A112959 -id:A112959 - cp's OEIS frontend

A112957 a(1) = a(2) = a(3) = 1; for n > 1, a(n+3) = a(n)^2 + a(n+1)^2 + a(n+2)^2.

1, 1, 1, 3, 11, 131, 17291, 298995963, 89398586189293211, 7992107212644486930829797919966571, 63873777698404030240264509605345282496735163325301838600463378485931

Offset: 1

Jonathan Vos Post, Jan 02 2006; definition corrected Jan 02 2006

A quadratic tribonacci sequence.

This is to A000283 as a tribonacci (A000213) is to Fibonacci. Two oddities about this sequence: (a) its first 7 terms are identical to terms numbered 2 through 8 of A072878; (b) only one of the first 9 terms are composite. Primes in the sequence begin 3, 11, 131, 17291 and 89398586189293211. What is the next prime?

Seiichi Manyama, Table of n, a(n) for n = 1..14

Cf. A000213, A000283, A072878, A112958, A112959, A112960.

Join[{a=1,b=1,c=1},Table[d=a*a+b*b+c*c;a=b;b=c;c=d,{n,10}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

A112958 a(1) = a(2) = a(3) = a(4) = 1; for n>1: a(n+4) = a(n)^2 + a(n+1)^2 + a(n+2)^2 + a(n+3)^2.

Original entry on oeis.org

1, 1, 1, 1, 4, 19, 379, 144019, 20741616379, 430214650034342688004, 185084645104171955001009752069374428191659

Offset: 1

Jonathan Vos Post, Jan 02 2006

A quadratic tetranacci sequence.

This is to A000283 as a tetranacci (A000288) is to Fibonacci. Primes in this begin 19, 379.

			1^2 + 4^2 + 19^2 + 379^2 = 144019.

Seiichi Manyama, Table of n, a(n) for n = 1..15

Cf. A000283, A000288, A112957, A112959, A112960.

RecurrenceTable[{a[1] == a[2] == a[3] == a[4] == 1, a[n] == a[n-1]^2 + a[n-2]^2 + a[n-3]^2 + a[n-4]^2}, a, {n, 15}] (* Vincenzo Librandi, Aug 21 2016 *)

A112960 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1; for n>1: a(n+6) = (a(n))^2 + (a(n+1))^2 + (a(n+2))^2 + (a(n+3))^2 + (a(n+4))^2 + (a(n+5))^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 41, 1721, 2963561, 8782696764281, 77135762453320729974211241, 5949925849255124079413733148488788342637650064971321

Offset: 1

Jonathan Vos Post, Jan 02 2006

A quadratic hexanacci sequence.

This is to A000283 as a hexanacci (A000383) is to Fibonacci. Primes in this begin a(8) = 41, a(9) = 1721. Semiprimes begin a(7), a(10), a(12).

			1^2 + 6^2 + 41^2 + 1721^2 + 2963561^2 + 8782696764281^2 = 77135762453320729974211241.

Seiichi Manyama, Table of n, a(n) for n = 1..17

Cf. A000283, A000383, A112957, A112958, A112959.

RecurrenceTable[{a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == 1, a[n] == a[n-1]^2 + a[n-2]^2 + a[n-3]^2 + a[n-4]^2 + a[n-5]^2 + a[n-6]^2}, a, {n, 17}] (* Vincenzo Librandi, Aug 21 2016 *)
nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,a^2+b^2+c^2+d^2+e^2+f^2}; NestList[ nxt,{1,1,1,1,1,1},12][[All,1]] (* Harvey P. Dale, Apr 12 2020 *)

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

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.

cp's OEIS Frontend

A112957 a(1) = a(2) = a(3) = 1; for n > 1, a(n+3) = a(n)^2 + a(n+1)^2 + a(n+2)^2.

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A112958 a(1) = a(2) = a(3) = a(4) = 1; for n>1: a(n+4) = a(n)^2 + a(n+1)^2 + a(n+2)^2 + a(n+3)^2.

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A112960 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1; for n>1: a(n+6) = (a(n))^2 + (a(n+1))^2 + (a(n+2))^2 + (a(n+3))^2 + (a(n+4))^2 + (a(n+5))^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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A113592 Array of quadratic pseudofibonacci sequences, read by antidiagonals.

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