A212336 -id:A212336 - cp's OEIS frontend

A097826 Partial sums of Chebyshev sequence S(n,11) = U(n,11/2) = A004190(n).

1, 12, 132, 1441, 15720, 171480, 1870561, 20404692, 222581052, 2427986881, 26485274640, 288910034160, 3151525101121, 34377866078172, 375005001758772, 4090677153268321, 44622443684192760, 486756203372852040, 5309695793417179681, 57919897524216124452

Offset: 0

Wolfdieter Lang, Aug 31 2004

Colin Barker, Table of n, a(n) for n = 0..963
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (12,-12,1).

Cf. A076765, A097784.

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

a:=[1,12,132];; for n in [4..30] do a[n]:=12*a[n-1]-12*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019

I:=[1,12,132]; [n le 3 select I[n] else 12*Self(n-1)-12*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, May 24 2019

LinearRecurrence[{12,-12,1}, {1,12,132}, 30] (* G. C. Greubel, May 24 2019 *)

Vec(1/((1-x)*(1-11*x+x^2)) + O(x^30)) \\ Colin Barker, Jun 15 2015

(1/((1-x)*(1 - 11*x + x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

a(n) = Sum_{k=0..n} S(k, 11), with S(k, 11) = U(k, 11/2) = A004190(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1 - 11*x + x^2)) = 1/(1 - 12*x + 12*x^2 - x^3).
a(n) = 12*a(n-1) - 12*a(n-2) + a(n-3) with n >= 2, a(-1)=0, a(0)=1, a(1)=12.
a(n) = 11*a(n-1) - a(n-2) + 1 with n >= 1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 11) - S(n, 11) - 1)/9.
a(n) = (2^(-n)*(-13*2^n + (65 - 18*sqrt(13))*(11 - 3*sqrt(13))^n + (11 + 3*sqrt(13))^n*(65 + 18*sqrt(13))))/117. - Colin Barker, Mar 06 2016

A097828 Partial sums of Chebyshev sequence S(n,13)= U(n,13/2)=A078362(n).

Original entry on oeis.org

1, 14, 182, 2353, 30408, 392952, 5077969, 65620646, 847990430, 10958254945, 141609323856, 1829962955184, 23647909093537, 305592855260798, 3949059209296838, 51032176865598097, 659469240043478424

Offset: 0

Wolfdieter Lang, Aug 31 2004

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (14, -14, 1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

a(n) = sum(S(k, 13), k=0..n) with S(k, 13)=U(k, 13/2)=A078362(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-13*x+x^2)) = 1/(1-14*x+14*x^2-x^3).
a(n) = 14*a(n-1)-14*a(n-2)+a(n-3) with n>=2, a(-1)=0, a(0)=1, a(1)=14.
a(n) = 13*a(n-1)-a(n-2)+1 with n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 13) - S(n, 13) -1)/11.

A097830 Partial sums of Chebyshev sequence S(n,16) = U(n,16/2) = A077412(n).

Original entry on oeis.org

1, 17, 272, 4336, 69105, 1101345, 17552416, 279737312, 4458244577, 71052175921, 1132376570160, 18046972946640, 287619190576081, 4583860076270657, 73054142029754432, 1164282412399800256, 18555464456367049665, 295723148889472994385, 4713014917775200860496

Offset: 0

Wolfdieter Lang, Aug 31 2004

Harvey P. Dale, Table of n, a(n) for n = 0..800
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (17, -17, 1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

LinearRecurrence[{17,-17,1},{1,17,272},30] (* or *) Accumulate[ ChebyshevU[Range[0,30],8]] (* Harvey P. Dale, Nov 09 2011 *)

Vec(1/((1-x)*(1-16*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 04 2016

a(n) = sum(S(k, 16), k=0..n) with S(k, 16) = U(k, 8) = A077412(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-16*x+x^2)) = 1/(1-17*x+17*x^2-x^3).
a(n) = 17*a(n-1)-17*a(n-2)+a(n-3) with n>=2, a(-1)=0, a(0)=1, a(1)=17.
a(n) = 16*a(n-1)-a(n-2)+1 with n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 16) - S(n, 16) -1)/14.
a(n) = (-6+(45-17*sqrt(7))*(8-3*sqrt(7))^n+(8+3*sqrt(7))^n*(45+17*sqrt(7)))/84. - Colin Barker, Mar 04 2016

A097829 Partial sums of Chebyshev sequence S(n,15)= U(n,15/2)=A078364(n).

Original entry on oeis.org

1, 16, 240, 3585, 53536, 799456, 11938305, 178275120, 2662188496, 39754552321, 593656096320, 8865086892480, 132382647290881, 1976874622470736, 29520736689770160, 440834175724081665, 6582991899171454816

Offset: 0

Wolfdieter Lang, Aug 31 2004

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16, -16, 1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

a(n) = sum(S(k, 15), k=0..n) with S(k, 15) = U(k, 15/2) = A078364(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-15*x+x^2)) = 1/(1-16*x+16*x^2-x^3).
a(n) = 16*a(n-1)-16*a(n-2)+a(n-3) with n>=2, a(-1)=0, a(0)=1, a(1)=16.
a(n) = 15*a(n-1)-a(n-2)+1 with n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 15) - S(n, 15) -1)/13.

A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 24, 552, 12673, 290928, 6678672, 153318529, 3519647496, 80798573880, 1854847551745, 42580695116256, 977501140122144, 22439945527693057, 515141245996818168, 11825808712399124808, 271478459139183052417, 6232178751488811080784, 143068632825103471805616

Offset: 0

Francesca Arici, Sep 11 2020

Michael De Vlieger, Table of n, a(n) for n = 0..735
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A097778 (first differences).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

CoefficientList[Series[x/((1 - x) (x^2 - 23 x + 1)), {x, 0, 18}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = A004254(n)*A004254(n+1)/5 = A160695(n+1)/5.
G.f.: x/((1-x)*(x^2-23*x+1)). - Alois P. Heinz, Sep 11 2020
From Klaus Purath, Jun 18 2025: (Start)
a(n) = (A004253(n+1)^2 - 1) / 15.
a(n) = (A030221(n)^2 - 1) / 35.
a(n) + a(n+1) = A004253(n+1)^2. (End)

a(13)-a(14) corrected and more terms added by Alois P. Heinz, Sep 11 2020

A212335 Expansion of 1/(1-22x+22x^2-x^3).

Original entry on oeis.org

1, 22, 462, 9681, 202840, 4249960, 89046321, 1865722782, 39091132102, 819048051361, 17160917946480, 359560228824720, 7533603887372641, 157846121406000742, 3307234945638642942, 69294087737005501041, 1451868607531476878920

Offset: 0

Bruno Berselli, Jun 12 2012

Partial sums of A092499 (after 0).

Bruno Berselli, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (22,-22,1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

m:=17; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-22*x+22*x^2-x^3)));

a:= n-> (<<0|1|0>, <0|0|1>, <1|-22|22>>^n. <<1, 22, 462>>)[1, 1]:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 15 2012

CoefficientList[Series[1/(1 - 22 x + 22 x^2 - x^3), {x, 0, 16}], x]
LinearRecurrence[{22,-22,1},{1,22,462},20] (* Harvey P. Dale, Nov 04 2017 *)

makelist(coeff(taylor(1/(1-22*x+22*x^2-x^3), x, 0, n), x, n), n, 0, 16);

PARI
```
Vec(1/(1-22*x+22*x^2-x^3)+O(x^17))
```

G.f.: 1/((1-x)*(1-21*x+x^2)).
a(n) = (((230-11*sqrt(437))*(21-sqrt(437))^n+(230+11*sqrt(437))*(21+sqrt(437))^n)/2^n-23)/437.
a(n) = a(-n-3) = 23*a(n-1)-23*a(n-2)+a(n-3).
a(n)*a(n+2) = a(n+1)*(a(n+1)-1).

A097831 Partial sums of Chebyshev sequence S(n,17)= U(n,17/2)=A078366(n).

Original entry on oeis.org

1, 18, 306, 5185, 87840, 1488096, 25209793, 427078386, 7235122770, 122570008705, 2076455025216, 35177165419968, 595935357114241, 10095723905522130, 171031371036761970, 2897437583719431361, 49085407552193571168

Offset: 0

Wolfdieter Lang, Aug 31 2004

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (18,-18,1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

LinearRecurrence[{18,-18,1},{1,18,306},20] (* Harvey P. Dale, Nov 20 2022 *)

a(n) = sum(S(k, 17), k=0..n) with S(k, 17) = U(k, 17/2) = A078366(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-17*x+x^2)) = 1/(1-18*x+18*x^2-x^3).
a(n) = 18*a(n-1)-18*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=18.
a(n) = 17*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 17) - S(n, 17) -1)/15.

A097832 Partial sums of Chebyshev sequence S(n,19)= U(n,19/2)=A078368(n).

Original entry on oeis.org

1, 20, 380, 7201, 136440, 2585160, 48981601, 928065260, 17584258340, 333172843201, 6312699762480, 119608122643920, 2266241630472001, 42938982856324100, 813574432639685900, 15414975237297708001

Offset: 0

Wolfdieter Lang, Aug 31 2004

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (20, -20, 1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

a(n) = sum(S(k, 19), k=0..n) with S(k, 19) = U(k, 19/2) = A078368(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-19*x+x^2)) = 1/(1-20*x+20*x^2-x^3).
a(n) = 20*a(n-1)-20*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=20.
a(n) = 19*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 19) - S(n, 19) -1)/17.

A097833 Partial sums of Chebyshev sequence S(n,20)= U(n,10)=A075843(n+1).

Original entry on oeis.org

1, 21, 420, 8380, 167181, 3335241, 66537640, 1327417560, 26481813561, 528308853661, 10539695259660, 210265596339540, 4194772231531141, 83685179034283281, 1669508808454134480, 33306490990048406320

Offset: 0

Wolfdieter Lang, Aug 31 2004

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (21, -21, 1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

LinearRecurrence[{21, -21, 1},{1, 21, 420},16] (* Ray Chandler, Aug 11 2015 *)

a(n) = sum(S(k, 20), k=0..n) with S(k, 20) = U(k, 10) = A075843(k+1) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-20*x+x^2)) = 1/(1-21*x+21*x^2-x^3).
a(n) = 20*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 20) - S(n, 20) -1)/18.
a(n) = 21*a(n-1)-21*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=21.
a(n) = (((10+3*sqrt(11))^(-n)*(33+10*sqrt(11)-11*(10+3*sqrt(11))^n*(1257+379*sqrt(11))+(10+3*sqrt(11))^(2*n)*(262680+79201*sqrt(11)))))/(198*(1257+379*sqrt(11))). - Colin Barker, Mar 03 2016

A097827 Partial sums of Chebyshev sequence S(n,12)= U(n,6)=A004191(n).

Original entry on oeis.org

1, 13, 156, 1860, 22165, 264121, 3147288, 37503336, 446892745, 5325209605, 63455622516, 756142260588, 9010251504541, 107366875793905, 1279392258022320, 15245340220473936, 181664690387664913

Offset: 0

Wolfdieter Lang, Aug 31 2004

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (13, -13, 1).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

a(n) = sum(S(k, 12), k=0..n) with S(k, 12) = U(k, 6) = A004191(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-12*x+x^2)) = 1/(1-13*x+13*x^2-x^3).
a(n) = 13*a(n-1)-13*a(n-2)+a(n-3) with n>=2, a(-1)=0, a(0)=1, a(1)=13.
a(n) = 12*a(n-1)-a(n-2)+1 with n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 12) - S(n, 12) -1)/10.

cp's OEIS Frontend

A097826 Partial sums of Chebyshev sequence S(n,11) = U(n,11/2) = A004190(n).

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A097828 Partial sums of Chebyshev sequence S(n,13)= U(n,13/2)=A078362(n).

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A097830 Partial sums of Chebyshev sequence S(n,16) = U(n,16/2) = A077412(n).

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A097829 Partial sums of Chebyshev sequence S(n,15)= U(n,15/2)=A078364(n).

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A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

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A212335 Expansion of 1/(1-22*x+22*x^2-x^3).

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A097831 Partial sums of Chebyshev sequence S(n,17)= U(n,17/2)=A078366(n).

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A097832 Partial sums of Chebyshev sequence S(n,19)= U(n,19/2)=A078368(n).

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A097833 Partial sums of Chebyshev sequence S(n,20)= U(n,10)=A075843(n+1).

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A212335 Expansion of 1/(1-22x+22x^2-x^3).