A147703 -id:A147703 - cp's OEIS frontend

A147721 a(n) = C(2,n) DELTA C(0,n).

1, 1, 1, 3, 4, 1, 11, 17, 7, 1, 41, 72, 40, 10, 1, 153, 301, 208, 72, 13, 1, 571, 1244, 1021, 446, 113, 16, 1, 2131, 5093, 4819, 2525, 813, 163, 19, 1, 7953, 20688, 22104, 13452, 5218, 1336, 222, 22, 1, 29681, 83481, 99192, 68568, 30986, 9586, 2042, 290, 25, 1

Offset: 0

Paul Barry, Nov 11 2008

Triangle T equal to [1,2,1,0,0,0,...] DELTA [1,0,0,0,...] for Deléham DELTA as in A084938.

T = A147720*A007318. Row sums are A147722.

			Triangle begins
    1;
    1,   1;
    3,   4,   1;
   11,  17,   7,   1;
   41,  72,  40,  10,   1;
  153, 301, 208,  72,  13,   1;

Indranil Ghosh, Rows 0..100, flattened

Cf. A129267, A007318, A147703.

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - 3*x)/(1 - 4*x + (1 + y)*x^2 - y*x), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017, after Philippe Deléham *)

Riordan array ((1-3x)/(1-4x+x^2), x(1-x)/(1-4x+x^2)).
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), n > 1. - Philippe Deléham, Feb 13 2012
G.f.: (1-3*x)/(1-4*x+(1+y)*x^2-y*x). - Philippe Deléham, Feb 13 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001835(n), A147722(n), A084120(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Feb 13 2012

A147841 a(n) = 11a(n-1) - 9a(n-2) with a(0)=1, a(1)=9.

Original entry on oeis.org

1, 9, 90, 909, 9189, 92898, 939177, 9494865, 95990922, 970446357, 9810991629, 99186890706, 1002756873105, 10137643587801, 102489267607866, 1036143151396317, 10475171256888693, 105901595463208770, 1070641008783298233, 10823936737447401633, 109427535032871733866, 1106287454724562457829, 11184314186674341431325

Offset: 0

Philippe Deléham, Nov 14 2008

G. C. Greubel, Table of n, a(n) for n = 0..990
Index entries for linear recurrences with constant coefficients, signature (11,-9).

Cf. A147703, A190872, A333344, A333345 (growth power).

A147841:= n-> simplify( 3^n*(ChebyshevU(n, 11/6) - (2/3)*ChebyshevU(n-1, 11/6)) ):
seq(A147841(n), n=0..25); # G. C. Greubel, May 28 2020

Table[3^n*(ChebyshevU[n, 11/6] - (2/3)*ChebyshevU[n-1, 11/6]), {n,0,25}] (* G. C. Greubel, May 28 2020 *)
LinearRecurrence[{11,-9},{1,9},30] (* Harvey P. Dale, Feb 28 2023 *)

a(n) = polcoeff(lift(('x-2)*Mod('x,'x^2-11*'x+9)^n), 1); \\ Kevin Ryde, Apr 11 2020

a(n) = Sum_{k=0..n} A147703(n,k)*8^k.
G.f.: (1-2*x)/(1 -11*x +9*x^2).
a(n) = 9*A333344(n-1) = A190872(n+1) - 2*A190872(n) = A333344(n) - A190872(n). - Kevin Ryde, Apr 11 2020
a(n) = 3^n*(ChebyshevU(n, 11/6) - (2/3)*ChebyshevU(n-1, 11/6)). - G. C. Greubel, May 28 2020
E.g.f.: exp(11*x/2)*(85*cosh(sqrt(85)*x/2) + 7*sqrt(85)*sinh(sqrt(85)*x/2))/85. - Stefano Spezia, Mar 02 2023

Entries corrected by Paolo P. Lava, Nov 18 2008

Terms a(18) onward added by G. C. Greubel, May 28 2020

A152174 a(n) = -2a(n-1)+4a(n-2), n>1 ; a(0) = 1, a(1) = -4.

Original entry on oeis.org

1, -4, 12, -40, 128, -416, 1344, -4352, 14080, -45568, 147456, -477184, 1544192, -4997120, 16171008, -52330496, 169345024, -548012032, 1773404160, -5738856448, 18571329536, -60098084864, 194481487872, -629355315200

Offset: 0

Philippe Deléham, Nov 27 2008

Signed version of A087206.

Index entries for linear recurrences with constant coefficients, signature (-2,4).

Cf. A087206, A147703.

LinearRecurrence[{-2,4},{1,-4},30] (* Harvey P. Dale, Feb 13 2023 *)

Vec((1-2*x)/(1+2*x-4*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

G.f.: (1-2x)/(1+2x-4x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*(-5)^k.
a(n) = (-1)^n * A087206(n).

A152167 a(n)=-a(n-1)+3*a(n-2), n>1 ; a(0)=1, a(1)=-3 .

Original entry on oeis.org

1, -3, 6, -15, 33, -78, 177, -411, 942, -2175, 5001, -11526, 26529, -61107, 140694, -324015, 746097, -1718142, 3956433, -9110859, 20980158, -48312735, 111253209, -256191414, 589951041, -1358525283, 3128378406, -7203954255, 16589089473

Offset: 0

Philippe Deléham, Nov 27 2008

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

Cf. A105476, A147703.

LinearRecurrence[{-1,3},{1,-3},30] (* Harvey P. Dale, Dec 24 2013 *)

Vec((1-2*x)/(1+x-3*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

G.f.: (1-2*x)/(1+x-3*x^2).
a(n) = (-1)^n*A105476(n+1).
a(n) = Sum{k=0..n} A147703(n,k)*(-4)^k.

a(19) corrected by Charles R Greathouse IV, Jan 11 2012

A152185 a(n) = -3a(n-1) + 5a(n-2), n > 1; a(0)=1, a(1)=-5.

Original entry on oeis.org

1, -5, 20, -85, 355, -1490, 6245, -26185, 109780, -460265, 1929695, -8090410, 33919705, -142211165, 596232020, -2499751885, 10480415755, -43940006690, 184222098845, -772366329985, 3238209484180, -13576460102465

Offset: 0

Philippe Deléham, Nov 28 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,5).

Cf. A152163, A152166, A152167, A152174.

LinearRecurrence[{-3, 5}, {1, -5}, 25] (* Paolo Xausa, Jan 19 2024 *)

Vec((1-2*x)/(1+3*x-5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

G.f.: (1-2x)/(1+3x-5x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*(-6)^k.
a(n) = (-1)^n*A152187(n). - Philippe Deléham, Nov 29 2008

A152223 a(n) = -4a(n-1) + 6a(n-2) for n > 1 with a(0) = 1 and a(1) = -6.

Original entry on oeis.org

1, -6, 30, -156, 804, -4152, 21432, -110640, 571152, -2948448, 15220704, -78573504, 405618240, -2093913984, 10809365376, -55800945408, 288059973888, -1487045568000, 7676542115328, -39628441869312, 204573020169216, -1056062731892736, 5451689048586240

Offset: 0

Philippe Deléham, Nov 29 2008

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

Cf. A147703.

a152223 n = a152223_list !! n
a152223_list = 1 : -6 : zipWith (-)
   (map (* 6) $ a152223_list) (map (* 4) $ tail a152223_list)
-- Reinhard Zumkeller, Jan 12 2012

LinearRecurrence[{-4, 6}, {1, -6}, 23] (* Bruno Berselli, Jan 12 2012 *)

Vec((1-2*x)/(1+4*x-6*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

G.f.: (1 - 2*x)/(1 + 4*x - 6*x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*(-7)^k.
a(n) = (1/2)*((-2 - sqrt(10))^n + (-2 + sqrt(10))^n) + (1/5)*sqrt(10)*((-2 - sqrt(10))^n - (-2 + sqrt(10))^n). - Bruno Berselli, Jan 12 2012

a(17)-a(23) corrected by Charles R Greathouse IV, Jan 12 2012

A152239 a(n) = -5a(n-1) + 7a(n-2) for n > 1 with a(0) = 1 and a(1) = -7.

Original entry on oeis.org

1, -7, 42, -259, 1589, -9758, 59913, -367871, 2258746, -13868827, 85155357, -522858574, 3210380369, -19711911863, 121032221898, -743144492531, 4562948015941, -28016751527422, 172024393748697, -1056239229435439

Offset: 0

Philippe Deléham, Nov 30 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-5,7).

Cf. A147703.

LinearRecurrence[{-5,7},{1,-7},20] (* Harvey P. Dale, Apr 14 2022 *)

Vec((1-2*x)/(1+5*x-7*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 17 2012

G.f.: (1 - 2*x)/(1 + 5*x - 7*x^2).

a(n) = Sum_{k=0..n} A147703(n,k)*(-8)^k.

Several terms corrected by Johannes W. Meijer, Aug 17 2010

A147838 a(n)=8a(n-1)-6a(n-2), a(0)=1, a(1)=6 .

Original entry on oeis.org

1, 6, 42, 300, 2148, 15384, 110184, 789168, 5652240, 40482912, 289949856, 2076701376, 14873911872, 106531086720, 763005222528, 5464855259904, 39140810744064, 280337354393088, 2007853970680320, 14380807639084032

Offset: 0

Philippe Deléham, Nov 14 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -6).

LinearRecurrence[{8,-6},{1,6},40] (* Harvey P. Dale, Mar 01 2012 *)

a(n)=Sum_{k, 0<=k<=n}A147703(n,k)*5^k . G.f.: (1-2x)/(1-8x+6*x^2).

a(n)= ((5+sqrt(10))/10)*(4+sqrt(10))^n + ((5-sqrt(10))/10)*(4-sqrt(10))^n [From Richard Choulet, Nov 20 2008]

A152594 a(n) = -5a(n-1)-2a(n-2), n>1 ; a(0)=1, a(1)=-1 .

Original entry on oeis.org

1, -1, 3, -13, 59, -269, 1227, -5597, 25531, -116461, 531243, -2423293, 11053979, -50423309, 230008587, -1049196317, 4785964411, -21831429421, 99585218283, -454263232573, 2072145726299, -9452202166349, 43116719379147

Offset: 0

Philippe Deléham, Dec 09 2008

Index entries for linear recurrences with constant coefficients, signature (-5,-2).

Cf. A052984, A147703.

LinearRecurrence[{-5,-2},{1,-1},30] (* Harvey P. Dale, Jun 18 2015 *)

G.f.: (1+4*x)/(1+5*x+2*x^2).
a(n) = Sum_{k=0..n-1} A147703(n,k)*(-2)^(n-k).
a(n) = (-1)^n*A052984(n-1), n>=1 ; a(0)=1.
E.g.f.: exp(-5*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jun 17 2025

A152596 a(n) = 7a(n-1) - 6a(n-2), n>1; a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 15, 87, 519, 3111, 18663, 111975, 671847, 4031079, 24186471, 145118823, 870712935, 5224277607, 31345665639, 188073993831, 1128443962983, 6770663777895, 40623982667367, 243743896004199, 1462463376025191, 8774780256151143, 52648681536906855, 315892089221441127

Offset: 0

Philippe Deléham, Dec 09 2008

Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A147703.

Table[MatrixPower[{{3,2},{3,4}},n][[1]][[1]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{7,-6},{1,3},30] (* Harvey P. Dale, Jul 27 2021 *)

G.f.: (1-4*x)/(1 - 7*x + 6*x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*2^(n-k).
a(n) = (1/5)*(3 + 2*6^n), with n>=0. - Paolo P. Lava, Dec 12 2008
E.g.f.: exp(x)*(3 + 2*exp(5*x))/5. - Stefano Spezia, Sep 30 2023

a(21)-a(23) from Stefano Spezia, Sep 30 2023

cp's OEIS Frontend

A147721 a(n) = C(2,n) DELTA C(0,n).

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A147841 a(n) = 11*a(n-1) - 9*a(n-2) with a(0)=1, a(1)=9.

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A152174 a(n) = -2*a(n-1)+4*a(n-2), n>1 ; a(0) = 1, a(1) = -4.

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A152167 a(n)=-a(n-1)+3*a(n-2), n>1 ; a(0)=1, a(1)=-3 .

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A152185 a(n) = -3*a(n-1) + 5*a(n-2), n > 1; a(0)=1, a(1)=-5.

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A152223 a(n) = -4*a(n-1) + 6*a(n-2) for n > 1 with a(0) = 1 and a(1) = -6.

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A152239 a(n) = -5*a(n-1) + 7*a(n-2) for n > 1 with a(0) = 1 and a(1) = -7.

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Extensions

A147838 a(n)=8*a(n-1)-6*a(n-2), a(0)=1, a(1)=6 .

A147841 a(n) = 11a(n-1) - 9a(n-2) with a(0)=1, a(1)=9.

A152174 a(n) = -2a(n-1)+4a(n-2), n>1 ; a(0) = 1, a(1) = -4.

A152185 a(n) = -3a(n-1) + 5a(n-2), n > 1; a(0)=1, a(1)=-5.

A152223 a(n) = -4a(n-1) + 6a(n-2) for n > 1 with a(0) = 1 and a(1) = -6.

A152239 a(n) = -5a(n-1) + 7a(n-2) for n > 1 with a(0) = 1 and a(1) = -7.

A147838 a(n)=8a(n-1)-6a(n-2), a(0)=1, a(1)=6 .

A152594 a(n) = -5a(n-1)-2a(n-2), n>1 ; a(0)=1, a(1)=-1 .

A152596 a(n) = 7a(n-1) - 6a(n-2), n>1; a(0)=1, a(1)=3.