A000283 -id:A000283 - cp's OEIS frontend

A001042 a(n) = a(n-1)^2 - a(n-2)^2.

1, 2, 3, 5, 16, 231, 53105, 2820087664, 7952894429824835871, 63248529811938901240357985099443351745, 4000376523371723941902615329287219027543200136435757892789536976747706216384

Offset: 0

N. J. A. Sloane, R. K. Guy

The next term has 152 digits. - Franklin T. Adams-Watters, Jun 11 2009

Archimedeans Problems Drive, Eureka, 27 (1964), 6.
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).

Reinhard Zumkeller, Table of n, a(n) for n = 0..13
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!)
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001146, A000278, A000283, A058182, A007018.

Cf. A064236 (numbers of digits).

a001042 n = a001042_list !! n
a001042_list = 1 : 2 : zipWith (-) (tail xs) xs
               where xs = map (^ 2) a001042_list
-- Reinhard Zumkeller, Dec 16 2013

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]^2-a[n-2]^2},a,{n,0,12}] (* Harvey P. Dale, Jan 11 2013 *)

a(n) ~ c^(2^n), where c = 1.1853051643868354640833201434870139866230288004895868726506278977814490371... . - Vaclav Kotesovec, Dec 17 2014

More terms from James Sellers, Sep 19 2000.

A112980 a(0) = 0, a(1) = 1; for n>=2: a(n) = a(n-1)^5 + a(n-2)^5.

Original entry on oeis.org

0, 1, 1, 2, 33, 39135425, 91801604643057285538237803582627026018

Offset: 0

Jonathan Vos Post, Jan 02 2006

			a(3) = 1^5 + 1^5 = 2.
a(4) = 1^5 + 2^5 = 33.
a(5) = 2^5 + 33^5 = 39135425.
a(6) = 33^5 + 39135425^5 = 91801604643057285538237803582627026018.

Eric Weisstein's World of Mathematics, Quintic Equation

Cf. A000045, A000283, A112961, A112969.

RecurrenceTable[{a[1]==a[2]==1,a[n]==a[n-1]^5+a[n-2]^5},a,{n,7}] (* Harvey P. Dale, May 01 2012 *)

a(0)=0 prepended by Alois P. Heinz, Sep 15 2023

A112982 a(1) = a(2) = a(3) = a(4) = 1; for n>4: a(n) = a(n-1)^4 + a(n-2)^4 + a(n-3)^4 + a(n-4)^4.

Original entry on oeis.org

1, 1, 1, 1, 4, 259, 4499860819, 410011770879070587605284428972195139939

Offset: 1

Jonathan Vos Post, Jan 03 2006

A quartic tetranacci sequence.

This is a quartic (biquadratic) analog of a tetranacci sequence A000288, similarly to A000283 being the quadratic analog of the Fibonacci sequence A000045. a(5), a(6) a(7) and a(8) are semiprime. a(9) has 155 digits.

			a(5) = 1^4 + 1^4 + 1^4 + 1^4 = 4.
a(6) = 1^4 + 1^4 + 1^4 + 4^4 = 259.
a(7) = 1^4 + 1^4 + 4^4 + 259^4 = 4499860819.
a(8) = 1^4 + 4^4 + 259^4 + 4499860819^4.

Indranil Ghosh, Table of n, a(n) for n = 1..10
Eric Weisstein's World of Mathematics, Quartic Equation.

Cf. A000045, A000283, A000288.

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==a[n-1]^4+ a[n-2]^4+ a[n-3]^4+ a[n-4]^4},a,{n,10}] (* Harvey P. Dale, May 19 2019 *)

A113494 a(0) = 2, a(1) = 3; for n>1, a(n) = a(n-2)^2 + a(n-1)^2.

Original entry on oeis.org

2, 3, 13, 178, 31853, 1014645293, 1029505071621669458, 1059880692494738761409621021617069613, 1123347082323126985848870489739619316533660939418089557571160604297983533

Offset: 0

Michael Hogan (Michael(AT)michaelhogan.com), Jan 10 2006

Harper's Magazine, Feb 2006, Page 14.

Cf. A000283.

a[1] = 2; a[2] = 3; a[n_] := a[n - 2]^2 + a[n - 1]^2; Array[a, 9] (* Robert G. Wilson v *)

Corrected and extended by Robert G. Wilson v, Jan 11 2006

A113848 a(1) = a(2) = 1, a(n+2) = 2*a(n) + a(n+1)^2.

Original entry on oeis.org

1, 1, 3, 11, 127, 16151, 260855055, 68045359719085327, 4630170979299719971778494028407039, 21438483297549327871400796194793048411084076762817293736211302918175

Offset: 1

Jonathan Vos Post, Jan 24 2006

In this sequence the primes begin a(3) = 3, a(4) = 11, a(5) = 127, a(9) = 4630170979299719971778494028407039.

			a(1) = 1 by definition.
a(2) = 1 by definition.
a(3) = 2*1 + 1^2 = 3.
a(4) = 2*1 + 3^2 = 11.
a(5) = 2*3 + 11^2 = 127.
a(6) = 2*11 + 127^2 = 16151.

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000278, A000283, A014253, A063827, A072878, A112957, A112958, A112959, A112960, A112961, A112969, A113785.

Join[{a=1,b=1},Table[c=1*b^2+2*a;a=b;b=c,{n,10}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
RecurrenceTable[{a[1]==1, a[2]==1, a[n] == 2*a[n-2] + a[n-1]^2}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(1) = a(2) = 1, for n>2: a(n) = 2*a(n-2) + a(n-1)^2. a(1) = a(2) = 1, for n>0: a(n+2) = 2*a(n) + a(n+1)^2.

a(n) ~ c^(2^n), where c = 1.163464453662702696843453679269882816346479873363677551158525103156732040997... . - Vaclav Kotesovec, Dec 18 2014

A114955 A 2/3-power Fibonacci sequence.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Feb 21 2006

a(n) is also the minimum number of distinct palindromes (not counting the empty string) occurring as substrings of an n-bit binary string. For example, the string 11001 contains the five distinct palindromes 0, 00, 1, 11, and 1001. In fact, every 5-bit binary string contains five distinct palindromes, so a(5) = 5. - Austin Shapiro, Feb 15 2023

			a(2) = ceiling(a(0)^(2/3) + a(1)^(2/3)) = ceiling(1^(2/3) + 1^(2/3)) = 2.
a(3) = ceiling(a(1)^(2/3) + a(2)^(2/3)) = ceiling(1^(2/3) + 2^(2/3)) = ceiling(2.58740105) = 3.
a(4) = ceiling(2^(2/3) + 3^(2/3)) = ceiling(3.66748488) = 4.
a(5) = ceiling(3^(2/3) + 4^(2/3)) = ceiling(4.59992592) = 5.
a(6) = ceiling(4^(2/3) + 5^(2/3)) = ceiling(5.44385984) = 6.
a(7) = ceiling(5^(2/3) + 6^(2/3)) = ceiling(6.22594499) = 7.
a(8) = ceiling(6^(2/3) + 7^(2/3)) = ceiling(6.96123296) = 7.

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

Cf. A000283, A112961, A112969, A114793.

nxt[{a_,b_}]:={b,Ceiling[b^(2/3)+a^(2/3)]}; Transpose[NestList[nxt,{1,1},80]][[1]] (* Harvey P. Dale, Jan 03 2013 *)

{a(n)=if(n<1, n==0, if(n>8, 8, n-(n>7)))} /* Michael Somos, Aug 31 2006 */

a(0) = a(1) = 1, for n>1 a(n) = ceiling(a(n-1)^(2/3) + a(n-2)^(2/3)).
a(n) = 8 for all n>8.
Euler transform of length 8 sequence [ 1, 1, 1, 0, 0, -1, 0, -1]. - Michael Somos, Aug 31 2006
G.f.: (1-x^6)(1-x^8)/((1-x)(1-x^2)(1-x^3)). - Michael Somos, Aug 31 2006

A228898 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,x+y) and (y,x^2 + y^2) are edges.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 12, 13, 16, 19, 21, 29, 31, 34, 39, 45, 50, 55, 63, 73, 74, 81, 89, 97, 112, 119, 131, 144, 155, 160, 178, 185, 186, 191, 193, 205, 212, 233, 236, 246, 257, 283, 297, 312, 343, 369, 377, 391, 398, 417, 425, 441, 469, 479, 482, 505, 524, 555

Offset: 1

Clark Kimberling, Sep 08 2013

The tree has infinitely many branches which are essentially linear recurrence sequences (and infinitely many which are not). The extreme branches are (1,2)->(2,3)->(3,5)->(5,8)->... and (1,2)->(2,5)->(5,29)->(29,866)->... These branches contribute to A228898, as subsequences, the Fibonacci numbers, A000045, and A000283.

			Taking the first generation of edges to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5)}, which grows G(3) = {(3,5), (3,13), (5,7), (5,29)}, ... Expelling duplicate nodes and sorting leave (1, 2, 3, 5, 7, 8, 12, 13, 16, 19,...).

Cf. A228853, A000045.

f[x_, y_] := {{y, x + y}, {y, x^2 + y^2}}; x = 1; y = 2; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {18}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A112981 a(1) = a(2) = a(3) = 1; for n>3: a(n) = a(n-1)^3 + a(n-2)^3 + a(n-3)^3.

Original entry on oeis.org

1, 1, 1, 3, 29, 24417, 14557168544129, 3084826414596074361107793217201624802791

Offset: 1

Jonathan Vos Post, Jan 02 2006

A cubic tribonacci sequence.

This is a cubic analog of a tribonacci sequence A000213, similarly to A000283 being the quadratic analog of the Fibonacci sequence A000045. a(4) and a(5) are primes; a(7) is semiprime; a(6) and a(8) have 3 prime factors. a(9) has 119 digits.

			a(6) = 1^3 + 3^3 + 29^3 = 24417.

Cf. A000045, A000213, A000283.

RecurrenceTable[{a[1]==a[2]==a[3]==1,a[n]==a[n-1]^3+a[n-2]^3+a[n-3]^3},a,{n,10}] (* Harvey P. Dale, Jan 25 2018 *)

A113592 Array of quadratic pseudofibonacci sequences, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 11, 1, 4, 11, 40, 127, 1, 5, 18, 127, 1612, 16151, 1, 6, 27, 332, 16151, 2598264, 260855055, 1, 7, 38, 739, 110260

Offset: 1

Jonathan Vos Post, Jan 26 2006

Row 1 is A113848. Column 1 is A000012 (the simplest sequence of positive numbers: the all 1's sequence). Column 2 is A000027 (the natural numbers) = n. Column 3 is A010000 = A059100(n+1) = n^2 + 2. Column 4 is 2*n + (n^2 + 2)^2 = n^4 + 4*n^2 + 2*n + 4. Column 5 is 2*(n^2 + 2) + (n^4 + 4*n^2 + 2*n + 4)^2 = n^8 + 8*n^6 + 4*n^5 + 24*n^4 + 16*n^3 + 38*n^2 + 16*n + 20.

			Table (upper left corner):
1...1...3...11...127....16151...260855055...
1...2...6...40...1612...2598624.675284696600...
1...3...11..127..16151..260855055...
1...4...18..332..110260.12157268264...
1...5...27..739..546175...
1...6...38..1456.2120012...
1...7...51..2615.6838327...
1...8...66..4372.19114516...
1...9...83..6907.47706815
1..10..102..10424.108659980...

Cf. A000012, A000027, A000278, A000283, A010000, A014253, A059100, A063827, A072878, A112957, A112958, A112959, A112960, A112961, A112969, A113785.

Antidiagonals of table: T(i, j) = j-th iteration of a(i, 0) = 1, a(i, 1) = i and for j>1: a(i, j) = 2*a(i, j-2) + a(i, j-1)^2.

A114950 a(n) = a(n-1)^4 + a(n-2)^2, with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 17, 83525, 48670514501156640914, 5611303368570568119463158581109807779153712597124269146443734128560476495542441

Offset: 0

Jonathan Vos Post, Feb 21 2006

a(7) has 315 digits.

			a(2) = a(1)^4 + a(0)^2 = 1^4 + 1^2 = 2.
a(3) = a(2)^4 + a(1)^2 = 2^4 + 1^2 = 17.
a(4) = a(3)^4 + a(2)^2 = 17^4 + 2^2 = 83525.
a(5) = a(4)^4 + a(3)^2 = 83525^4 + 17^2 = 48670514501156640914.

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000283, A112969, A114793.

RecurrenceTable[{a[0] ==1, a[1] == 1, a[n] == a[n-1]^4 + a[n-2]^2}, a, {n, 0, 8}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(n) ~ c^(4^n), where c = 1.045263645117629170027922399491730015846213509999461317320720034161754262379... . - Vaclav Kotesovec, Dec 18 2014

Formula corrected by Vaclav Kotesovec, Dec 18 2014

Missing a(3) added from Vaclav Kotesovec, Dec 18 2014

cp's OEIS Frontend

A001042 a(n) = a(n-1)^2 - a(n-2)^2.

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A112980 a(0) = 0, a(1) = 1; for n>=2: a(n) = a(n-1)^5 + a(n-2)^5.

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A112982 a(1) = a(2) = a(3) = a(4) = 1; for n>4: a(n) = a(n-1)^4 + a(n-2)^4 + a(n-3)^4 + a(n-4)^4.

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A113494 a(0) = 2, a(1) = 3; for n>1, a(n) = a(n-2)^2 + a(n-1)^2.

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A113848 a(1) = a(2) = 1, a(n+2) = 2*a(n) + a(n+1)^2.

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A114955 A 2/3-power Fibonacci sequence.

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A228898 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,x+y) and (y,x^2 + y^2) are edges.

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A112981 a(1) = a(2) = a(3) = 1; for n>3: a(n) = a(n-1)^3 + a(n-2)^3 + a(n-3)^3.

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