A003095 -id:A003095 - cp's OEIS frontend

A193637 a(n) = a(n-1)^2 - n^(n+1).

0, -1, -7, -32, 0, -15625, 243860689, 59468035633789920, 3536447262141707692104062559388672, 12506459237909580203511583184455022770672120296396568887010875139183

Offset: 0

Arkadiusz Wesolowski, Aug 01 2011

Example of a recursive sequence which produces a table containing two zeros.

			a(2) = -7 because a(1) = -1 and (-1)^2 - 2^(2+1) = -7.

Vincenzo Librandi, Table of n, a(n) for n = 0..12
Eric Weisstein's World of Mathematics, Recursive Sequence

Cf. A003095, A007778.

RecurrenceTable[{a[n] == a[n - 1]^2 - n^(n + 1), a[0] == 0}, a, {n, 10}]

a=0; for(n=0, 10, print1(a=a^2-n^(n+1), ", "));

a(0) = 0, a(n) = a(n-1)^2 - n^(n+1).

A193925 a(n) = a(n-1)^2 - n^(n-2) + n.

Original entry on oeis.org

0, 0, 1, 1, -11, 1, -1289, 1644721, 2705106905705, 7317603371292879756764065, 53547319099556919431874542743248407878119975324235

Offset: 0

Arkadiusz Wesolowski, Aug 09 2011

Example of a recursive sequence which produces a table containing three ones.

			a(4) = -11 because a(3) = 1 and 1^2 - 4^(4-2) + 4 = -11.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..14
Eric Weisstein's World of Mathematics, Recursive Sequence

Cf. A003095, A000272.

RecurrenceTable[{a[n] == a[n - 1]^2 - n^(n - 2) + n, a[0] == 0}, a, {n, 10}]

print1(a=0, ", "); for(n=1, 10, print1(a=a^2-n^(n-2)+n, ", "));

a(0) = 0, a(n) = a(n-1)^2 - n^(n-2) + n.

A256344 Moduli n for which A248218(n) = 4 (length of the terminating cycle of 0 under x -> x^2+1 modulo n).

Original entry on oeis.org

13, 26, 39, 47, 52, 78, 79, 91, 94, 104, 113, 141, 143, 156, 158, 169, 173, 182, 188, 197, 208, 226, 237, 247, 273, 282, 286, 299, 312, 316, 329, 338, 339, 346, 353, 364, 376, 377, 394, 403, 416, 429, 439, 452, 474, 481, 494, 507, 517, 519, 546, 553, 559, 564, 572, 591, 598

Offset: 1

M. F. Hasler, Mar 25 2015

nonn

If x is a member and y is a member of this sequence or A248219 or A256342, then LCM(x,y) is a member. - Robert Israel, Mar 09 2021

			See A256342 or A256349.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A248218, A248219, A256342 - A256349, A003095, A247981.

filter:= proc(n) local x, k, R,p;
  x:= 0; R[0]:= 0;
  for k from 1 do
    x:= x^2+1 mod n;
    if assigned(R[x]) then return evalb(k-R[x] = 4)
    else R[x]:= k
    fi
  od;
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 09 2021

for(i=1,600,A248218(i)==4&&print1(i","))

A256345 Moduli n for which A248218(n) = 5 (length of the terminating cycle of 0 under x -> x^2+1 modulo n).

Original entry on oeis.org

83, 151, 167, 223, 249, 257, 283, 359, 373, 453, 501, 563, 581, 607, 669, 677, 771, 821, 849, 953, 1057, 1077, 1119, 1169, 1321, 1561, 1577, 1689, 1743, 1799, 1821, 1981, 1987, 2017, 2031, 2463, 2513, 2573, 2611, 2833, 2859, 2869

Offset: 1

M. F. Hasler, Mar 25 2015

nonn

If x is a member and y is a member of this sequence or A248219, then LCM(x,y) is a member. - Robert Israel, Mar 09 2021

			See A256342 or A256349.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A248218, A248219, A256342 - A256349, A003095, A247981.

filter:= proc(n) local x, k, R,p;
  x:= 0; R[0]:= 0;
  for k from 1 do
    x:= x^2+1 mod n;
    if assigned(R[x]) then return evalb(k-R[x] = 5)
    else R[x]:= k
    fi
  od;
end proc:
select(filter, [$1..3000]); # Robert Israel, Mar 09 2021

for(i=1,2900,A248218(i)==5&&print1(i","))

A256346 Moduli n for which A248218(n) = 6.

Original entry on oeis.org

10, 17, 18, 20, 30, 34, 36, 40, 49, 50, 51, 54, 55, 60, 68, 70, 72, 73, 80, 85, 90, 98, 99, 100, 102, 108, 110, 115, 118, 119, 120, 126, 136, 140, 144, 145, 146, 147, 150, 153, 160, 165, 170, 180, 187, 190, 194, 196, 198, 199, 200, 204, 207, 210, 211, 216, 219, 220, 230, 236, 238, 240, 245, 250

Offset: 1

M. F. Hasler, Mar 25 2015

nonn

If x is a member of this sequence, and y is a member of this sequence or A248219 or A256342 or A256343, then LCM(x,y) is a member of this sequence. - Robert Israel, Mar 09 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A248218, A248219, A256342 - A256349, A003095, A247981.

filter:= proc(n) local x, k, R,p;
  x:= 0; R[0]:= 0;
  for k from 1 do
    x:= x^2+1 mod n;
    if assigned(R[x]) then return evalb(k-R[x] = 6)
    else R[x]:= k
    fi
  od;
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 09 2021

for(i=1,250,A248218(i)==6&&print1(i","))

A256347 Moduli n for which A248218(n) = 7.

Original entry on oeis.org

41, 123, 131, 287, 317, 393, 503, 547, 727, 779, 861, 917, 951, 1091, 1237, 1271, 1277, 1509, 1517, 1627, 1637, 1641, 1681, 1763, 2089, 2181, 2219, 2239, 2337, 2357, 2383, 2489, 2531, 2671, 2751, 2789

Offset: 1

M. F. Hasler, Mar 25 2015

nonn

If x is a member, and y is a member of this sequence or A248219, then LCM(x,y) is a member. - Robert Israel, Mar 09 2021

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A248218, A248219, A256342 - A256349, A003095, A247981.

filter:= proc(n) local x, k, R,p;
  x:= 0; R[0]:= 0;
  for k from 1 do
    x:= x^2+1 mod n;
    if assigned(R[x]) then return evalb(k-R[x] = 7)
    else R[x]:= k
    fi
  od;
end proc:
select(filter, [$1..10000]); # Robert Israel, Mar 09 2021

for(i=1,3000,A248218(i)==7&&print1(i","))

A328699 Start with 0, a(n) is the smallest number of iterations: x -> (x^2+1) mod n needed to run into a cycle.

Original entry on oeis.org

0, 0, 2, 1, 0, 2, 3, 2, 2, 0, 4, 2, 0, 3, 2, 3, 2, 2, 5, 1, 3, 4, 6, 2, 1, 0, 2, 3, 9, 2, 4, 3, 4, 2, 3, 2, 5, 5, 2, 2, 0, 3, 7, 4, 2, 6, 10, 3, 3, 1, 2, 1, 7, 2, 4, 3, 5, 9, 8, 2, 5, 4, 3, 4, 0, 4, 10, 2, 6, 3, 7, 2, 3, 5, 2, 5, 4, 2, 4, 3, 2, 0, 6, 3, 2, 7, 9, 4, 2, 2, 3

Offset: 1

Jianing Song, Oct 26 2019

nonn

Let f(0) = 0, f(k+1) = (f(k)^2+1) mod n, then a(n) is the smallest i such that f(i) = f(j) for some j > i.
Obviously a(n) <= A000224(n): f(1), f(2), ..., f(A000224(n)+1) are all of the form (s^2+1) mod n, so there must exists 0 <= i < j <= A000224(n)+1 such that f(i) = f(j), and a(n) <= i <= A000224(n). The equality seems to hold only for n = 3.
k divides A003095(m) for some m > 0 if and only if a(k) = 0, in which case all the indices m such that k divides A003095(m) are m = t*A248218(k), t = 0, 1, 2, 3, ...

			A003095(n) mod 3: 0, 1, (2). {A003095(n) mod 3} enters into the cycle (2) from the 2nd term on, so a(3) = 2.
A003095(n) mod 7: 0, 1, 2, (5). {A003095(n) mod 7} enters into the cycle (5) from the 3rd term on, so a(7) = 3.
A003095(n) mod 29: 0, 1, 2, 5, 26, 10, 14, 23, 8, (7, 21). {A003095(n) mod 29} enters into the cycle (7, 21) from the 9th term on, so a(29) = 9.
A003095(n) mod 37: 0, 1, 2, 5, 26, (11). {A003095(n) mod 37} enters into the cycle (11) from the 5th term on, so a(37) = 5.
A003095(n) mod 41: (0, 1, 2, 5, 26, 21, 32). {A003095(n) mod 41} enters into the cycle (0, 1, 2, 5, 26, 21, 32) from the very beginning, so a(41) = 0.

Cf. A003095, A248218 (cycle length), A328700, A000224.

a(n) = my(v=[0],k); for(i=2, n+1, k=(v[#v]^2+1)%n; v=concat(v, k); for(j=1, i-1, if(v[j]==k, return(j-1))))

A363257 a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 31, 257, 16642, 69247363, 1198799355237125, 359279973529237254190922184970, 32270524844792355518177347536627638351478874995525184567711

Offset: 0

Harry Richman, May 23 2023

nonn

Iterated application of A033638, with a shift.

Andrew Howroyd, Table of n, a(n) for n = 0..16
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Cf. A003095, A006894, A014980, A033638.

a(n) = if(n < 1, 0, floor( ((a(n-1) + 1) / 2)^2 ) + 1) \\ Andrew Howroyd, Jan 01 2024

a(n) = A033638(a(n-1)+1) for n > 0.

log a(n) ~ C * 2^n for some constant C.

A135378 Main diagonal of "square and add k" array.

Original entry on oeis.org

2, 5, 38, 2707, 21418388, 3000279372337641, 255122481276683701099886061668842

Offset: 0

Jonathan Vos Post, Dec 09 2007

Array of recurrence "start with 2, square and add k" begins:
k..|.A[k,n]=A[k,n-1]^2 + k
-1.|.2..3...8....63.......3968..15745023.247905749270528.............A003096
.|.2..4..16...256......65536..4294967296.18446744073709551616......A001146
.|.2..5..26...677.....458330..210066388901.44127887745906175987802.A003095
.|.2..6..38..1446....2090918.4371938082726...19113842599189892819591078...
.|.2..7..52..2707....7327852.53697414933907..2883412370584178505178284652.
.|.2..8..68..4628...21418388.458747344518548.210449126102819371741916028308.
.|.2..9..86..7401...54774806.3000279372337641.9001676312074749038996905444886.
.|.2.10.106.11242..126382570.1597255405035792810...
.|.2.11.128.16391..268664888.72180822044052551...
.|.2.12.152.23112..534164552.285331768613360712..
.|.2.13.178.31693.1004446258.1008912285210202573.
|.2.14.206.42446.1801662926.3245989298922881486.

Cf. A001146, A003095, A003096, A062013.

A[k_,0] = 2; A[k_,n_] := A[k,n] = A[k, n-1]^2 + k; a[n_] := A[n, n]; a /@ Range[0, 6] (* Giovanni Resta, Jun 20 2016 *)

a(n) = A[n,n] where A[k,n] = n-th term of recurrence A[k,0] = 2, A[k,n] = A[k,n-1]^2 + k.

Corrected and edited by Giovanni Resta, Jun 20 2016

A248545 a(n+1) = a(n)^2 + 75.

Original entry on oeis.org

0, 75, 5700, 32490075, 1055604973505700, 1114301860089969598947932490075, 1241668635399966182910364962424859894024949485439789973505700

Offset: 0

Vaclav Kotesovec, Oct 08 2014

nonn

Dedicated to N. J. A. Sloane for his 75th birthday!

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A003095.

RecurrenceTable[{a[n+1]==a[n]^2+75,a[0]==0},a,{n,0,10}]
NestList[#^2+75&,0,10] (* Harvey P. Dale, Jan 14 2016 *)

a(n) ~ c^(2^n), where c = 8.688980833378203252201919626948141475048572223268...

cp's OEIS Frontend

A193637 a(n) = a(n-1)^2 - n^(n+1).

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A193925 a(n) = a(n-1)^2 - n^(n-2) + n.

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A256344 Moduli n for which A248218(n) = 4 (length of the terminating cycle of 0 under x -> x^2+1 modulo n).

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A256345 Moduli n for which A248218(n) = 5 (length of the terminating cycle of 0 under x -> x^2+1 modulo n).

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A256346 Moduli n for which A248218(n) = 6.

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A256347 Moduli n for which A248218(n) = 7.

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A328699 Start with 0, a(n) is the smallest number of iterations: x -> (x^2+1) mod n needed to run into a cycle.

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A363257 a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.

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