A161007 -id:A161007 - cp's OEIS frontend

A133356 a(n) = 2a(n-1) + 16a(n-2) for n>1, a(0)=1, a(1)=1.

1, 1, 18, 52, 392, 1616, 9504, 44864, 241792, 1201408, 6271488, 31765504, 163874816, 835997696, 4293992448, 21963948032, 112631775232, 576686718976, 2955481841664, 15137951186944, 77563611840512, 397334442672128

Offset: 0

Philippe Deléham, Dec 21 2007

Binomial transform of A001026 (powers of 17), with interpolated zeros .

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,16).

First differences of A161007.

Cf. A001026, A098158, A161007.

[n le 2 select 1 else 2*(Self(n-1) +8*Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 15 2022

LinearRecurrence[{2,16},{1,1},30] (* Harvey P. Dale, Dec 12 2012 *)

Vec((1-x)/(1-2*x-16*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

A133356=BinaryRecurrenceSequence(2,16,1,1)
[A133356(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

G.f.: (1-x)/(1-2*x-16*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*17^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=17, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i<=j), A[i,j] = -1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = (4*i)^(n-1)*(4*i*ChebyshevU(n, -i/4) - ChebyshevU(n-1, -i/4)) = A161007(n) - A161007(n-1). - G. C. Greubel, Oct 15 2022

A123004 Expansion of g.f. x^2/(1 - 2x - 25x^2).

Original entry on oeis.org

0, 1, 2, 29, 108, 941, 4582, 32689, 179928, 1177081, 6852362, 43131749, 257572548, 1593438821, 9626191342, 59088353209, 358831489968, 2194871810161, 13360530869522, 81592856993069, 497198985724188, 3034219396275101

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 25).

Sequences of the form (m*i)^(n-2)*ChebyshevU(n-2, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), this sequence (m=5), A123005 (m=7), A123006 (m=11).

[n le 2 select n-1 else 2*Self(n-1) +25*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 12 2021

Rest@CoefficientList[Series[x^2/(1 -2*x -25*x^2), {x,0,40}], x]
Join[{a=0,b=1},Table[c=2*b+25*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

[(5*i)^(n-2)*chebyshev_U(n-2, -i/5) for n in [1..30]] # G. C. Greubel, Jul 12 2021

a(n) = 2*a(n-1) + 25*a(n-2).
a(n+1) = ((1+sqrt(26))^n - (1-sqrt(26))^n)/(2*sqrt(26)). - Rolf Pleisch, Jul 06 2009
a(n) = (5*i)^(n-2)*ChebyshevU(n-2, -i/5). - G. C. Greubel, Jul 12 2021

Definition replaced by generating function - the Assoc. Eds. of the OEIS, Mar 27 2010

A123005 Expansion of g.f. x^2/(1-2x-49x^2).

Original entry on oeis.org

0, 1, 2, 53, 204, 3005, 16006, 179257, 1142808, 11069209, 78136010, 698663261, 5225991012, 44686481813, 345446523214, 2880530655265, 22687940948016, 186521884004017, 1484752874460818, 12109078065118469, 96971046978817020

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,49).

Sequences of the form (m*i)^(n-1)*ChebyshevU(n-1, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), A123004 (m=5), this sequence (m=7), A123006 (m=11).

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) -49*Self(n-2): n in [1..31]]; // G. C. Greubel, Jul 12 2021

CoefficientList[Series[x^2/(1-2x-49x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 12 2020 *)

[(7*i)^(n-2)*chebyshev_U(n-2, -i/7) for n in [1..30]] # G. C. Greubel, Jul 12 2021

a(n) = 2*a(n-1) + 49*a(n-2).

a(n) = (7*i)^(n-2)*ChebyshevU(n-2, -i/7). - G. C. Greubel, Jul 12 2021

Definition replaced by generating function - the Assoc. Eds. of the OEIS, Mar 27 2010

A123006 Expansion of x^2/(1 -2x -121x^2).

Original entry on oeis.org

0, 1, 2, 125, 492, 16109, 91750, 2132689, 15367128, 288789625, 2437001738, 39817548101, 374512306500, 5566947933221, 56449884952942, 786500469825625, 8403437018957232, 111973430886815089, 1240762741067455250

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..800
Index entries for linear recurrences with constant coefficients, signature (2, 121).

Sequences of the form (m*i)^(n-1)*ChebyshevU(n-1, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), A123004 (m=5), A123005 (m=7), this sequence (m=11).

[n le 2 select n-1 else 2*Self(n-1) + 121*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 12 2021

Rest@CoefficientList[Series[x^2/(1 -2*x -121*x^2), {x,0,30}], x]

[(11*i)^(n-2)*chebyshev_U(n-2, -i/11) for n in [1..30]] # G. C. Greubel, Jul 12 2021

a(n) = 2*a(n-1) + 121*a(n-2).

a(n) = (11*i)^(n-2)*ChebyshevU(n-2, -i/11). - G. C. Greubel, Jul 12 2021

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).

Original entry on oeis.org

5, 7, 101, 11, 13, 269, 17, 19, 509, 23, 709, 821, 29, 31, 46957, 55399, 37, 168846239, 41, 43, 9177868096974864412935432937651459122761, 47, 485329129, 2789, 53, 3229, 3461, 59, 61, 1563353111, 139237612541, 67, 5021, 71, 73, 484639, 6221, 79, 6869, 83, 7549

Offset: 1

XU Pingya, Sep 24 2017

nonn

			For k = {1, 2, 3, 4, 5}, ((1 + sqrt(6))^k - (1 - sqrt(6))^k)/(2*sqrt(6)) = {1, 2, 9, 28, 101}. 101 is odd prime, so a(3) = 101.

Cf. A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914, A079773, A161007, A099134.

g[n_, k_] := ((1 + Sqrt[n])^k - (1 - Sqrt[n])^k)/(2Sqrt[n]);
Table[k = 3; While[! PrimeQ[Expand@g[2n, k]], k++]; Expand@g[2n, k], {n, 41}]

g(n,k) = ([0,1;2*n-1,2]^k*[0;1])[1,1]
a(n) = for(k=3,oo,if(ispseudoprime(g(n,k)),return(g(n,k)))) \\ Jason Yuen, Apr 12 2025

When 2*n + 3 = p is prime, a(n) = p.

cp's OEIS Frontend

A133356 a(n) = 2*a(n-1) + 16*a(n-2) for n>1, a(0)=1, a(1)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A123004 Expansion of g.f. x^2/(1 - 2*x - 25*x^2).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A123005 Expansion of g.f. x^2/(1-2*x-49*x^2).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A123006 Expansion of x^2/(1 -2*x -121*x^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2*n))^k - (1 - sqrt(2*n))^k)/(2*sqrt(2*n)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A133356 a(n) = 2a(n-1) + 16a(n-2) for n>1, a(0)=1, a(1)=1.

A123004 Expansion of g.f. x^2/(1 - 2x - 25x^2).

A123005 Expansion of g.f. x^2/(1-2x-49x^2).

A123006 Expansion of x^2/(1 -2x -121x^2).

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).