849, A359850.

a(11) from Daniel Suteu, Jan 17 2023

A357453 Number of compositions (ordered partitions) of n into tetranacci numbers 1,2,4,8,15,29, ... (A000078).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 31, 56, 98, 174, 306, 542, 956, 1690, 2984, 5273, 9313, 16453, 29062, 51340, 90689, 160203, 282994, 499908, 883078, 1559948, 2755624, 4867776, 8598858, 15189770, 26832521, 47399291, 83730207, 147908288, 261277998, 461544073

Offset: 0

Ilya Gutkovskiy, Sep 29 2022

nonn

Cf. A000078, A076739, A287656, A357451, A357452, A357455.

A000078[0] = A000078[1] = A000078[2] = 0; A000078[3] = 1; A000078[n_] := A000078[n] = A000078[n - 1] + A000078[n - 2] + A000078[n - 3] + A000078[n - 4]; nmax = 36; CoefficientList[Series[1/(1 - Sum[x^A000078[k], {k, 4, 20}]), {x, 0, nmax}], x]

G.f.: 1 / (1 - Sum_{k>=4} x^A000078(k)).

A359877 a(n) is the smallest tetranacci number (A000078) with exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 4, 8, 56, 108, 5536, 28074040, 39648, 147312, 18566888967365603514688, 9966792788887776, 2775641472, 2505471397838180985096739296, 1445523368993397560000765219760086502994234237205516083525719052320, 44092571484448511101335177770183225655413760

Offset: 0

Ilya Gutkovskiy, Jan 16 2023

nonn

			a(6) = 5536, because 5536 is a tetranacci number with 6 prime factors (counted with multiplicity) {2, 2, 2, 2, 2, 173} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Tetranacci Number

Cf. A000078, A001222, A072397, A359849, A359876, A359879.

A359879 a(n) is the index of the smallest tetranacci number (A000078) with exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 17, 30, 20, 22, 82, 60, 37, 100, 236, 157, 156, 242, 240

Offset: 0

Ilya Gutkovskiy, Jan 16 2023

a(19) > 400. - Daniel Suteu, Jan 18 2023

Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Tetranacci Number

Cf. A000078, A001222, A072396, A359851, A359877, A359878.

a(18) from Daniel Suteu, Jan 18 2023

A303264 Indices of primes in tetranacci sequence A000078.

Original entry on oeis.org

5, 9, 13, 14, 38, 58, 403, 2709, 8419, 14098, 31563, 50698, 53194, 155184

Offset: 1

M. F. Hasler, Apr 18 2018

T = A000078 is defined by T(n) = Sum_{k=1..4} T(n-k), T(3) = 1, T(n) = 0 for n < 3.

The largest terms correspond to unproven probable primes T(a(n)).

Cf. A000045, A000073, A000078, A001591, A001592, A122189 (or A066178), ... (Fibonacci, tribonacci, tetranacci numbers).
Cf. A005478, A092836, A104535, A105757, A105759, A105761, ... (primes in Fibonacci numbers and above generalizations).
Cf. A001605, A303263, A303264, A248757, A249635, ... (indices of primes in A000045, A000073, A000078, ...).
Cf. A247027: Indices of primes in the tetranacci sequence A001631 (starting 0, 0, 1, 0...), A104534 (a variant: a(n) - 2), A105756 (= A248757 - 3), A105758 (= A249635 - 4).

a(n,N=5,S=vector(N,i,i>N-2))={for(i=N,oo,ispseudoprime(S[i%N+1]=2*S[(i-1)%N+1]-S[i%N+1])&&!n--&&return(i))}

a(n) = A104534(n) + 2.

A357452 Number of partitions of n into tetranacci numbers 1,2,4,8,15,29, ... (A000078).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, 26, 27, 36, 37, 46, 48, 60, 62, 74, 78, 94, 98, 114, 120, 140, 147, 168, 178, 204, 215, 242, 256, 288, 304, 338, 358, 398, 420, 462, 488, 537, 567, 619, 654, 714, 753, 816, 860, 932, 982, 1058, 1114

Offset: 0

Ilya Gutkovskiy, Sep 29 2022

nonn

Cf. A000078, A003107, A240844, A287656, A357453, A357454.

A000078[0] = A000078[1] = A000078[2] = 0; A000078[3] = 1; A000078[n_] := A000078[n] = A000078[n - 1] + A000078[n - 2] + A000078[n - 3] + A000078[n - 4]; nmax = 55; CoefficientList[Series[Product[1/(1 - x^A000078[k]), {k, 4, 20}], {x, 0, nmax}], x]

G.f.: Product_{k>=4} 1 / (1 - x^A000078(k)).

cp's OEIS Frontend

A086088 Decimal expansion of the limit of the ratio of consecutive terms in the tetranacci sequence A000078.

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A104411 Number of prime factors, with multiplicity, of the tetranacci numbers A000078.

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A287656 Number of partitions of n into distinct tetranacci numbers (with a single type of 1) (A000078).

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A359849 a(n) is the smallest tetranacci number (A000078) with exactly n distinct prime factors.

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A359851 a(n) is the index of the smallest tetranacci number (A000078) with exactly n distinct prime factors.

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A357453 Number of compositions (ordered partitions) of n into tetranacci numbers 1,2,4,8,15,29, ... (A000078).

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A359877 a(n) is the smallest tetranacci number (A000078) with exactly n prime factors (counted with multiplicity).

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A359879 a(n) is the index of the smallest tetranacci number (A000078) with exactly n prime factors (counted with multiplicity).

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A303264 Indices of primes in tetranacci sequence A000078.

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