A350923 -id:A350923 - 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)) ); }

A350922 a(0) = 2, a(1) = 5, and a(n) = 7*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

2, 5, 29, 194, 1325, 9077, 62210, 426389, 2922509, 20031170, 137295677, 941038565, 6449974274, 44208781349, 303011495165, 2076871684802, 14235090298445, 97568760404309, 668746232531714, 4583654867317685, 31416837838692077, 215334210003526850, 1475922632185995869

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.
From William P. Orrick, Dec 20 2023: (Start)
Every term is a Markov number (see A002559) and, for n > 1, corresponds to a node of the Markov tree A368546 whose sibling and ancestors are all odd-indexed Fibonacci numbers. For n > 1, a(n) is the label of the node obtained from the root by going left n - 2 times and then right. Its Farey index, described in the comments to A368546, is 2 / (2*n - 1).
For instance, a(3) = 194 comes from going left once from the root node of the Markov tree and then right, which corresponds to the sequence of Markov numbers 5, 13, 194.  The corresponding sequence of Farey indices is 1/2, 1/3, 2/5. The sibling of the final node corresponds to Markov number 34 and Farey index 1/4. (End)

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A049684, A064170, A350916, A002559, A000045.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350920, A350921, A350923, A350924, A350925, A350926.

CoefficientList[Series[(2 - x)*(1 - 5*x)/((1 - x)*(1 - 7*x + x^2)), {x, 0, 22}],x] (* James C. McMahon, Dec 22 2023 *)

G.f.: (2 - x)*(1 - 5*x)/((1 - x)*(1 - 7*x + x^2)). - Stefano Spezia, Jan 22 2022
a(n) = 3*A049684(n) + 2 = 3*A064170(n+2) - 1. - Hugo Pfoertner, Jan 22 2022
a(n) = 3*A000045(2*n - 1) * A000045(2*n + 1) - 1 = A000045(2*n - 1)^2 + A000045(2*n + 1)^2. - William P. Orrick, Jan 08 2023

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

A253175 Indices of hexagonal numbers (A000384) which are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 7, 67, 661, 6541, 64747, 640927, 6344521, 62804281, 621698287, 6154178587, 60920087581, 603046697221, 5969546884627, 59092422149047, 584954674605841, 5790454323909361, 57319588564487767, 567405431320968307, 5616734724645195301, 55599941815130984701

Offset: 1

Colin Barker, Jan 08 2015

Also positive integers x in the solutions to 4*x^2-6*y^2-2*x+6*y-2 = 0, the corresponding values of y being A253475.

			7 is in the sequence because the 7th hexagonal number is 91, which is also the 6th centered hexagonal number.

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

Cf. A000384, A003215, A253475, A006244, A350923.

LinearRecurrence[{11, -11, 1}, {1, 7, 67}, 25] (* Paolo Xausa, May 30 2025 *)

Vec(-x*(x^2-4*x+1)/((x-1)*(x^2-10*x+1)) + O(x^100))

a(n) = 11*a(n-1)-11*a(n-2)+a(n-3).
G.f.: -x*(x^2-4*x+1) / ((x-1)*(x^2-10*x+1)).
a(n) = (2+(5-2*sqrt(6))^n*(3+sqrt(6))-(-3+sqrt(6))*(5+2*sqrt(6))^n)/8. - Colin Barker, Mar 05 2016
4*a(n) = 1+3*A072256(n). - R. J. Mathar, Feb 07 2022
a(n) = A350923(n)/2. - Paolo Xausa, May 30 2025

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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A350922 a(0) = 2, a(1) = 5, and a(n) = 7*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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