A350922 -id:A350922 - cp's OEIS frontend

A350916 Positive integers k such that (k+1)^4 has a divisor congruent to -1 modulo k.

1, 2, 3, 5, 9, 11, 14, 17, 29, 35, 41, 43, 59, 65, 69, 125, 134, 139, 174, 194, 339, 386, 449, 461, 681, 901, 937, 1169, 1322, 1325, 1715, 1971, 2211, 3054, 6395, 7989, 8857, 9077, 10849, 11483, 12545, 13082, 20909, 21506, 23861, 35233, 54734, 62210, 66923, 89045, 129494, 143289, 172899, 174725, 203321, 332315, 375129, 390051, 426389, 493697, 561513, 982094

Offset: 1

Max Alekseyev, Jan 21 2022

nonn

For (k+1)^3 similar sequence is finite {1, 2, 3, 5, 9, 11, 14}, while for (k+1)^2 it is just {1, 2, 3, 5}. Starting with power 4 (this sequence), the number of values of k is infinite. One series of values for power 6 is given by A001570.
Formed by the union of 10 linear recurrent sequences satisfying b(n) = q*b(n-1) - b(n-2) - 4: A350919 (q=3), A350920 (q=4), A350921 (q=6), A350922 (q=7), A350923 (q=10), A103974 (q=14), A350924 (q=16), A350925 (q=16), A350926 (q=23), A350917 (q=23). Each of them give identities (b(n)+1)^4 = (b(n)*b(n-1)-1) * (b(n)*b(n+1)-1).
Only terms 1, 2, 5, 9, 11, 14, 29 are shared between two or more sequences, all others come from exactly one sequence.

Max Alekseyev, Table of n, a(n) for n = 1..5000

Union of A350917, A350919, A350920, A350921, A350922, A350923, A103974, A350924, A350925, A350926.

Cf. A001570.

{ for(k=1,10^6, fordiv((k+1)^4,d, if(Mod(d,k)==-1, print1(k,", "); break)) ); }

A350923 a(0) = 2, a(1) = 2, and a(n) = 10*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

2, 2, 14, 134, 1322, 13082, 129494, 1281854, 12689042, 125608562, 1243396574, 12308357174, 121840175162, 1206093394442, 11939093769254, 118184844298094, 1169909349211682, 11580908647818722, 114639177128975534, 1134810862641936614, 11233469449290390602, 111199883630261969402

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Essentially the same as A157085. - R. J. Mathar, Feb 07 2022

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Cf. A031138, A054318, A097784, A253175, A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350920, A350921, A350922, A350924, A350925, A350926.

LinearRecurrence[{11, -11, 1}, {2, 2, 14}, 25] (* Paolo Xausa, May 30 2025 *)

G.f.: 2*(1 - 10*x + 7*x^2)/((1 - x)*(1 - 10*x + x^2)). - Stefano Spezia, Jan 22 2022
From Hugo Pfoertner, Jan 22 2022: (Start)
a(n) = A031138(n) + 1.
a(n) = 3*A054318(n) - 1.
a(n) = 12*A097784(n-2) + 2 for n >= 2. (End)
a(n) = 2 * A253175(n) for n>=1. - Alois P. Heinz, Jan 22 2022

A350917 a(0) = 1, a(1) = 2, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

1, 2, 41, 937, 21506, 493697, 11333521, 260177282, 5972743961, 137112933817, 3147624733826, 72258255944177, 1658792261982241, 38079963769647362, 874180374439907081, 20068068648348215497, 460691398537569049346, 10575834097715739919457, 242783492848924449098161, 5573444501427546589338242, 127946440039984647105681401, 2937194676418219336841333977

Offset: 0

Max Alekseyev, Jan 21 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.
Other properties for all n:
(a(n)+1)*(a(n+2)+1) = (a(n+1)+1)*(a(n+1)+26);
((105*a(n) - 20)^2 - 50^2) / 21 is an integer square.

Sebastian et al., Are there infinitely many positive integer solutions to (3+3k+l)^2=m(kl-k^3-1)?, MathOverflow, 2022.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350919, A350920, A350921, A350922, A350923, A350924, A350925, A350926.

a(n) = 17/42*A090731(n) - 15/2*A097778(n-1) + 4/21.

G.f.: ( -1+22*x-17*x^2 ) / ( (x-1)*(x^2-23*x+1) ). - R. J. Mathar, Feb 07 2022

A350919 a(0) = 9, a(1) = 9, and a(n) = 3*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

9, 9, 14, 29, 69, 174, 449, 1169, 3054, 7989, 20909, 54734, 143289, 375129, 982094, 2571149, 6731349, 17622894, 46137329, 120789089, 316229934, 827900709, 2167472189, 5674515854, 14856075369, 38893710249, 101825055374, 266581455869, 697919312229, 1827176480814, 4783610130209, 12523653909809, 32787351599214, 85838400887829, 224727851064269

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A032908, A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350920, A350921, A350922, A350923, A350924, A350925, A350926.

nxt[{a_,b_}]:={b,3b-a-4}; NestList[nxt,{9,9},40][[;;,1]] (* or *) LinearRecurrence[{4,-4,1},{9,9,14},40] (* Harvey P. Dale, Jul 19 2024 *)

a(n) = 5*A032908(n) - 1. - Hugo Pfoertner, Jan 22 2022
G.f.: (3 - 2*x)*(3 - 7*x)/((1 - x)*(1 - 3*x + x^2)). - Stefano Spezia, Jan 22 2022
a(n) = 5*A001519(n) +4. - R. J. Mathar, Feb 07 2022

A350920 a(0) = 5, a(1) = 5, and a(n) = 4*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

5, 5, 11, 35, 125, 461, 1715, 6395, 23861, 89045, 332315, 1240211, 4628525, 17273885, 64467011, 240594155, 897909605, 3351044261, 12506267435, 46674025475, 174189834461, 650085312365, 2426151414995, 9054520347611, 33791929975445, 126113199554165, 470660868241211, 1756530273410675, 6555460225401485, 24465310628195261, 91305782287379555

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001835, A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350921, A350922, A350923, A350924, A350925, A350926.

a(n) = 3*A001835(n) + 2. - Hugo Pfoertner, Jan 22 2022

G.f.: (5 - 20*x + 11*x^2)/((1 - x)*(1 - 4*x + x^2)). - Stefano Spezia, Jan 22 2022

A350921 a(0) = 3, a(1) = 3, and a(n) = 6*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

3, 3, 11, 59, 339, 1971, 11483, 66923, 390051, 2273379, 13250219, 77227931, 450117363, 2623476243, 15290740091, 89120964299, 519435045699, 3027489309891, 17645500813643, 102845515571963, 599427592618131, 3493720040136819, 20362892648202779, 118683635849079851, 691738922446276323

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A001653, A011900, A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350920, A350922, A350923, A350924, A350925, A350926.

G.f.: (3 - 18*x + 11*x^2)/((1 - x)*(1 - 6*x + x^2)). - Stefano Spezia, Jan 22 2022

a(n) = 2*A001653(n) + 1 = 4*A011900(n-1) - 1 for n >= 1. - Hugo Pfoertner, Jan 22 2022

A350924 a(0) = 1, a(1) = 3, and a(n) = 16*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

1, 3, 43, 681, 10849, 172899, 2755531, 43915593, 699893953, 11154387651, 177770308459, 2833170547689, 45152958454561, 719614164725283, 11468673677149963, 182779164669674121, 2912997961037635969, 46425188211932501379, 739890013429882386091, 11791815026666185676073

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Index entries for linear recurrences with constant coefficients, signature (17,-17,1).

Cf. A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350920, A350921, A350922, A350923, A350925, A350926.

nxt[{a_,b_}]:={b,16b-a-4}; NestList[nxt,{1,3},20][[All,1]] (* or *) LinearRecurrence[ {17,-17,1},{1,3,43},20] (* Harvey P. Dale, Jan 08 2023 *)

a350924 = [1, 3]
for k in range(2, 100): a350924.append(16*a350924[k-1]-a350924[k-2]-4)
print(a350924) # Karl-Heinz Hofmann, Jan 22 2022

G.f.: (1 - 14*x + 9*x^2)/((1 - x)*(1 - 16*x + x^2)). - Stefano Spezia, Jan 22 2022

7*a(n) = 2 +5*A077412(n) -61*A077412(n-1). - R. J. Mathar, Feb 07 2022

A350925 a(0) = 1, a(1) = 9, and a(n) = 16*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

1, 9, 139, 2211, 35233, 561513, 8948971, 142622019, 2273003329, 36225431241, 577333896523, 9201116913123, 146640536713441, 2337047470501929, 37246118991317419, 593600856390576771, 9460367583257910913, 150772280475735997833, 2402896120028518054411

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Index entries for linear recurrences with constant coefficients, signature (17,-17,1).

Cf. A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350920, A350921, A350922, A350923, A350924, A350926.

LinearRecurrence[{17,-17,1},{1,9,139},20] (* Harvey P. Dale, Feb 09 2025 *)

G.f.: (1 - 8*x + 3*x^2)/((1 - x)*(1 - 16*x + x^2)). - Stefano Spezia, Jan 22 2022

7*a(n) = 2+5*A077412(n)-19*A077412(n-1). - R. J. Mathar, Feb 07 2022

A350926 a(0) = 1, a(1) = 17, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

1, 17, 386, 8857, 203321, 4667522, 107149681, 2459775137, 56467678466, 1296296829577, 29758359401801, 683145969411842, 15682598937070561, 360016629583211057, 8264699881476783746, 189728080644382815097, 4355481154939327963481, 99986338482960160344962, 2295330303953144359970641

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350920, A350921, A350922, A350923, A350925, A350925.

LinearRecurrence[{24,-24,1},{1,17,386},20] (* Harvey P. Dale, Jun 12 2022 *)

G.f.: (1 - 7*x + 2*x^2)/((1 - x)*(1 - 23*x + x^2)). - Stefano Spezia, Jan 22 2022

21*a(n) = 4+17*A097778(n)-38*A097778(n-1). - R. J. Mathar, Feb 07 2022

A368546 Alternative version of the Markov tree A327345.

Original entry on oeis.org

5, 13, 29, 34, 194, 433, 169, 89, 1325, 7561, 2897, 6466, 37666, 14701, 985, 233, 9077, 135137, 51641, 294685, 4400489, 1686049, 43261, 96557, 8399329, 48928105, 3276509, 1278818, 7453378, 499393, 5741, 610, 62210, 2423525, 925765, 13782649, 537169541

Offset: 0

William P. Orrick, Jan 04 2024

The Markov tree is a complete, infinite binary tree. Vertices are labeled by triples. The root vertex is (1, 5, 2). The left child of (a, b, c) is (a, 3*a*b - c, b); its right child is (b, 3*b*c - a, c). The sequence is a triangle read by rows consisting of the middle element of each triple, which is always the largest element of the triple. Row r contains 2^r elements.
The tree contains contains exactly one representative of each class of permutation equivalent nonsingular solutions to Markov's equation, a^2 + b^2 + c^2 = 3 * a * b * c. Nonsingular solutions are those in which a, b, and c are three distinct numbers. The two singular triples (1, 1, 1) and (1, 2, 1) are omitted in this sequence.
A consequence of Markov's equation is that the recurrence for the tree may be reformulated as follows: the left child of (a, b, c) is (a, (a^2 + b^2) / c, b); its right child is (b, (b^2 + c^2) / a, c).
An open problem is to prove the uniqueness conjecture, which asserts that the largest element of a triple determines the other two.
Frobenius proposed assigning a rational number index in (0,1) to each vertex of the tree, and hence to each term in this sequence. This is the Farey index, obtained by assigning the triple (0/1, 1/2, 1/1) to the root vertex and using the following rules to assign triples to the rest of the tree: the vertex labeled (u/v, w/x, y/z) with w = u + u and x = v + z has left child (u/v, (u+w)/(v+x), w/x) and right child (w/x, (w+y)/(x+z), y/z). The Farey index is the center element of the triple. Each rational number in (0, 1) appears as the Farey index of exactly one vertex of the tree. The index of a(n) is A007305(n+2) / A007306(n+2).
A sequence of leftward steps in the tree produces odd-indexed Fibonacci numbers, A001519, which have Farey indices of the form 1 / n. A sequence of rightward steps in the tree produces odd-indexed Pell numbers, A001653, which have Farey indices of the form (n - 1) / n. A sequence of leftward steps followed by a single rightward step produces A350922, corresponding to Farey indices of the form 2 / (2 * n + 1). Alternating steps right, left, right, left, right, ... produces A064098, which corresponds to Farey indices of the form F(n) / F(n + 1), where F(n) is the n-th Fibonacci number.

			The initial levels of the tree are as follows. (See p. 47 of Aigner's book.)
                               (1,5,2)
             (1,13,5)                              (5,29,2)
   (1,34,13)         (13,194,5)         (5,433,29)             (29,169,2)
(1,        (34,     (13,     (194,    (5,       (433,       (29,       (169,
 89,        1325,    7561,    2897,    6466,     37666,      14701,     985,
 ,34)        13)      194)     5)       433)      29)         169)       2)

Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013. x+257 pp. ISBN: 978-3-319-00887-5; 978-3-319-00888-2 MR3098784.

John Tyler Rascoe, Rows n = 0..10, flattened
Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings, [archive.org copy of the book].
Robert A. Gore, Patterns Within the Markov Tree, arXiv:2506.04299 [math.GM], 2025.

Other presentations of the Markov numbers, Markov triples, or the Markov tree: A002559, A253809, A291694, A305313, A305314, A327345.
Subsequences in the Markov tree: A001519, A001653, A350922, A064098.
Farey tree: A007305, A007306.

def Mtree(x): return(x[0],(3*x[0]*x[1])-x[2],x[1]), (x[1],(3*x[1]*x[2])-x[0],x[2])
def A368546_rowlist(maxrow):
    A,B = [[(1,5,2)]],[]
    for n in range(maxrow+1):
        A.append([])
        for j in A[n]:
            B.append(max(j))
            for k in Mtree(j):
                A[n+1].append(k)
    return(B) # John Tyler Rascoe, Feb 09 2024

def stripUpToFirst1(w):
    x = w
    while x % 2 == 0:
        x = x // 2
    return(x // 2)
def stripUpToFirst0(w):
    x = w
    while x % 2 == 1:
        x = x // 2
    if x == 0:
        return(None)
    else:
        return(x // 2)
@CachedFunction
def markovNumber(w):
    if w == None:
        return(2)
    elif w == 0:
        return(1)
    elif w == 1:
        return(5)
    elif w % 2 == 0:
        return(3*markovNumber(stripUpToFirst1(w))*markovNumber(w//2) - markovNumber(stripUpToFirst0(w//2)))
    else:
        return(3*markovNumber(stripUpToFirst0(w))*markovNumber(w//2) - markovNumber(stripUpToFirst1(w//2)))
[markovNumber(w) for w in range(1,38)]

cp's OEIS Frontend

A350916 Positive integers k such that (k+1)^4 has a divisor congruent to -1 modulo k.

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A350923 a(0) = 2, a(1) = 2, and a(n) = 10*a(n-1) - a(n-2) - 4 for n >= 2.

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A350917 a(0) = 1, a(1) = 2, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

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A350919 a(0) = 9, a(1) = 9, and a(n) = 3*a(n-1) - a(n-2) - 4 for n >= 2.

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A350920 a(0) = 5, a(1) = 5, and a(n) = 4*a(n-1) - a(n-2) - 4 for n >= 2.

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A350921 a(0) = 3, a(1) = 3, and a(n) = 6*a(n-1) - a(n-2) - 4 for n >= 2.

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A350924 a(0) = 1, a(1) = 3, and a(n) = 16*a(n-1) - a(n-2) - 4 for n >= 2.

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A350925 a(0) = 1, a(1) = 9, and a(n) = 16*a(n-1) - a(n-2) - 4 for n >= 2.

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A350926 a(0) = 1, a(1) = 17, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

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