A165806 -id:A165806 - cp's OEIS frontend

A165808 Expansion of x(403+2967x+1047*x^2-x^3)/(1-x)^4.

403, 4579, 16945, 41917, 83911, 147343, 236629, 356185, 510427, 703771, 940633, 1225429, 1562575, 1956487, 2411581, 2932273, 3522979, 4188115, 4932097, 5759341, 6674263, 7681279, 8784805, 9989257, 11299051, 12718603, 14252329, 15904645, 17679967, 19582711

Offset: 1

A.K. Devaraj, Sep 29 2009

Old name was: As mentioned in short description of A165806, polynomials have the following unique property: let f(x) be a polynomial in x. Then f(x+k*f(x)) is congruent to 0 (mod(f(x)); here k belongs to N. The present case pertains to f(x) = x^3 + 2x + 11 when x is complex (2 + 3i). The quotient f(x+k*f(x))/f(x), for any given k, consists of two parts: a) a rational integer part and b) rational integer coefficient of sqrt(-1). This sequence pertains to a.

			f(x) = x^3 + 2*x + 11. When x = 2 + 3*i, we get f(x) = -31 + 15*i. x + f(x) = -29 + 18*i. f(-29 + 18*i) = 3752 + 39618*i. When this value is divided by (-31 + 15*i) we get 403 - 1083*i; needless to say, PARI takes care of necessary rationalization.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A165806, A165809.

LinearRecurrence[{4, -6, 4, -1}, {403, 4579, 16945, 41917}, 100](* G. C. Greubel, Apr 08 2016 *)

Vec((403+2967*x+1047*x^2-x^3)/(1-x)^4+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

From R. J. Mathar, Sep 30 2009: (Start)
a(n) = 1-13*n-321*n^2+736*n^3.
G.f.: x*(403+2967*x+1047*x^2-x^3)/(1-x)^4. (End)
From G. C. Greubel, Apr 08 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (1 + 402*x + 1887*x^2 + 736*x^3)*exp(x) - 1. [corrected by Jason Yuen, Aug 14 2025] (End)

More terms from R. J. Mathar, Sep 30 2009

Edited by Jon E. Schoenfield, Dec 12 2013

A168244 a(n) = 1 + 3n - 2n^2.

Original entry on oeis.org

1, 2, -1, -8, -19, -34, -53, -76, -103, -134, -169, -208, -251, -298, -349, -404, -463, -526, -593, -664, -739, -818, -901, -988, -1079, -1174, -1273, -1376, -1483, -1594, -1709, -1828, -1951, -2078, -2209, -2344, -2483, -2626, -2773, -2924, -3079, -3238, -3401, -3568, -3739, -3914, -4093, -4276, -4463, -4654, -4849

Offset: 0

A.K. Devaraj, Nov 21 2009

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients f(x + n*f(x))/f(x), as in A168235 and A168240. a(n) is the real part of the quotient at x = 1+sqrt(-5).
The imaginary part of the quotient is sqrt(5)*A045944(n).
As stated in short description of A168244 the quotient is in two parts: rational integers (cf. A168244) and rational integer multiples of sqrt(-5). It so happens that the sequence of rational integer coefficients of sqrt(-5) is A045944. - A.K. Devaraj, Nov 22 2009
This sequence contains half of all integers m such that -8*m +17 is an odd square. The other half are found in A091823 multiplied by -1. The squares resulting from A168244 are (4*n - 3)^2, those from A091823 are (4*n + 3)^2. - Klaus Purath, Jul 11 2021

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

Cf. A168235, A168240, A165806, A165808, A165809, A045944, A091823.

[1+3*n-2*n^2: n in [1..50]]; // Vincenzo Librandi, Jul 10 2012

Table[-(n + (n + 1)^2 - 3)/2, {n, -1, 200, 2}]  (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
LinearRecurrence[{3,-3,1},{2,-1,-8},60] (* Harvey P. Dale, Jun 06 2015 *)

a(n) = 1 + 3*n - 2*n^2; \\ Altug Alkan, Apr 09 2016

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 1 + x*(2-7*x+x^2)/(1-x)^3.
a(-n) = -A091823(n), a(0) = 1. - Michael Somos, May 11 2014
E.g.f.: (1 + x - 2*x^2)*exp(x). - G. C. Greubel, Apr 09 2016
a(n) = a(n-2) + (-2)*sqrt((-8)*a(n-1) + 17), n > 1. - Klaus Purath, Jul 08 2021

Edited, definition simplified, sequence extended beyond a(5) by R. J. Mathar, Nov 23 2009

a(0)=1 added by N. J. A. Sloane, Apr 09 2016

A168235 1+5n+7n^2.

Original entry on oeis.org

13, 39, 79, 133, 201, 283, 379, 489, 613, 751, 903, 1069, 1249, 1443, 1651, 1873, 2109, 2359, 2623, 2901, 3193, 3499, 3819, 4153, 4501, 4863, 5239, 5629, 6033, 6451, 6883, 7329, 7789, 8263, 8751, 9253, 9769, 10299, 10843, 11401, 11973, 12559, 13159, 13773

Offset: 1

A.K. Devaraj, Nov 21 2009

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients defined by f(x + n*f(x))/f(x). a(n) is the quotient at x=2.

See A168240 for x=3 or A168244 for x= 1+sqrt(-5).

			When x = 2, f(x) = 7. Hence at n=1, f( x + f(x))/f(x) = 13 = a(1).

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

Cf. A165806, A165808, A165809.

Table[1+5n+7n^2,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{13,39,79},60] (* Harvey P. Dale, Feb 07 2015 *)

a(n)=1+5*n+7*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(1)=13, a(2)=39, a(3)=79, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Feb 07 2015
From G. C. Greubel, Apr 09 2016: (Start)
G.f.: (1 + 10*x + 3*x^2)/(1-x)^3.
E.g.f.: (1 + 12*x + 7*x^2)*exp(x). (End)

Edited, definition simplified, sequence extended beyond a(8) by R. J. Mathar, Nov 23 2009

A166957 a(n) = 841n^3 + 261n^2 + 28*n + 1.

Original entry on oeis.org

1, 1131, 7829, 25141, 58113, 111791, 191221, 301449, 447521, 634483, 867381, 1151261, 1491169, 1892151, 2359253, 2897521, 3512001, 4207739, 4989781, 5863173, 6832961, 7904191, 9081909, 10371161, 11776993, 13304451

Offset: 0

A.K. Devaraj, Oct 25 2009

Polynomials in one variable have a certain property viz f(x+f(x)) == 0 (mod f(x)). This is true even when the polynomial is in two variables (not necessarily homogeneous). This sequence is a demonstration when the polynomial is x^3 + 2*x*y + y^2 (x = 2, y=3).

When x = 2 and y=3, f(x,y) = 29. Hence f((2 + 29), (3 + 29))/29 = 1131.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A165806, A165808, A165809.

List([0..40], n-> 1 +28*n +261*n^2 +841*n^3); # G. C. Greubel, Sep 02 2019

I:=[1131, 7829, 25141, 58113]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 01 2012

[1 +28*n +261*n^2 +841*n^3: n in [0..40]]; // G. C. Greubel, Sep 02 2019

seq(1 +28*n +261*n^2 +841*n^3, n=0..40); # G. C. Greubel, Sep 02 2019

CoefficientList[Series[(1131+3305*x+611*x^2-x^3)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 01 2012 *)
Table[841n^3+261n^2+28n+1,{n,30}] (* or *) LinearRecurrence[{4,-6,4,-1},{1131,7829,25141,58113},30] (* Harvey P. Dale, Apr 11 2013 *)

a(n) = ((2+n*29)^3 + 2*(2+n*29)*(3+n*29) + (3+n*29)^2)/29

[1 +28*n +261*n^2 +841*n^3 for n in (0..40)] # G. C. Greubel, Sep 02 2019

a(n) = 841*n^3 + 261*n^2 + 28*n + 1.
G.f.: x*(1131 + 3305*x + 611*x^2 - x^3)/(1-x)^4 . - R. J. Mathar, Sep 02 2011
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Jul 01 2012
E.g.f.: (1 + 1130*x + 2784*x^2 + 841*x^3)*exp(x). - G. C. Greubel, Apr 09 2016

Formula, description, editing, and program correction by Charles R Greathouse IV, Nov 04 2009

a(0)=1 added by N. J. A. Sloane, Apr 09 2016

A167190 a(n) = 1 + 85n + 2232n^2 + 15276*n^3.

Original entry on oeis.org

17594, 131307, 432796, 1013717, 1965726, 3380479, 5349632, 7964841, 11317762, 15500051, 20603364, 26719357, 33939686, 42356007, 52059976, 63143249, 75697482, 89814331, 105585452, 123102501, 142457134, 163741007, 187045776

Offset: 1

A.K. Devaraj, Oct 30 2009

As mentioned in A166957, polynomials in two variables, not necessarily homogeneous, also have a property similar to that in a single variable (cf. A165806, A165808 and A165809) viz f(x+k*f(x,y), y+k*f(x,y)) == 0 (mod f(x,y)). The quotient has two parts: a rational integer and a rational integer coefficient of sqrt(-1), when x belongs to Z(x = 5) and y is complex (sqrt(-1)). The polynomial considered is identical with that in A166957 viz x^3 + 2*x*y + y^2. The present sequence is only that of the rational integers and seq A167191 will consist of rational integer coefficients of sqrt(-1). Note: k belongs to N.

			When x = 5 and y = i, f(x,y) = x^3 + 2xy + y^2 = 124 + 10i. The quotient of f(x + f(x,y), y + f(x,y))/(124 + 10i) is 17594 + 2664i.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A165806, A165808, A165809, A166957.

List([0..40], n-> 1 +85*n +2232*n^2 +15276*n^3); # G. C. Greubel, Sep 01 2019

I:=[17594, 131307, 432796, 1013717]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012

seq(1 + 85*n + 2232*n^2 + 15276*n^3, n=1..40); # G. C. Greubel, Sep 01 2019

CoefficientList[Series[(17594+60931*x+13132*x^2-x^3)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)
Table[1 +85*n +2232*n^2 +15276*n^3, {n,40}] (* G. C. Greubel, Sep 01 2019 *)

vector(40, n, 1 +85*n +2232*n^2 +15276*n^3) \\ G. C. Greubel, Sep 01 2019

[1 +85*n +2232*n^2 +15276*n^3 for n in (0..40)] # G. C. Greubel, Sep 01 2019

G.f.: x*(17594 + 60931*x + 13132*x^2 - x^3)/(1-x)^4 . - R. J. Mathar, Sep 02 2011
a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - R. J. Mathar, Sep 02 2011
E.g.f.: (1 + 17593*x + 48060*x^2 + 15276*x^3)*exp(x) -1. - G. C. Greubel, Apr 09 2016

Extended beyond a(6) by R. J. Mathar, Nov 17 2009

A168240 a(n) = 13n^2 + 7n + 1.

Original entry on oeis.org

21, 67, 139, 237, 361, 511, 687, 889, 1117, 1371, 1651, 1957, 2289, 2647, 3031, 3441, 3877, 4339, 4827, 5341, 5881, 6447, 7039, 7657, 8301, 8971, 9667, 10389, 11137, 11911, 12711, 13537, 14389, 15267, 16171, 17101, 18057, 19039, 20047, 21081, 22141, 23227

Offset: 1

A.K. Devaraj, Nov 21 2009

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients defined by f(x + n*f(x))/f(x). a(n) is the quotient at x=3.

			f(x) = 13 when x = 3. Hence at n = 1, f(x + f(x))/f(x) = 21 = a(1).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A165806, A165808, A165809, A168235.

LinearRecurrence[{3, -3, 1}, {21, 67, 139}, 50] (* G. C. Greubel, Apr 09 2016 *)
Table[13n^2+7n+1,{n,50}] (* Harvey P. Dale, Mar 22 2019 *)

a(n)=13*n^2+7*n+1 \\ Charles R Greathouse IV, Jun 17 2017

From R. J. Mathar, Nov 23 2009: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(21+4*x+x^2)/(1-x)^3. (End)
E.g.f.: (13*x^2 + 20*x + 1)*exp(x). - G. C. Greubel, Apr 09 2016

Edited, definition simplified, sequence extended beyond a(8) by R. J. Mathar, Nov 23 2009

A167467 a(n) = 25n^3 - n(5*n+1)/2 + 1.

Original entry on oeis.org

23, 190, 652, 1559, 3061, 5308, 8450, 12637, 18019, 24746, 32968, 42835, 54497, 68104, 83806, 101753, 122095, 144982, 170564, 198991, 230413, 264980, 302842, 344149, 389051, 437698, 490240, 546827, 607609, 672736, 742358, 816625, 895687, 979694, 1068796

Offset: 1

A.K. Devaraj, Nov 05 2009

Also the real part of f(x+n*f(x,y,z), y+n*f(x,y,z), z+n*f(x,y,z))/f(x,y,z) for f(x,y,z) = x^3+y^2+z at x=(-1+i*sqrt(3))/2, y=i and z=5.
If f(x,y,z) is a trivariate polynomial, f(x+n*f(x,y,z),y+n*f(x,y,z),z+n*f(x,y,z)) is congruent to 0 (mod f(x,y,z)).
The ratio f(x+n*f,y+n*f,z+n*f)/f of these two functions is decomposed into the real part (this sequence here), and the imaginary part. The imaginary part is 2*n*i + sqrt(3)*A167469(n)*i, where i=sqrt(-1) is the imaginary unit.

			f(x +f(x,y,z), y + f(x,y,z), z + f(x,y,z)) = (23 + 2i + 6*sqrt(3)*i)* f(x,y,z) at n=1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A165806, A165808, A165809, A166715, A166957, A167190.

List([1..50], n-> 25*n^3 - n*(5*n+1)/2 + 1); # G. C. Greubel, Sep 01 2019

[25*n^3 - n*(5*n+1)/2 + 1: n in [1..50]]; // G. C. Greubel, Sep 01 2019

f := proc(x,y,z) x^3+y^2+z ; end proc:
A167467 := proc(n) local rho,a ,x,y,z; a := f(x+n*f(x,y,z),y+n*f(x,y,z),z+n*f(x,y,z))/f(x,y,z) ; rho := (-1+I*sqrt(3))/2 ; a := subs({x = rho, y=I,z=5},a) ; a := expand(a) ; Re(a) ; end:
seq(A167467(n),n=1..50) ; # R. J. Mathar, Nov 12 2009

LinearRecurrence[{4,-6,4,-1},{23,190,652,1559},50] (* Harvey P. Dale, Sep 28 2012 *)

a(n)=1+25*n^3-n*(5*n+1)/2 \\ Charles R Greathouse IV, Jul 07 2013

[25*n^3 - n*(5*n+1)/2 + 1 for n in (1..50)] # G. C. Greubel, Sep 01 2019

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(23 + 98*x + 30*x^2 - x^3)/(1-x)^4.
E.g.f.: (2 + 44*x + 145*x^2 + 50*x^3)*exp(x)/2 -1. - G. C. Greubel, Apr 09 2016

a(2) and a(3) corrected, definition simplified and sequence extended by R. J. Mathar, Nov 12 2009

A171216 a(n) = (4^(5*n+1) + 7)/11.

Original entry on oeis.org

373, 381301, 390451573, 399822410101, 409418147942773, 419244183493398901, 429306043897240473973, 439609388950774245347701, 450160014285592827236045173, 460963854628447055089710256501, 472026987139529784411863302656373

Offset: 1

A.K. Devaraj, Dec 05 2009

In A165806, A165808 & A165809 a congruence property of polynomial functions was demonstrated. In the present sequence a congruence property of exponential functions is demonstrated. Let the function be f(n) = 2^n + 7. Then f(n + k*phi(f(n))) is congruent to 0 mod(f(n)). This is a sequence of quotients generated by (f(n + k*phi f(n)))/f(n) when n = 2.

A. K. Devaraj, "Euler's generalisation of Fermat's theorem - a further generalisation" - Hawaii International conference on Mathematics & Statistics (2004). [ISSN 15503747]

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1025,-1024).

[(4^(5*n+1) + 7)/11 : n in [1..15]]; // Vincenzo Librandi, Oct 06 2011

(4^(5*Range[15]+1)+7)/11 (* Paolo Xausa, Mar 20 2024 *)

a(n)=(4^(5*n+1) + 7)/11 \\ Charles R Greathouse IV, Oct 05 2011

G.f. -x*(-373+1024*x) / ( (1024*x-1)*(x-1) ). - R. J. Mathar, Oct 08 2011

A165987 a(n) is the trace of the matrix f(X + nf(X))/f(X), where X is the 2 X 2 matrix [-13, 17; 31, 97] and f(x) = x^3 - 5x + 67.

Original entry on oeis.org

1099258818702, 8792791182238, 29674231047422, 70337212371066, 137375369109982, 237382335220982, 376951744660878, 562677231386482, 801152429354606, 1098970972522062, 1462726494845662, 1899012630282218, 2414423012788542, 3015551276321446, 3708991054837742

Offset: 1

A.K. Devaraj, Oct 03 2009

Old name was: As mentioned in the short description (cf. A165806 & A165808) polynomials have the property: f(x + k*f(x)) is congruent to 0 mod(f(x)). This is true even if the variable is a square matrix. For this sequence let X be a 2x2 matrix (X belongs to Z): col1:-13, 31;col2: 17, 97. Let the polynomial be X^3 -5X + 67. The present sequence is a sequence of traces of the matrices resulting from the division of f(X + k*f(X))/f(X). Here k belongs to N.

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

Cf. A165806, A165808.

with(LinearAlgebra):
f:= x-> x^3-5*x+67:
a:= n-> (X-> Trace(f(X+n*f(X)).f(X)^(-1)))(<<-13|17>, <31|97>>):
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 13 2024

G.f.: 2*(549309615337*x^3+2197877953721*x^2+549629409347*x+1)/(x-1)^4. - Alois P. Heinz, Mar 13 2024

a(5)-a(15) added and edited by Alois P. Heinz, Mar 13 2024

cp's OEIS Frontend

A165808 Expansion of x*(403+2967*x+1047*x^2-x^3)/(1-x)^4.

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A168244 a(n) = 1 + 3*n - 2*n^2.

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A168235 1+5*n+7*n^2.

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A166957 a(n) = 841*n^3 + 261*n^2 + 28*n + 1.

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A167190 a(n) = 1 + 85*n + 2232*n^2 + 15276*n^3.

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A168240 a(n) = 13*n^2 + 7*n + 1.

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A165808 Expansion of x(403+2967x+1047*x^2-x^3)/(1-x)^4.

A168244 a(n) = 1 + 3n - 2n^2.

A168235 1+5n+7n^2.

A166957 a(n) = 841n^3 + 261n^2 + 28*n + 1.

A167190 a(n) = 1 + 85n + 2232n^2 + 15276*n^3.

A168240 a(n) = 13n^2 + 7n + 1.

A167467 a(n) = 25n^3 - n(5*n+1)/2 + 1.

A165987 a(n) is the trace of the matrix f(X + nf(X))/f(X), where X is the 2 X 2 matrix [-13, 17; 31, 97] and f(x) = x^3 - 5x + 67.