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.

Showing 1-5 of 5 results.

A081342 a(n) = (8^n + 2^n)/2.

Original entry on oeis.org

1, 5, 34, 260, 2056, 16400, 131104, 1048640, 8388736, 67109120, 536871424, 4294968320, 34359740416, 274877911040, 2199023263744, 17592186060800, 140737488388096, 1125899906908160, 9007199254872064, 72057594038190080, 576460752303947776, 4611686018428436480
Offset: 0

Views

Author

Paul Barry, Mar 18 2003

Keywords

Comments

Binomial transform of A034494.
5th binomial transform of {1, 0, 9, 0, 81, 0, 729, 0, ...}.

Links

Crossrefs

Cf. A074603, A025551, A081343.

Programs

Formula

a(n) = (8^n + 2^n)/2.
a(n) = 10*a(n-1) - 16*a(n-2), a(0)=1, a(1)=5.
G.f.: (1-5*x)/((1-2*x)*(1-8*x)).
E.g.f.: exp(5*x)*cosh(3*x).
a(n) = ((5+sqrt(9))^n + (5-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = A074603(n)/2. - Michel Marcus, Jan 09 2020

A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 1, 3, 12, 0, 1, 6, 25, 8, 16, 1, 10, 45, 40, 80, 0, 1, 15, 75, 121, 252, 48, 64, 1, 21, 119, 287, 644, 336, 448, 0, 1, 28, 182, 588, 1457, 1360, 1888, 256, 256, 1, 36, 270, 1092, 3033, 4176, 6240, 2304, 2304, 0, 1, 45, 390, 1890, 5925, 10801, 17780, 11680, 12160, 1280, 1024
Offset: 0

Views

Author

Clark Kimberling, Jul 05 2009

Keywords

Examples

			First six rows:
  1;
  1,  0;
  1,  1,  4;
  1,  3, 12,  0;
  1,  6, 25,  8, 16;
  1, 10, 48, 40, 80, 0;

Links

Crossrefs

Cf. A162514, A162515, A162517.
Cf. A000217, A026150, A084057, A125818, A199572.
For fixed k, the sequences P(n,k), for n=1,2,3,4,5, are A084057, A084059, A146963, A081342, A081343, respectively.

Programs

  • Magma
    m:=12;
p:= func< n,x | ((x+Sqrt(x+4))^n + (x-Sqrt(x+4))^n)/2 >;
R:=PowerSeriesRing(Rationals(), m+1);
T:= func< n,k | Coefficient(R!( p(n,x) ), n-k) >;
[T(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 09 2023
  • Mathematica
    P[n_, x_]:= P[n, x]= ((x+Sqrt[x+4])^n + (x-Sqrt[x+4])^n)/2;
T[n_, k_]:= Coefficient[Series[P[n, x], {x,0,n-k+1}], x, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 08 2020; Jul 09 2023 *)
  • SageMath
    def p(n,x): return ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2
def T(n,k):
    P. = PowerSeriesRing(QQ)
    return P( p(n,x) ).list()[n-k]
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 09 2023

Formula

P(n,x) = 2*x*P(n-1,x) - (x^2 -x -4)*P(n-2,x).
From G. C. Greubel, Jul 09 2023: (Start)
T(n, k) = [x^(n-k)] ( ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2 ).
T(n, 1) = A000217(n-1), n >= 1.
T(n, n) = A199572(n).
Sum_{k=0..n} T(n, k) = A084057(n).
Sum_{k=0..n} 2^k*T(n, k) = A125818(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A026150(n).
Sum_{k=0..n} (-2)^k*T(n, k) = A133343(n). (End)

A025551 a(n) = 3^n*(3^n + 1)/2.

Original entry on oeis.org

1, 6, 45, 378, 3321, 29646, 266085, 2392578, 21526641, 193720086, 1743421725, 15690618378, 141215033961, 1270933711326, 11438398618965, 102945573221778, 926510115949281, 8338590914403366, 75047317842209805, 675425859417626778
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A081342, A081343.

Programs

  • GAP
    List([0..20], n-> Binomial(3^n+1,2) ); # G. C. Greubel, Jan 08 2020
  • Magma
    [Binomial(3^n+1,2): n in [0..20]]; // G. C. Greubel, Jan 08 2020
  • Maple
    seq( binomial(3^n +1,2), n=0..20); # G. C. Greubel, Jan 08 2020
  • Mathematica
    LinearRecurrence[{12,-27}, {1,6}, 20] (* G. C. Greubel, Jan 08 2020 *)
Table[3^n(3^n+1)/2,{n,0,20}] (* Harvey P. Dale, Mar 13 2022 *)
  • PARI
    Vec( (1-6*x)/((1-3*x)*(1-9*x)) + O(x^66) ) \\ Joerg Arndt, Sep 01 2013
  • Sage
    [binomial(3^n+1,2) for n in (0..20)] # G. C. Greubel, Jan 08 2020

Formula

From Philippe Deléham, Jul 11 2005: (Start)
Binomial transform of A081342.
6th binomial transform of (1, 0, 9, 0, 81, 0, 729, 0, . . ).
Inverse binomial transform of A081343.
a(n) = 12*a(n-1) - 27*a(n-2), a(0) = 1, a(1) = 6.
G.f.: (1-6*x)/((1-3*x)*(1-9*x)).
E.g.f.: exp(7*x)*cosh(3*x). (End)
a(n) = ((6+sqrt(9))^n + (6-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = Sum_{k=1..3^n} k. - Joerg Arndt, Sep 01 2013

A152429 a(n) = (11^n + 5^n)/2.

Original entry on oeis.org

1, 8, 73, 728, 7633, 82088, 893593, 9782648, 107374753, 1179950408, 12973595113, 142680249368, 1569336258673, 17261966423528, 189877968549433, 2088639343496888, 22974941225731393, 252723895719373448
Offset: 0

Views

Author

Philippe Deléham, Dec 03 2008

Keywords

Comments

Binomial transform of A081343.
Inverse binomial transform of A143079.

Links

Crossrefs

Cf. A162516.

Programs

  • GAP
    List([0..20], n-> (11^n+5^n)/2); # G. C. Greubel, Jan 08 2020
  • Magma
    [(11^n+5^n)/2: n in [0..20]]; // Vincenzo Librandi, Jun 01 2011
  • Maple
    seq( (11^n+5^n)/2, n=0..20); # G. C. Greubel, Jan 08 2020
  • Mathematica
    LinearRecurrence[{16,-55}, {1,8}, 20] (* G. C. Greubel, Jan 08 2020 *)
  • PARI
    vector(21, n, (11^(n-1) + 5^(n-1))/2 ) \\ G. C. Greubel, Jan 08 2020
  • Sage
    [(11^n+5^n)/2 for n in (0..20)] # G. C. Greubel, Jan 08 2020

Formula

a(n) = 16*a(n-1) - 55*a(n-2), with a(0)=1, a(1)=8.
G.f.: (1-8*x)/(1 - 16*x + 55*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*8^(2k-n)*9^(n-k).
a(n) = ((8 + sqrt(9))^n + (8 - sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
E.g.f.: (exp(11*x) + exp(5*x))/2. - G. C. Greubel, Jan 08 2020

A045594 Numbers k that divide 10^k + 4^k.

Original entry on oeis.org

1, 2, 4, 7, 8, 16, 32, 49, 58, 64, 128, 136, 256, 343, 512, 1024, 1552, 1682, 2048, 2312, 2401, 2564, 4096, 6176, 8192, 11179, 16384, 16807, 24059, 32768, 32896, 39304, 48778, 65536, 78253, 117649, 131072, 150544, 168413, 183944, 262144, 524288
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Cf. A081343.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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