cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A106565 a(n) = 5*a(n-1) + 5*a(n-2) with a(0) = 0, a(1) = 5.

Original entry on oeis.org

0, 5, 25, 150, 875, 5125, 30000, 175625, 1028125, 6018750, 35234375, 206265625, 1207500000, 7068828125, 41381640625, 242252343750, 1418169921875, 8302111328125, 48601406250000, 284517587890625, 1665594970703125
Offset: 0

Views

Author

Roger L. Bagula, May 30 2005

Keywords

Links

Crossrefs

Cf. A057088.

Programs

  • Magma
    I:=[0,5]; [n le 2 select I[n] else 5*(Self(n-1) +Self(n-2)): n in [1..41]]; // G. C. Greubel, Sep 06 2021
  • Mathematica
    LinearRecurrence[{5,5}, {0,5}, 40] (* G. C. Greubel, Sep 06 2021 *)
  • Sage
    [5*lucas_number1(n, 5, -5) for n in (0..40)] # G. C. Greubel, Sep 06 2021

Formula

Equals 5*A057088(n). - T. D. Noe, Feb 17 2006
From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 5*a(n-1) + 5*a(n-2), n > 1; a(0)=0, a(1)=5.
G.f.: 5*x/(1-5*x-5*x^2). (End)
a(n) = (1/6)*5^((n+1)/2)*((1-(-1)^n)*Lucas(2*n) + (1+(-1)^n)*sqrt(5)*Fibonacci(2*n)). - G. C. Greubel, Sep 06 2021

Extensions

Name changed by G. C. Greubel, Sep 06 2021

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42
Offset: 0

Views

Author

Philippe Deléham, Nov 12 2011

Keywords

Examples

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Crossrefs

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

Formula

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A368156 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 3, 10, 14, 12, 5, 20, 41, 44, 29, 8, 40, 98, 148, 131, 70, 13, 76, 224, 408, 497, 376, 169, 21, 142, 482, 1044, 1542, 1588, 1052, 408, 34, 260, 1003, 2492, 4351, 5456, 4894, 2888, 985, 55, 470, 2026, 5684, 11359, 16790, 18400, 14672, 7813
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    2
   2    4    5
   3   10   14    12
   5   20   41    44    29
   8   40   98   148   131    70
  13   76  224   408   497   376   169
  21  142  482  1044  1542  1588  1052  408
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 14*x^2 + 12*x^3, so (T(4,k)) = (3,10,14,12), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A000129, (p(n,n-1)); A007482 (row sums), (p(n,1)); A077925 (alternating row sums), (p(n,-1)); A057088, (p(n,2)); A015523, (p(n,-2)); A015568, (p(n,3)); A180250, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 8*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A087603 a(n) = (1/8)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*8^k.

Original entry on oeis.org

1, 10, 155, 2100, 29525, 410750, 5731375, 79905000, 1114275625, 15537531250, 216660471875, 3021168937500, 42128015328125, 587444444843750, 8191485291484375, 114224297381250000, 1592774664844140625, 22210083004410156250, 309703436610529296875
Offset: 0

Views

Author

Benoit Cloitre, Oct 25 2003

Keywords

Comments

More generally a(n)=(1/x)*sum(k=0,n,binomial(n,k)*Fibonacci(k)*x^k) satisfies the recurrence formula a(n)=(x+2)*a(n-1)+(x^2-x-1)*a(n-2).

Links

Crossrefs

Cf. A014445, A057088, A015553.

Programs

  • Mathematica
    LinearRecurrence[{10,55},{1,10},30] (* Harvey P. Dale, Nov 26 2014 *)
  • PARI
    Vec(1/(1-10*x-55*x^2) + O(x^50)) \\ Colin Barker, Mar 30 2016

Formula

a(n) = 10*a(n-1)+55*a(n-2).
G.f.: -1/(-1+10*x+55*x^2). - R. J. Mathar, Dec 05 2007
a(n) = ((-(5-4*sqrt(5))^(1+n)+(5+4*sqrt(5))^(1+n)))/(8*sqrt(5)). - Colin Barker, Mar 30 2016

A087579 a(n) = (1/6)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*6^k.

Original entry on oeis.org

1, 8, 93, 976, 10505, 112344, 1203397, 12885152, 137979729, 1477507240, 15821470061, 169419470448, 1814178395353, 19426591805816, 208023907911765, 2227562425662784, 23853192734743457, 255424852222168392, 2735141407084907389, 29288451971122142480
Offset: 0

Views

Author

Benoit Cloitre, Oct 25 2003

Keywords

Links

Crossrefs

Cf. A014445, A057088, A015553.

Programs

Formula

a(n) = 8*a(n-1) + 29*a(n-2).
G.f.: 1 / (-29*x^2-8*x+1). - Colin Barker, Aug 08 2013

Extensions

More terms from Colin Barker, Aug 08 2013

A087584 a(n) = (1/7)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*7^k.

Original entry on oeis.org

1, 9, 122, 1467, 18205, 223992, 2762333, 34044669, 419657674, 5172750495, 63760719089, 785929242096, 9687552661513, 119411072879553, 1471889315038010, 18142857823403763, 223633182327192277, 2756555811704284776, 33977962780753446341, 418820453306656692885
Offset: 0

Views

Author

Benoit Cloitre, Oct 25 2003

Keywords

Links

Crossrefs

Cf. A014445, A057088, A015553.

Formula

a(n) = 9*a(n-1) + 41*a(n-2).
G.f.: 1 / (-41*x^2-9*x+1). - Colin Barker, Aug 08 2013

Extensions

More terms from Colin Barker, Aug 08 2013

A370175 a(n) = floor(x*a(n-1)) for n > 0 where x = (5+3*sqrt(5))/2, a(0) = 1.

Original entry on oeis.org

1, 5, 29, 169, 989, 5789, 33889, 198389, 1161389, 6798889, 39801389, 233001389, 1364013889, 7985076389, 46745451389, 273652638889, 1601990451389, 9378215451389, 54901029513889, 321396224826389, 1881486271701389, 11014412482638889, 64479493771701389
Offset: 0

Views

Author

Philippe Deléham, Mar 18 2024

Keywords

Comments

x = A090550 = 1 + 3*phi = 5.854101966..., where phi is the golden ratio.

Examples

			a(0) = 1, a(1) = floor(x) = 5 where x = (5+3*sqrt(5))/2.
a(2) = floor(5*x) = 29, a(3) = floor(29*x) = 169.

Links

Crossrefs

Cf. A057088, A090550, A370174.

Programs

  • Mathematica
    NestList[Floor[#*(5 + 3*Sqrt[5])/2] &, 1, 30] (* or *)
LinearRecurrence[{6, 0, -5}, {1, 5, 29}, 30] (* Paolo Xausa, May 25 2024 *)

Formula

a(n) = 6*a(n-1) - 5*a(n-3), a(0) = 1, a(1) = 5, a(2) = 29.
a(n) = 5*a(n-1) + 5*a(n-2) - 1.
a(n) = (4*(5-2*sqrt(5))*((5-3*sqrt(5))/2)^n + 4*(5+2*sqrt(5))*((5+3*sqrt(5))/2)^n + 5)/45.
G.f.: (1 - x - x^2)/(1 - 6*x + 5*x^3).
a(n) = Sum_{k = 0..n} A370174(n,k)*4^k.
a(n) = (8*A057088(n) + 4*A057088(n-1) + 1)/9.

A225799 a(n) = Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k).

Original entry on oeis.org

0, 11, 143, 3058, 55341, 1052755, 19717984, 371084087, 6973353387, 131101759514, 2464418392865, 46327530894271, 870879506447808, 16371134451297043, 307750614069672631, 5785211638097121890, 108752568228856901349, 2044371455527726003547, 38430858858805840293152
Offset: 0

Views

Author

John Molokach, Jul 27 2013

Keywords

Comments

This sequence is part of a family of Fibonacci-like sequences, where:
Sum_{k=0..n} binomial(n,k)*m^(n-k)*Fibonacci(n+k) produces a sequence whose terms are divisible by (m+1); m>=1.
A recurrence relation for a(n) (m not equal to zero) is:
a(n) = (m+3)*a(n-1) + (m^2+m-1)*a(n-2); a(0)=0, a(1)=m+1.
Notable values of m include:
m = 1: Fibonacci(3n),
m = 0: Fibonacci(2n) (using recurrence relation only - the sum above is undefined for m=0),
m = -1: the zero sequence,
m = -2: (-1)*Fibonacci(n), or A152163(n+2).
For any value of m, the sequence gives a(n*k) divisible by a(n); n>=1, k>=1, m not equal to -1 (zero is not divisible by zero).
Equivalent sequences are given by: Sum_{k=0..n} binomial(n,k) * (m+1)^k * Fibonacci(k).
When these sequences are divided by m+1, we obtain the family of sequences A057088, A015553, A087567, A087579, A087584, A087603, and so on.
Another interesting value of m, m = -3, gives a(2n-1)= -2 * 5^(n-1); a(2n)=0.

Links

Crossrefs

Cf. A000045, A027941, A152163, A014445, A057088, A015553, A087567, A087579, A087584, A087603.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 0, 25}]
FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 0, 25}]]
LinearRecurrence[{13,109},{0,11},30] (* Harvey P. Dale, Jul 31 2018 *)

Formula

a(n) = ((13 + 11*sqrt(5))^n - (13 - 11*sqrt(5))^n)/(2^n*sqrt(5)).
a(n) = 13*a(n-1) + 109*a(n-2); a(0)=0, a(1)=11.
G.f.: 11*x*/(1 - 13*x - 109*x^2). - Corrected by Georg Fischer, May 10 2019
Previous Showing 21-28 of 28 results.

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