A030119 -id:A030119 - cp's OEIS frontend

A192951 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 3, 9, 20, 40, 74, 131, 225, 379, 630, 1038, 1700, 2773, 4511, 7325, 11880, 19252, 31182, 50487, 81725, 132271, 214058, 346394, 560520, 906985, 1467579, 2374641, 3842300, 6217024, 10059410, 16276523, 26336025, 42612643, 68948766

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + 3n - 1, with p(0,x)=1.  For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
...
The list of examples at A192744 is extended here; the recurrence is given by p(n,x) = x*p(n-1,x) + v(n), with p(0,x)=1, and the reduction of p(n,x) by x^2 -> x+1 is represented by u1 + u2*x:
...
If v(n)=        n, then u1=A001595, u2=A104161.
If v(n)=      n-1, then u1=A001610, u2=A066982.
If v(n)=     3n-1, then u1=A171516, u2=A192951.
If v(n)=     3n-2, then u1=A192746, u2=A192952.
If v(n)=     2n-1, then u1=A111314, u2=A192953.
If v(n)=      n^2, then u1=A192954, u2=A192955.
If v(n)=   -1+n^2, then u1=A192956, u2=A192957.
If v(n)=    1+n^2, then u1=A192953, u2=A192389.
If v(n)=   -2+n^2, then u1=A192958, u2=A192959.
If v(n)=    2+n^2, then u1=A192960, u2=A192961.
If v(n)=    n+n^2, then u1=A192962, u2=A192963.
If v(n)=   -n+n^2, then u1=A192964, u2=A192965.
If v(n)= n(n+1)/2, then u1=A030119, u2=A192966.
If v(n)= n(n-1)/2, then u1=A192967, u2=A192968.
If v(n)= n(n+3)/2, then u1=A192969, u2=A192970.
If v(n)=     2n^2, then u1=A192971, u2=A192972.
If v(n)=   1+2n^2, then u1=A192973, u2=A192974.
If v(n)=  -1+2n^2, then u1=A192975, u2=A192976.
If v(n)=  1+n+n^2, then u1=A027181, u2=A192978.
If v(n)=  1-n+n^2, then u1=A192979, u2=A192980.
If v(n)=  (n+1)^2, then u1=A001891, u2=A053808.
If v(n)=  (n-1)^2, then u1=A192981, u2=A192982.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (3, -2, -1, 1).

Cf. A000045, A192232, A192744.

F:=Fibonacci;; List([0..40], n-> F(n+4)+2*F(n+2)-(3*n+5)); # G. C. Greubel, Jul 12 2019

I:=[0, 1, 3, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-1*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011

F:=Fibonacci; [F(n+4)+2*F(n+2)-(3*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 3n - 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A171516 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192951 *)
(* Additional programs *)
LinearRecurrence[{3,-2,-1,1},{0,1,3,9},40] (* Vincenzo Librandi, Nov 16 2011 *)
With[{F=Fibonacci}, Table[F[n+4]+2*F[n+2]-(3*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,-1,-2,3]^n*[0;1;3;9])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(3*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [f(n+4)+2*f(n+2)-(3*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From Bruno Berselli, Nov 16 2011: (Start)
G.f.: x*(1+2*x^2)/((1-x)^2*(1 - x - x^2)).
a(n) = ((25+13*t)*(1+t)^n + (25-13*t)*(1-t)^n)/(10*2^n) - 3*n - 5 = A000285(n+2) - 3*n - 5  where t=sqrt(5). (End)
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (3*n+5). - G. C. Greubel, Jul 12 2019

A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 2, 6, 14, 30, 59, 110, 197, 343, 585, 983, 1634, 2695, 4420, 7220, 11760, 19116, 31029, 50316, 81535, 132061, 213827, 346141, 560244, 906685, 1467254, 2374290, 3841922, 6216618, 10058975, 16276058, 26335529, 42612115, 68948205, 111560915

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n(n+1)/2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000045, A192232, A192744, A192951.

F:=Fibonacci;; List([0..40], n-> F(n+4) +2*F(n+2) -(n^2+5*n+10)/2); # G. C. Greubel, Jul 11 2019

I:=[0, 1, 2, 6, 14]; [n le 5 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011

F:=Fibonacci; [F(n+4) +2*F(n+2) -(n^2+5*n+10)/2: n in [0..40]]; // G. C. Greubel, Jul 11 2019

q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n(n+1)/2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A030119 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192966 *)
LinearRecurrence[{4,-5,1,2,-1},{0,1,2,6,14},40] (* Vincenzo Librandi, Nov 16 2011 *)
Table[Fibonacci[n+4] +2*Fibonacci[n+2] -(n^2+5*n+10)/2, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(n^2+5*n+10)/2) \\ G. C. Greubel, Jul 11 2019

f=fibonacci; [f(n+4) +2*f(n+2) -(n^2+5*n+10)/2 for n in (0..40)] # G. C. Greubel, Jul 11 2019

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 - 2*x + 3*x^2 - x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n^2 + 5*n + 10)/2. - G. C. Greubel, Jul 11 2019

A210677 a(n) = a(n-1) + a(n-2) + n + 1, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 5, 10, 20, 36, 63, 107, 179, 296, 486, 794, 1293, 2101, 3409, 5526, 8952, 14496, 23467, 37983, 61471, 99476, 160970, 260470, 421465, 681961, 1103453, 1785442, 2888924, 4674396, 7563351, 12237779, 19801163, 32038976, 51840174, 83879186, 135719397, 219598621, 355318057

Offset: 0

Alex Ratushnyak, May 09 2012

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

Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
Cf. A030119: a(n)=a(n-1)+a(n-2)+n, a(0)=a(1)=1.
Cf. A210678: a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

LinearRecurrence[{3, -2, -1, 1}, {1, 1, 5, 10}, 39] (* Jean-François Alcover, Oct 05 2017 *)

From Colin Barker, Jun 30 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 - 2*x + 4*x^2 - 2*x^3)/((1 - x)^2*(1 - x - x^2)). (End)
E.g.f.: exp(x/2)*(25*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x)*(4 + x). - Stefano Spezia, Feb 24 2023

A210678 a(n) = a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 6, 12, 24, 43, 75, 127, 212, 350, 574, 937, 1525, 2477, 4018, 6512, 10548, 17079, 27647, 44747, 72416, 117186, 189626, 306837, 496489, 803353, 1299870, 2103252, 3403152, 5506435, 8909619, 14416087, 23325740, 37741862, 61067638, 98809537, 159877213, 258686789, 418564042

Offset: 0

Alex Ratushnyak, May 09 2012

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

Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
Cf. A030119: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=1.
Cf. A210677: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=1.

LinearRecurrence[{3,-2,-1,1},{1,1,6,12},40] (* Harvey P. Dale, Dec 10 2014 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,1},40][[;;,2]] (* Harvey P. Dale, Mar 19 2023 *)

From Colin Barker, Jun 30 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -2*x + 5*x^2 - 3*x^3)/((1 - x)^2*(1 - x - x^2)). (End)

A271388 a(n) = 4*a(n-1) + a(n-2) - n for n > 1, with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 6, 22, 89, 372, 1570, 6644, 28137, 119182, 504854, 2138586, 9059185, 38375312, 162560418, 688616968, 2917028273, 12356730042, 52343948422, 221732523710, 939274043241, 3978828696652, 16854588829826, 71397184015932, 302443324893529, 1281170483590022

Offset: 0

Ilya Gutkovskiy, Apr 06 2016

Index entries for linear recurrences with constant coefficients, signature (6,-8,2,1).

Cf. A030119, A033887, A048776, A098317.

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == 4 a[n - 1] + a[n - 2] - n}, a, {n, 28}]
LinearRecurrence[{6, -8, 2, 1}, {0, 1, 2, 6}, 29]
nxt[{n_,a_,b_}]:={n+1,b,4b+a-n-1}; NestList[nxt,{1,0,1},30][[;;,2]] (* Harvey P. Dale, Feb 07 2025 *)

x='x+O('x^99); concat(0, Vec(x*(1-4*x+2*x^2)/((1-x)^2*(1-4*x-x^2)))) \\ Altug Alkan, Apr 06 2016

a(n) = (3*fibonacci(3*n-2) + 2*n+3) >> 3; \\ Kevin Ryde, May 16 2021

G.f.: x*(1 - 4*x + 2*x^2)/((1 - x)^2*(1 - 4*x - x^2)).
E.g.f.: (1/80)*(10*exp(x)*(2*x + 3) - 3*(5 + 3*sqrt(5))*exp((2 - sqrt(5))*x) + 3*(3*sqrt(5) - 5)*exp((2 + sqrt(5))*x)).
a(n) = 6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4).
a(n) = (1/80)*(20*n - 3*(5 + 3*sqrt(5))*(2 - sqrt(5))^n + 3*(3*sqrt(5) - 5)*(2 + sqrt(5))^n + 30).
Lim_{n->infinity} a(n + 1)/a(n) = 2 + sqrt(5) = phi^3 = A098317, where phi is the golden ratio (A001622).
a(n) = (2*n + 3 + 3*A033887(n-1))/8. - R. J. Mathar, Mar 12 2017

A340326 a(n) = a(n-2) + (-1)^na(n-1) + n(1-(-1)^n) with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 5, 7, 8, 15, 7, 22, 3, 25, 0, 25, 1, 26, 5, 31, 8, 39, 7, 46, 3, 49, 0, 49, 1, 50, 5, 55, 8, 63, 7, 70, 3, 73, 0, 73, 1, 74, 5, 79, 8, 87, 7, 94, 3, 97, 0, 97, 1, 98, 5, 103, 8, 111, 7, 118, 3, 121, 0, 121, 1, 122, 5, 127, 8, 135, 7, 142, 3, 145, 0, 145, 1, 146, 5, 151

Offset: 0

José María Grau Ribas, Feb 07 2021

If n > 0 then a(2n-1) = A021067(n).

a(n) = 0 if and only if n == -1 (mod 12).

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-4,0,3,0,-1).

Cf. A030119, A065220.

a[0] = 1; a[1] = 1; a[n_] := a[n] =  a[n - 2] + (-1)^n a[n - 1] + n (1 - (-1)^n); Array[a,100]

G.f.: -(x^6+3*x^5-5*x^4-2*x^3+x^2-x-1)/(x^8-3*x^6+4*x^4-3*x^2+1). - Alois P. Heinz, Feb 07 2021

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A192951 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A210677 a(n) = a(n-1) + a(n-2) + n + 1, a(0) = a(1) = 1.

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A210678 a(n) = a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

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A271388 a(n) = 4*a(n-1) + a(n-2) - n for n > 1, with a(0) = 0, a(1) = 1.

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A340326 a(n) = a(n-2) + (-1)^n*a(n-1) + n*(1-(-1)^n) with a(0) = a(1) = 1.

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A340326 a(n) = a(n-2) + (-1)^na(n-1) + n(1-(-1)^n) with a(0) = a(1) = 1.