Kurmang. Aziz. Rashid - cp's OEIS frontend

A089061 a(0) = 5, a(1) = 7; for n>1, a(n) = a(n-1)+a(n-2)-(2n-2).

5, 7, 10, 13, 17, 22, 29, 39, 54, 77, 113, 170, 261, 407, 642, 1021, 1633, 2622, 4221, 6807, 10990, 17757, 28705, 46418, 75077, 121447, 196474, 317869, 514289, 832102, 1346333, 2178375, 3524646, 5702957, 9227537, 14930426, 24157893

Offset: 0

Kurmang. Aziz. Rashid, Dec 02 2003

nonn

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A088981, A000045.

RecurrenceTable[{a[0]==5,a[1]==7,a[n]==a[n-1]+a[n-2]-(2n-2)},a,{n,40}] (* or *) LinearRecurrence[{3,-2,-1,1},{5,7,10,13},40] (* Harvey P. Dale, Apr 23 2018 *)

a(n) = Fibonacci(n+1)+2n+4. - Ralf Stephan, Feb 24 2004

A090698 Primes of the form 2*n^2+1.

Original entry on oeis.org

3, 19, 73, 163, 883, 1153, 1459, 1801, 2179, 2593, 3529, 4051, 8713, 10369, 11251, 15139, 17299, 18433, 19603, 20809, 22051, 30259, 34849, 36451, 46819, 48673, 52489, 62659, 69193, 71443, 80803, 83233, 95923, 103969, 112339, 115201, 130051

Offset: 1

Kurmang. Aziz. Rashid, Dec 20 2003

A prime p can be expressed as either the sum of two squares or the sum of two squares - 1, p = X^2 + Y^2 or p = X^2 + Y^2 - 1, if and only if p is of the form 2*(m^2)+1 where m is either 1 or a multiple of 3.
Conjecture: 2^(a(n)-1) - 3 is not prime. - Vincenzo Librandi, Feb 04 2013.
Primes in A058331. - Vincenzo Librandi, Apr 10 2015

			19 = 2^2 + 4^2 - 1 = 2*(3^2)+1
73 = 5^2 + 7^2 - 1 = 2*(6^2)+1
163= 8^2 + 10^2 -1 = 2*(9^2)+1
883= 10^2+ 28^2 -1 = 2*(21^2)+1

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A058331, A089001, A089008, A090612.

[a: n in [0..400] | IsPrime(a) where a is 2*n^2+1];// Vincenzo Librandi, Dec 02 2011

Select[Table[2n^2+1,{n,0,900}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

is(n)=isprime(2*n^2+1) \\ Charles R Greathouse IV, Jan 05 2013

a(n)=2*A089001(n)^2+1 = A000040(A090612(n)).

Extended by Ray Chandler, Dec 21 2003

A088981 a(n+2) = a(n+1) + a(n) - (2*n + 1) where a(0)=7, a(1)=11.

Original entry on oeis.org

7, 11, 17, 25, 37, 55, 83, 127, 197, 309, 489, 779, 1247, 2003, 3225, 5201, 8397, 13567, 21931, 35463, 57357, 92781, 150097, 242835, 392887, 635675, 1028513, 1664137, 2692597, 4356679, 7049219, 11405839, 18454997

Offset: 0

Kurmang. Aziz. Rashid, Dec 01 2003

J. Baylis and R. Haggarty, Alice in Numberland, A Student's Guide to the Enjoyment of Higher Mathematics, Macmillan Education 1988.
G. Buckwell, Mastering Mathematics, Palgrave Master Series, 2nd Ed. 1997.
R. P. C. Forman, Additional Mathematics Pure & Applied, Stanley Thornes, 1989.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

LinearRecurrence[{3,-2,-1,1},{7,11,17,25},40] (* Harvey P. Dale, Jun 08 2018 *)

a=[7,11];for(n=2,10,a=concat(a,a[#a]+a[#a-1]-2*n+3)); a

a(n) = (2*alpha^(n+3) - 2*beta^(n+3) + 2*sqrt(5)*n + 3*sqrt(5)) / sqrt(5) where alpha = (1 + sqrt(5)) / 2 and beta = (1 - sqrt(5)) / 2.

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User: Kurmang. Aziz. Rashid

A089061 a(0) = 5, a(1) = 7; for n>1, a(n) = a(n-1)+a(n-2)-(2n-2).

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A090698 Primes of the form 2*n^2+1.

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A088981 a(n+2) = a(n+1) + a(n) - (2*n + 1) where a(0)=7, a(1)=11.

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