Jamel Ghanouchi - cp's OEIS frontend

A245906 Numbers of the form 4n^2 + 1 or 4n^2 + 8n + 1.

5, 13, 17, 33, 37, 61, 65, 97, 101, 141, 145, 193, 197, 253, 257, 321, 325, 397, 401, 481, 485, 573, 577, 673, 677, 781, 785, 897, 901, 1021, 1025, 1153, 1157, 1293, 1297, 1441, 1445, 1597, 1601, 1761, 1765, 1933, 1937, 2113, 2117, 2301, 2305, 2497, 2501, 2701

Offset: 1

Jamel Ghanouchi, Nov 13 2014

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

[IsEven(n) select n^2+4*n+1 else (n+1)^2+1: n in [1..50]]; // Bruno Berselli, Dec 02 2014

fn[n_]:=Module[{c=4n^2+1},{c,c+8n}]; Flatten[Array[fn,30]] (* or *) LinearRecurrence[{1,2,-2,-1,1},{5,13,17,33,37},60] (* Harvey P. Dale, May 21 2015 *)

list(lim)=Set(concat(vector(sqrtint((lim-1)\4),n,4*n^2+1), vector(sqrtint(lim\1+3)\2-1,n,4*n^2+8*n+1))) \\ Charles R Greathouse IV, Nov 13 2014

a(n)=if(n%2,(n+1)^2+1,n^2+4*n+1) \\ Charles R Greathouse IV, Nov 20 2014

a(n) = n^2 + O(n). - Charles R Greathouse IV, Nov 20 2014
G.f.: x*(5+8*x-6*x^2+x^4)/((1+x)^2*(1-x)^3). [Bruno Berselli, Dec 02 2014]
a(n) = (2*n*(n+5)-(2n+1)*(-1)^n+11)/2. [Bruno Berselli, Dec 02 2014]

Extended by Charles R Greathouse IV, Nov 13 2014

A181355 a(3n+1) = 4^(2^n), a(3n+2) = 3^(2^n), a(3*n+3) = 4^(2^n) - 3^(2^n).

Original entry on oeis.org

4, 3, 1, 16, 9, 7, 256, 81, 175, 65536, 6561, 58975, 4294967296, 43046721, 4251920575, 18446744073709551616, 1853020188851841, 18444891053520699775, 340282366920938463463374607431768211456, 3433683820292512484657849089281

Offset: 1

Jamel Ghanouchi, Jan 27 2011

Previous name was: Consider pairs of fractions (x,y) starting (4,3) and updated via z:=1/(1/x+1/y), x->x-z, y->y-z. The sequence shows the triples (numerator(x), numerator(y), numerator(x)-numerator(y)) after each update.

			(x=4,y=3) is shown as the first triple (4,3,1) in the sequence. This generates z=12/7 which generates the new pair (x,y) = (16/7,9/7) shown as (16,9,7). - _R. J. Mathar_, Feb 09 2011

x := 4 ; y := 3 ;
for loo from 1 to 7 do printf("%d, %d, %d, ", numer(x), numer(y), numer(x)-numer(y)) ; z := 1/(1/x+1/y) ; x := x-z ; y := y-z ; end do: # R. J. Mathar, Feb 09 2011

a(3*n+1) = 4^(2^n), a(3*n+2) = 3^(2^n), a(3*n+3) = 4^(2^n) - 3^(2^n). - Philippe Deléham , Oct 29 2013

Corrected by Philippe Deléham, Oct 29 2013

New name using Philippe Deléham's formula, Joerg Arndt, Nov 14 2014

A167541 a(n) = -(n - 4)(n - 5)(n - 12)/6.

Original entry on oeis.org

2, 5, 8, 10, 10, 7, 0, -12, -30, -55, -88, -130, -182, -245, -320, -408, -510, -627, -760, -910, -1078, -1265, -1472, -1700, -1950, -2223, -2520, -2842, -3190, -3565, -3968, -4400, -4862, -5355, -5880, -6438, -7030, -7657, -8320, -9020, -9758, -10535, -11352

Offset: 6

Jamel Ghanouchi, Nov 06 2009

Essentially the same as A111396.

The coefficient of x^(n-6) of the Polynomial B_n(x) defined in A135929.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A135929, A137276.

LinearRecurrence[{4,-6,4,-1},{2,5,8,10},50] (* Harvey P. Dale, May 27 2012 *)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x^6*(2 - 3*x)/(x - 1)^4.
a(n) = -A111396(n-12) for n > 11. - Bruno Berselli, Oct 02 2018

Minor edits by N. J. A. Sloane, Nov 09 2009

Definition simplified, sequence extended by R. J. Mathar, Nov 12 2009

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

Original entry on oeis.org

-2, -7, -15, -25, -35, -42, -42, -30, 0, 55, 143, 273, 455, 700, 1020, 1428, 1938, 2565, 3325, 4235, 5313, 6578, 8050, 9750, 11700, 13923, 16443, 19285, 22475, 26040, 30008, 34408, 39270, 44625, 50505, 56943, 63973, 71630, 79950, 88970, 98728, 109263, 120615

Offset: 8

Jamel Ghanouchi, Nov 06 2009

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A111373, A135929, A137276.

[Binomial(n-5,3)*(n-16)/4: n in [8..60]]; // G. C. Greubel, Jul 30 2022

Table[(n-5)*(n-6)*(n-7)*(n-16)/24, {n,8,60}] (* G. C. Greubel, Jun 15 2016 *)

[binomial(n-5,3)*(n-16)/4 for n in (8..60)] # G. C. Greubel, Jul 30 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^8*(-2+3*x)/(1-x)^5.
E.g.f.: (1/24)*( -3360 - 1800*x - 420*x^2 - 52*x^3 - 3*x^4 + (3360 - 1560*x + 300*x^2 - 28*x^3 + x^4)*exp(x) ). - G. C. Greubel, Jul 30 2022

Definition simplified, sequence extended by R. J. Mathar, Nov 12 2009

A167375 a(n)=3*a(n-1)-a(n-2) with a(0)=1, a(1)=3, a(2)=11.

Original entry on oeis.org

1, 3, 11, 30, 79, 207, 542, 1419, 3715, 9726, 25463, 66663, 174526, 456915, 1196219, 3131742, 8199007, 21465279, 56196830, 147125211, 385178803, 1008411198, 2640054791, 6911753175, 18095204734, 47373861027, 124026378347, 324705274014, 850089443695

Offset: 0

Jamel Ghanouchi, Nov 02 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1).

Cf. A135929, A138034.

I:=[1,3,11]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jun 26 2014

Join[{1},LinearRecurrence[{3,-1},{3,11},30]] (* Harvey P. Dale, Jun 25 2014 *)
CoefficientList[Series[(3 x^2 + 1)/(1 - 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 26 2014 *)
Table[3LucasL[2n+1]-Fibonacci[2n], {n,0,20}] (* Rigoberto Florez, Dec 24 2018 *)

a(n) = (-1)^n*A098150(n-1), n>0.
G.f.: (3*x^2+1)/(1-3*x+x^2).
a(n) = 3*L(2n+1)-F(2n), where F(n) is the n-th Fibonacci number and L(n) is the n-th Lucas number. - Rigoberto Florez, Dec 24 2018

Edited by R. J. Mathar, Nov 03 2009

A167536 a(n) = 3^(2^n) - 2^(2^n).

Original entry on oeis.org

1, 5, 65, 6305, 42981185, 1853015893884545, 3433683820274065740584139537665, 11790184577738583171520532579045597727214748217668409340885505

Offset: 0

Jamel Ghanouchi, Nov 06 2009

nonn

Cf. A118004.

[3^(2^n)-2^(2^n): n in [0..10]]; // Vincenzo Librandi Jun 15 2016

A167536 := proc(n) 3^(2^n)-2^(2^n) ; end proc: seq(A167536(n),n=1..10) ; # R. J. Mathar, Nov 07 2009

a[n_]:= 2^n; Table[3^a[n] - 2^a[n], {n, 0, 10}] (* G. C. Greubel, Jun 14 2016 *)
Table[3^(2^n) - 2^(2^n), {n, 0, 10}] (* Vincenzo Librandi, Jun 15 2016 *)

a(n)= A001047(2^n).

Edited by R. J. Mathar, Nov 11 2009

Offset changed from 1 to 0 from Vincenzo Librandi, Jun 15 2016

A167373 Expansion of (1+x)(3x+1)/(1+x+x^2).

Original entry on oeis.org

1, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1

Offset: 0

Jamel Ghanouchi, Nov 02 2009

Bisection of A138034.

Also row 2n of A137276 or A135929.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 22.

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

Cf. A135929, A138034, A137276.

A167373 := proc(n)
    option remember;
    if n < 4 then
        op(n+1,[1,3,-1,-2]) ;
    else
        procname(n-3) ;
    end if;
end proc:
seq(A167373(n),n=0..20) ; # R. J. Mathar, Feb 06 2020

CoefficientList[Series[(1 + x)*(3*x + 1)/(1 + x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2016 *)
LinearRecurrence[{-1,-1},{1,3,-1},120] (* Harvey P. Dale, Apr 05 2023 *)

G.f.: (1+x)*(3*x+1)/(1+x+x^2).
a(n) = a(n-3), n>4.
a(n) = - a(n-1) - a(n-2) for n>2.
a(n) = 4*sin(2*n*Pi/3)/sqrt(3)-2*cos(2*n*Pi/3) for n>0 with a(0)=1. - Wesley Ivan Hurt, Jun 12 2016

Edited by R. J. Mathar, Nov 03 2009
Further edited and extended by Simon Plouffe, Nov 23 2009
Recomputed by N. J. A. Sloane, Dec 20 2009

A167380 a(1)=1, a(2)=2, and continued periodically with 4, 5, 1, -4, -5, -1 .

Original entry on oeis.org

1, 2, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5, -1, 4, 5, 1, -4, -5

Offset: 1

Jamel Ghanouchi, Nov 02 2009

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

A167380 := proc(n) if n <= 2 then n; else op(1+ (n mod 6),[-4,-5,-1,4,5,1]) ; end if; end proc: seq(A167380(n),n=1..120) ; # R. J. Mathar, Nov 02 2009

PadRight[{1,2},120,{-5,-1,4,5,1,-4}] (* or *) Join[{1,2},LinearRecurrence[ {1,-1},{4,5},120]] (* Harvey P. Dale, Mar 08 2015 *)

a(1)=1. a(2)=2. a(6k-3)=4. a(6k-2)=5. a(6k-1)=1. a(6k)=-4. a(6k+1)=-5. a(6k+2)=-1.
From R. J. Mathar, Nov 03 2009: (Start)
a(n) = a(n-1) - a(n-2), n > 4.
G.f.: x*(1+x)*(3*x^2+1)/(1-x+x^2). (End)

Unrelated material removed, and values corrected according to definition, by R. J. Mathar, Nov 05 2009

A167386 a(n) = (-1)^nn(n+1)(2n-5)/6.

Original entry on oeis.org

1, -1, -2, 10, -25, 49, -84, 132, -195, 275, -374, 494, -637, 805, -1000, 1224, -1479, 1767, -2090, 2450, -2849, 3289, -3772, 4300, -4875, 5499, -6174, 6902, -7685, 8525, -9424, 10384, -11407, 12495, -13650, 14874, -16169, 17537, -18980, 20500, -22099, 23779

Offset: 1

Jamel Ghanouchi, Nov 02 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-4,-6,-4,-1).

[(-1)^n*n*(n+1)*(2*n-5)/6: n in [1..50]]; // Vincenzo Librandi, Aug 25 2011

Table[n (-1)^n*(n + 1) (2*n - 5)/6, {n, 60}] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)

a(n)=(-1)^n*n*(n+1)*(2*n-5)/6 \\ Charles R Greathouse IV, Aug 24 2011

a(n) = -4*a(n-1) - 6*a(n-2) - 4*a(n-3) - a(n-4).

G.f.: x*(1+3*x)/(1+x)^4.

Note that for a sequence in which every other term is zero, the OEIS policy is to omit those zeros. - N. J. A. Sloane, Nov 08 2009

Zeros removed by skipping even-indexed polynomials - R. J. Mathar, Nov 12 2009

A167387 a(n) = (-1)^(n+1) * n(n-1)(n-4)*(n+1)/12.

Original entry on oeis.org

1, -2, 0, 10, -35, 84, -168, 300, -495, 770, -1144, 1638, -2275, 3080, -4080, 5304, -6783, 8550, -10640, 13090, -15939, 19228, -23000, 27300, -32175, 37674, -43848, 50750, -58435, 66960, -76384, 86768, -98175, 110670, -124320, 139194, -155363, 172900

Offset: 2

Jamel Ghanouchi, Nov 02 2009

The coefficient of [x^4] of the Polynomial B_{2n}(x) defined in A137276.

Essentially the same as A052472.

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (-5,-10,-10,-5,-1).

Cf. A052472, A137276, A135929, A138034.

List([2..40], n-> (-1)^(n+1)*(n-4)*Binomial(n+1,3)/2); # G. C. Greubel, May 19 2019

[(-1)^(n+1)*n*(n-1)*(n-4)*(n+1)/12: n in [2..40]]; // Vincenzo Librandi, Jun 13 2016

Table[(-1)^(n+1)*(n+1)*n*(n-1)*(n-4)/12, {n, 2, 40}] (* G. C. Greubel, Jun 12 2016 *)
LinearRecurrence[{-5, -10, -10, -5, -1}, {1, -2, 0, 10, -35}, 40] (* Vincenzo Librandi, Jun 13 2016 *)

vector(40, n, n++; (-1)^(n+1)*(n-4)*binomial(n+1,3)/2) \\ G. C. Greubel, May 19 2019

[(-1)^(n+1)*(n-4)*binomial(n+1,3)/2 for n in (2..40)] # G. C. Greubel, May 19 2019

a(n) = -5*a(n-1) -10*a(n-2) -10*a(n-3) -5*a(n-4) -a(n-5).
G.f.: x^2*(1+3*x)/(1+x)^5.
E.g.f.: x^2*(6 + 2*x - x^2)*exp(-x)/12. - G. C. Greubel, May 19 2019

cp's OEIS Frontend

User: Jamel Ghanouchi

A245906 Numbers of the form 4n^2 + 1 or 4n^2 + 8n + 1.

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A181355 a(3*n+1) = 4^(2^n), a(3*n+2) = 3^(2^n), a(3*n+3) = 4^(2^n) - 3^(2^n).

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A167541 a(n) = -(n - 4)*(n - 5)*(n - 12)/6.

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A167543 a(n) = (n-5)*(n-6)*(n-7)*(n-16)/24.

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A167375 a(n)=3*a(n-1)-a(n-2) with a(0)=1, a(1)=3, a(2)=11.

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A167536 a(n) = 3^(2^n) - 2^(2^n).

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A167373 Expansion of (1+x)*(3*x+1)/(1+x+x^2).

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A181355 a(3n+1) = 4^(2^n), a(3n+2) = 3^(2^n), a(3*n+3) = 4^(2^n) - 3^(2^n).

A167541 a(n) = -(n - 4)(n - 5)(n - 12)/6.

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

A167373 Expansion of (1+x)(3x+1)/(1+x+x^2).

A167386 a(n) = (-1)^nn(n+1)(2n-5)/6.

A167387 a(n) = (-1)^(n+1) * n(n-1)(n-4)*(n+1)/12.