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 1-10 of 10 results.

A213787 a(n) = Sum_{1<=i

Original entry on oeis.org

0, 0, 0, 2, 17, 102, 518, 2442, 11010, 48444, 209979, 902132, 3854708, 16416204, 69769244, 296148174, 1256077725, 5324954250, 22567665834, 95626443110, 405154147310, 1716454353240, 7271524823255, 30804002164872, 130491325800072, 552779233930872, 2341634254967448, 9919384305913082, 42019349641680905
Offset: 0

Views

Author

N. J. A. Sloane, Jun 20 2012

Keywords

Links

Crossrefs

Cf. A000045, A190173.

Programs

  • Maple
    a:= n-> (Matrix(9, (i, j)-> `if`(i=j-1, 1, `if`(i=9,
         [1, 2, -14, -11, 40, 16, -29, -2, 6][j], 0)))^(n+3).
         <<0, -1, 0, 0, 0, 0, 2, 17, 102>>)[1, 1]:
seq (a(n), n=0..30);  # Alois P. Heinz, Jun 20 2012
  • Mathematica
    LinearRecurrence[{6, -2, -29, 16, 40, -11, -14, 2, 1}, {0, 0, 0, 2, 17, 102, 518, 2442, 11010}, 30] (* Jean-François Alcover, Feb 13 2016 *)
  • PARI
    x='x+O('x^50); concat([0,0,0], Vec((x^4+2*x^3-4*x^2-5*x-2)*x^3 / ((x+1) * (x^2-x-1) * (x^2+4*x-1) * (x^2-3*x+1) * (x^2+x-1)))) \\ G. C. Greubel, Mar 05 2017

Formula

G.f.: (x^4+2*x^3-4*x^2-5*x-2)*x^3 / ((x+1) * (x^2-x-1) * (x^2+4*x-1) * (x^2-3*x+1) * (x^2+x-1)). - Alois P. Heinz, Jun 20 2012

A203006 (n-1)-st elementary symmetric function of the first n Fibonacci numbers.

Original entry on oeis.org

1, 2, 5, 17, 91, 758, 10094, 215094, 7378716, 408057060, 36439600740, 5258207000160, 1226732478115680, 462844011818878560, 282472779283129656000, 278884771717353348456000, 445462025196173918554440000, 1151206495594319717393795136000
Offset: 1

Views

Author

Clark Kimberling, Dec 29 2011

Keywords

Comments

From R. J. Mathar, Oct 01 2016: (Start)
The k-th elementary symmetric functions of the integers F(j), j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:
1;
1, 1;
1, 2, 1;
1, 4, 5, 2;
1, 7, 17, 17, 6;
which is the unsigned version of A158472. This here is the first subdiagonal. The diagonal seems to be A003266. The 2nd column is A000071, the 3rd A190173, the 4th A213787. (End)

Examples

			0th elementary symmetric function: 1
1st e.s.f. of {1,1}: 1+1=2
2nd e.s.f. of {1,1,2}: 1*1+1*2+2*2=5

Links

Crossrefs

Cf. A000045.

Programs

  • Maple
    f:= proc(n) local x,P,i;
P:= mul(x+combinat:-fibonacci(i),i=1..n);
coeff(P,x,1)
end proc:
map(f, [$1..20]); # Robert Israel, Aug 18 2024
  • Mathematica
    f[k_] := Fibonacci[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 18}]  (* A203006 *)

A203245 Second elementary symmetric function of the first n terms of (1,2,3,5,8,...).

Original entry on oeis.org

2, 11, 41, 129, 376, 1048, 2850, 7635, 20273, 53537, 140912, 370128, 970978, 2545243, 6668697, 17467233, 45743336, 119779496, 313622210, 821130915, 2149841377, 5628507841, 14735867616, 38579395104, 101002803266, 264429800363
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2011

Keywords

Crossrefs

Cf. A190173.

Programs

  • Mathematica
    f[k_] := Fibonacci[k + 1]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]  (* A203245 *)

Formula

Empirical g.f.: -x*(x^3-x^2-3*x-2) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, Aug 15 2014

A213786 a(n) = Sum_{1<=iA020985(k).

Original entry on oeis.org

0, 0, 1, -1, 0, 2, -1, 1, 4, 8, 13, 7, 2, -2, 1, -3, 0, 4, 9, 3, 8, 14, 7, 13, 6, 0, -5, -1, 4, 10, 3, 9, 16, 24, 33, 23, 32, 42, 31, 41, 52, 64, 77, 63, 50, 38, 49, 37, 26, 16, 7, 15, 6, -2, 5, -3, 4, 12, 21, 11, 2, -6, 1, -7, 0, 8, 17, 7, 16, 26, 15, 25, 36
Offset: 0

Views

Author

N. J. A. Sloane, Jun 20 2012

Keywords

Comments

Suggested by A190173.

Links

Crossrefs

Cf. A020985, A190173.

Programs

  • Maple
    b:= proc(n) option remember; local r;
      `if`(n=0, 1, `if`(irem(n, 2, 'r')=0, b(r), b(r)*(-1)^r))
    end:
s:= proc(j) option remember; `if`(j<2, 0, s(j-1)+b(j-1)) end:
a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+b(n)*s(n)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 23 2012
  • Mathematica
    b[0] = 1; b[1] = 1; b[n_?EvenQ] := b[n] = b[n/2]; b[n_?OddQ] := b[n] = (-1)^((n-1)/2)*b[(n-1)/2]; a[n_] := Sum[b[i]*b[j], {i, 1, n-1}, {j, i+1, n}]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Nov 27 2014 *)
  • Python
    def A213786(n): return sum((-1 if (i&(i>>1)).bit_count()&1 else 1)*sum(-1 if (j&(j>>1)).bit_count()&1 else 1 for j in range(i+1,n+1)) for i in range(1,n+1)) # Chai Wah Wu, Feb 12 2023

A242300 a(n) = Sum_{0<=iA000032(k) is the k-th Lucas number.

Original entry on oeis.org

0, 2, 11, 35, 105, 292, 796, 2130, 5655, 14927, 39281, 103160, 270600, 709282, 1858291, 4867275, 12746265, 33375932, 87388676, 228801650, 599034975, 1568333527, 4106014561, 10749789360, 28143481680, 73680863042, 192899442971, 505018008755, 1322155461705
Offset: 0

Views

Author

J. M. Bergot, May 10 2014

Keywords

Comments

This sequence does for Lucas numbers what A190173 does for Fibonacci numbers.

Examples

			For L(12) = a(13) the sum is (L(13)-1)^2 + L(11) + 1 = 520^2 + 200 = 270600 and for L(13) = a(14) the sum is (L(14)-1)^2 + l(12) - 4 = 842^2 + 322 - 4 = 709282.

Links

Crossrefs

Cf. A000032, A190173.

Programs

  • PARI
    concat(0, Vec(-x*(x^3+5*x^2-3*x-2)/((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, May 13 2014
  • Sage
    [(lucas_number2(i+1,1,-1)-1)^2+lucas_number2(i-1,1,-1)+(5*(-1)^i-3)/2 for i in [0..50]] # Tom Edgar, May 13 2014

Formula

The sums are (1) for L(2*k): (L(2*k+1)-1)^2 + L(2*k-1) + 1 and (2) for L(2*k+1): (L(2*k+2)-1)^2 + L(2*k) - 4.
G.f.: -x*(x^3+5*x^2-3*x-2) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, May 12 2014
a(n) = (L(n+1)-1)^2 + L(n-1) + (5*(-1)^n-3)/2. - Colin Barker, May 13 2014

Extensions

Typo in a(18) fixed by Colin Barker, May 12 2014
More terms from Colin Barker, May 12 2014

A213626 a(n) = Sum_{0<=iA020985(m).

Original entry on oeis.org

0, 0, 1, -2, -2, 0, -5, -4, 0, 8, 21, 2, -10, -16, -15, -20, -20, -16, -7, -22, -14, 0, -21, -8, -28, -40, -45, -46, -42, -32, -49, -40, -24, 0, 33, -10, 22, 64, 11, 52, 104, 168, 245, 154, 78, 16, 65, 4, -44, -80, -105, -90, -114, -128, -123, -136, -132, -120
Offset: 0

Views

Author

Alois P. Heinz, Jun 23 2012

Keywords

Links

Crossrefs

Cf. A020985, A190173, A213786, A213787.

Programs

  • Maple
    b:= proc(n) option remember; local r;
      `if`(n=0, 1, `if`(irem(n, 2, 'r')=0, b(r), b(r)*(-1)^r))
    end:
s:= proc(j) option remember; `if`(j<0, 0, s(j-1)+b(j)       ) end:
t:= proc(k) option remember; `if`(k<1, 0, t(k-1)+b(k)*s(k-1)) end:
a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+b(n)*t(n-1)) end:
seq(a(n), n=0..100);
  • Mathematica
    b[n_] := b[n] = Module[{q, r}, If[n==0, 1, {q, r}=QuotientRemainder[n, 2]; If[r == 0, b[q], b[q]*(-1)^q]]];
s[j_] := s[j] = If[j < 0, 0, s[j-1] + b[j]];
t[k_] := t[k] = If[k < 1, 0, t[k-1] + b[k]*s[k-1]];
a[n_] := a[n] = If[n < 2, 0, a[n-1] + b[n]*t[n-1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)
  • Python
    def A213626(n): return sum((-1 if (i&(i>>1)).bit_count()&1 else 1)*sum((-1 if (j&(j>>1)).bit_count()&1 else 1)*sum(-1 if (k&(k>>1)).bit_count()&1 else 1 for k in range(j+1,n+1)) for j in range(i+1,n+1)) for i in range(n+1)) # Chai Wah Wu, Feb 12 2023

A213785 a(n) = Sum(P(i)*P(j), 1<=iA000129(k).

Original entry on oeis.org

0, 0, 2, 17, 113, 693, 4123, 24234, 141738, 827298, 4824716, 28127435, 163955435, 955642695, 5569991317, 32464523892, 189217679988, 1102842830628, 6427842380918, 37464218883749, 218357488856453, 1272680757553593, 7417727160997231, 43233682460792670
Offset: 0

Views

Author

N. J. A. Sloane, Jun 20 2012

Keywords

Comments

Suggested by A190173.

Links

Crossrefs

Cf. A000129, A213788, A190173.

Programs

  • Magma
    I:=[0, 0, 2, 17, 113]; [n le 5 select I[n] else 9*Self(n-1)-20*Self(n-2)+8*Self(n-3)+5*Self(n-4)-Self(n-5): n in [1..30]]; // Vincenzo Librandi, Jun 20 2012
  • Mathematica
    LinearRecurrence[{9,-20,8,5,-1},{0,0,2,17,113},30] (* Vincenzo Librandi, Jun 20 2012 *)

Formula

G.f.: x^2*(2-x)/((1-x)*(1-6*x+x^2)*(1-2*x-x^2)). [Bruno Berselli, Jun 20 2012]
a(n) = 9*a(n-1) -20*a(n-2) +8*a(n-3) +5*a(n-4) -a(n-5). - Vincenzo Librandi, Jun 20 2012

A242496 a(n)=sum_{j=0..n} sum_{i=0..j} F(i)*L(j), where F(n)=A000045(n) and L(n)=A000032(n).

Original entry on oeis.org

0, 1, 7, 23, 72, 204, 564, 1521, 4059, 10747, 28336, 74504, 195576, 512865, 1344063, 3521007, 9221688, 24148468, 63230860, 165555665, 433454835, 1134839091, 2971111392, 7778574288, 20364739632, 53315851969, 139583151799, 365434146311, 956720165544
Offset: 0

Views

Author

J. M. Bergot, May 16 2014

Keywords

Examples

			For n=5, 0*(2+1+3+4+7+11) + 1*(1+3+4+7+11) + 1*(3+4+7+11) + 2*(4+7+11) + 3*(7+11) + 5*11 = 204 = F(2*5+3) - L(n+2) + 0 = 233-29 = 204.

Links

Crossrefs

Cf. A190173, A000045, A000032, A242300.

Programs

  • Maple
    A242496 := proc(n)
    add(add(A000045(i)*A000032(j),i=0..j),j=0..n) ;
end proc: # R. J. Mathar, May 17 2014
  • Mathematica
    LinearRecurrence[{4,-2,-6,4,2,-1},{0,1,7,23,72,204},30] (* Harvey P. Dale, Oct 03 2020 *)
  • PARI
    F(n) = fibonacci(n)
L(n) = if(n==0, 2, F(2*n)/F(n))
vector(30, n, sum(i=0, n-1, sum(j=i, n-1, F(i)*L(j)))) \\ Colin Barker, May 16 2014

Formula

a(n) = A001519(n+2) - A000032(n+2) + A059841(n).
a(n) = L(n)*F(n+3) - L(n+2) + (1-3*(-1)^n)/2. - Colin Barker, May 18 2014
G.f.: -x*(3*x^2-3*x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, May 16 2014

Extensions

Two terms corrected, and more terms added by Colin Barker, May 16 2014
Formula corrected by Colin Barker, May 17 2014

A359045 a(n) = Sum_{1<=iA020985(m).

Original entry on oeis.org

0, 0, 0, -1, -2, -2, -4, -5, -4, 0, 8, -5, -12, -14, -16, -17, -20, -20, -16, -25, -22, -14, -28, -21, -34, -40, -40, -45, -46, -42, -52, -49, -40, -24, 0, -33, -10, 22, -20, 11, 52, 104, 168, 91, 28, -22, 16, -33, -70, -96, -112, -105, -120, -126, -128, -133
Offset: 0

Views

Author

Chai Wah Wu, Feb 12 2023

Keywords

Crossrefs

Cf. A020985, A190173, A213626, A213786, A213787.

Programs

  • Python
    def A359045(n): return sum((-1 if (i&(i>>1)).bit_count()&1 else 1)*sum((-1 if (j&(j>>1)).bit_count()&1 else 1)*sum(-1 if (k&(k>>1)).bit_count()&1 else 1 for k in range(j+1,n+1)) for j in range(i+1,n+1)) for i in range(1,n+1))

Formula

a(n) = A213626(n)-A213786(n).

A242558 a(n) = Sum_{j=0..n} Sum_{i=0..j} L(i)*F(j) where L(i)=A000032(i) and F(j)=A000045(j).

Original entry on oeis.org

0, 3, 9, 29, 80, 220, 588, 1563, 4125, 10857, 28512, 74792, 196040, 513619, 1345281, 3522981, 9224880, 24153636, 63239220, 165569195, 433476725, 1134874513, 2971168704, 7778667024, 20364889680, 53316094755, 139583544633, 365434781933
Offset: 0

Views

Author

J. M. Bergot, May 17 2014

Keywords

Examples

			For n=5, a(n) = F(2*5+3) - F(5+2) - 0 = 233 - 13 = 220.

Links

Crossrefs

Cf. A190173, A000045, A000032, A242300, A242496.

Programs

  • Mathematica
    LinearRecurrence[{4,-2,-6,4,2,-1},{0,3,9,29,80,220},30] (* Harvey P. Dale, Aug 15 2016 *)

Formula

a(n) = A001519(n+2) - A000045(n+2) - A059841(n).
G.f.: -x*(-3+3*x+x^2) / ( (x-1)*(1+x)*(x^2-3*x+1)*(x^2+x-1) ). - R. J. Mathar, May 17 2014
Showing 1-10 of 10 results.

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