A102847 -id:A102847 - cp's OEIS frontend

A158983 Coefficients of polynomials (in descending powers of x) P(n,x) := 2 + P(n-1,x)^2, where P(1,x) = x + 2.

1, 2, 1, 4, 6, 1, 8, 28, 48, 38, 1, 16, 120, 544, 1628, 3296, 4432, 3648, 1446, 1, 32, 496, 4928, 35064, 189248, 800992, 2711424, 7419740, 16475584, 29610272, 42666880, 48398416, 41867904, 26125248, 10550016, 2090918, 1, 64, 2016, 41600, 631536

Offset: 1

Clark Kimberling, Apr 02 2009

			Row 1: 1 2 (from x+2)
Row 2: 1 4 6 (from x^2+4x+6)
Row 3: 1 8 28 48 38
Row 4: 1 16 120 544 1628 3296 4432 3648 1446

Clark Kimberling, Polynomials defined by a second-order recurrence, interlacing zeros, and Gray codes, The Fibonacci Quarterly 48 (2010) 209-218.

Cf. A158982, A158984, A158985, A158986. A102847 (row sums).

From Peter Bala, Jul 01 2015: (Start)
P(n,x) = P(n,-4 - x) for n >= 2.
P(n+1,x)= P(n,(2 + x)^2). Thus if alpha is a zero of P(n,x) then sqrt(alpha) - 2 is a zero of P(n+1,x).
Define a sequence of polynomials Q(n,x) by setting Q(1,x) = 2 + x^2 and Q(n,x) = Q(n-1, 2 + x^2) for n >= 2. Then P(n,x) = Q(n,sqrt(x)).
Q(n,x) = Q(k,Q(n-k,x)) for 1 <= k <= n-1; P(n,x) = P(k,P(n-k,x)^2) for 1 <= k <= n - 1.
n-th row sum = P(n,1) = A102847(n);
P(n,1) = P(n+1,-1) = P(n+1,-3); P(n,1) = P(n,-5) for n >= 2.
(End)

A065653 a(0) = 0, a(1) = 1, a(n) = a(n-2)*a(n-2) + 2 for n > 1.

Original entry on oeis.org

0, 1, 2, 3, 6, 11, 38, 123, 1446, 15131, 2090918, 228947163, 4371938082726, 52416803445748571, 19113842599189892819591078, 2747521283470239265968814548542043, 365338978906606237729724396156395693696687137202086

Offset: 0

Reinhard Zumkeller, Nov 10 2001

nonn

Harry J. Smith, Table of n, a(n) for n=0,...,24

Cf. A065652

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]^2+2},a,{n,20}] (* Harvey P. Dale, Aug 19 2013 *)

a(n)=if(n<2, n>0, 2+a(n-2)^2) /* Michael Somos, Mar 25 2006 */

{ for (n=0, 24, if (n>1, a=a2^2 + 2; a2=a1; a1=a, if (n, a=a1=1, a=a2=0)); write("b065653.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 25 2009

A102847(n)=a(2n+1), A072191(n)=a(2n).

One more term from Harvey P. Dale, Aug 19 2013

A378395 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

Original entry on oeis.org

3, 4, 5, 11, 60, 61, 123, 7564, 7565, 15131, 114473580, 114473581, 228947163, 26208401722874284, 26208401722874285, 52416803445748571, 1373760641735119632984407274271020, 1373760641735119632984407274271021

Offset: 1

Miguel-Ángel Pérez García-Ortega, Nov 24 2024

The only Pythagorean triple whose inradius is equal to r and such that its long leg and its hypotenuse are consecutive is (2r+1,2r^2+2r,2r^2+2r+1).

			Triples begin:
  3, 4, 5;
  11, 60, 61;
  123, 7564, 7565;
  15131, 114473580, 11447358;
...

J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2024.

Cf. A102847 (short leg), A365577.

{a0,b0,c0}={3,4,5};f[n_]:=Module[{fn0=2c0+1,fn1=((2c0+1)^2+1)/2},Do[{fn0,fn1}={2fn1+1,((2fn1+1)^2+1)/2},{n}];fn0];t[n_]:={f[n-1],(f[n-1]^2-1)/2,(f[n-1]^2+1)/2};ternas={a0,b0,c0};For[i=1,i<=6,i++,ternas=Join[ternas,t[i]]];ternas

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A158983 Coefficients of polynomials (in descending powers of x) P(n,x) := 2 + P(n-1,x)^2, where P(1,x) = x + 2.

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A065653 a(0) = 0, a(1) = 1, a(n) = a(n-2)*a(n-2) + 2 for n > 1.

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A378395 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

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