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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A213578 Antidiagonal sums of the convolution array A213576.

Original entry on oeis.org

1, 4, 13, 34, 80, 174, 359, 712, 1371, 2580, 4768, 8684, 15629, 27852, 49225, 86390, 150704, 261530, 451795, 777360, 1332791, 2277864, 3882048, 6599064, 11191705, 18940564, 31992709, 53943562, 90807056, 152631750, 256190783
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Links

Crossrefs

Cf. A213576, A213500.

Programs

  • GAP
    List([1..40], n-> n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5)); # G. C. Greubel, Jul 05 2019
  • Magma
    [n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5): n in [1..40]]; // Vincenzo Librandi, Jul 05 2019
  • Mathematica
    b[n_]:= n; c[n_]:= Fibonacci[n];
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213576 *)
r[n_] := Table[t[n, k], {k,40}]  (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n,1,40}] (* A213577 *)
s[n_] := Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* alternate program *)
LinearRecurrence[{4,-4,-2,4,0,-1},{1,4,13,34,80,174},40] (* Harvey P. Dale, Jul 04 2019 *)
  • PARI
    vector(40, n, n*fibonacci(n+4)-2*(fibonacci(n+5)-n-5)) \\ G. C. Greubel, Jul 05 2019
  • Sage
    [n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5) for n in (1..40)] # G. C. Greubel, Jul 05 2019

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
G.f.: (1 + x^2)/(1 - 2*x + x^3)^2.
a(n) = n*F(n+4) - 2*(F(n+5) - n - 5), F = A000045. - Ehren Metcalfe, Jul 05 2019

A213580 Principal diagonal of the convolution array A213579.

Original entry on oeis.org

1, 5, 15, 35, 74, 146, 277, 511, 925, 1651, 2916, 5108, 8889, 15385, 26507, 45491, 77806, 132678, 225645, 382835, 648121, 1095075, 1846920, 3109800, 5228209, 8777261, 14716167, 24643331, 41220050, 68873786, 114964741, 191719783
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Links

Crossrefs

Cf. A000045, A213500, A213579.

Programs

  • GAP
    F:=Fibonacci;; List([1..40], n-> F(n+3) +n*F(n+2) -2*(n+1)); # G. C. Greubel, Jul 08 2019
  • Magma
    F:=Fibonacci; [F(n+3) + n*F(n+2) -2*(n+1): n in [1..40]]; // G. C. Greubel, Jul 08 2019
  • Mathematica
    (* First program *)
b[n_]:= Fibonacci[n]; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213579 *)
r[n_]:= Table[T[n, k], {k, 40}]
d = Table[T[n, n], {n, 1, 40}] (* A213580 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A053808 *)
(* Second program *)
Table[Fibonacci[n+3] + n*Fibonacci[n+2] -2*(n+1), {n, 40}] (* G. C. Greubel, Jul 08 2019 *)
  • PARI
    vector(40, n, f=fibonacci; f(n+3) +n*f(n+2) -2*(n+1)) \\ G. C. Greubel, Jul 08 2019
  • Sage
    f=fibonacci; [f(n+3) +n*f(n+2) -2*(n+1) for n in (1..40)] # G. C. Greubel, Jul 08 2019

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).
G.f.: x*(1 + x - x^2 - 3*x^3)/(1 - 2*x + x^3)^2.
a(n) = Fibonacci(n+3) + n*Fibonacci(n+2) - 2*(n+1). - G. C. Greubel, Jul 08 2019

A213583 Principal diagonal of the convolution array A213582.

Original entry on oeis.org

1, 9, 38, 120, 327, 819, 1948, 4482, 10085, 22341, 48930, 106236, 229075, 491175, 1048184, 2227782, 4718097, 9960921, 20970910, 44039520, 92273951, 192937179, 402652308, 838859850, 1744829437, 3623877549, 7516191578, 15569255172, 32212253355, 66571991631
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Links

Crossrefs

Cf. A213500, A213582.

Programs

  • GAP
    List([1..40], n-> (n+1)*(2^(n+2) -3*n-4)/2); # G. C. Greubel, Jul 08 2019
  • Magma
    [(n+1)*(2^(n+2) -3*n-4)/2: n in [1..40]]; // G. C. Greubel, Jul 08 2019
  • Mathematica
    (* First program *)
b[n_]:= 2^n - 1; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213582 *)
r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213583 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A156928 *)
(* Second program *)
LinearRecurrence[{7,-19,25,-16,4},{1,9,38,120,327},40] (* Harvey P. Dale, Apr 06 2013 *)
Table[(n+1)*(2^(n+2)-3*n-4)/2, {n,40}] (* G. C. Greubel, Jul 08 2019 *)
  • PARI
    Vec(x*(1 + 2*x - 6*x^2) / ((1 - x)^3*(1 - 2*x)^2) + O(x^40)) \\ Colin Barker, Nov 04 2017
  • PARI
    vector(40, n, (n+1)*(2^(n+2) -3*n-4)/2) \\ G. C. Greubel, Jul 08 2019
  • Sage
    [(n+1)*(2^(n+2) -3*n-4)/2 for n in (1..40)] # G. C. Greubel, Jul 08 2019

Formula

a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5).
G.f.: x*(1 + 2*x - 6*x^2) / ((1 - x)^3*(1 - 2*x)^2).
a(n) = (n+1)*(2^(n+2) - 3*n -4)/2. - Colin Barker, Nov 04 2017
E.g.f.: (4*(1+2*x)*exp(2*x) - (3*x^2+10*x+4)*exp(x))/2. - G. C. Greubel, Jul 08 2019

A213585 Principal diagonal of the convolution array A213584.

Original entry on oeis.org

1, 7, 22, 54, 116, 232, 443, 821, 1490, 2664, 4710, 8256, 14373, 24883, 42878, 73594, 125880, 214664, 365087, 619425, 1048666, 1771852, 2988362, 5031744, 8459401, 14201887, 23811238, 39873726, 66695420, 111440104, 186016835
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Links

Crossrefs

Cf. A000045, A213500, A213584.

Programs

  • GAP
    F:=Fibonacci;; List([1..40], n-> F(n+4) +n*F(n+3) -(4*n+3)) # G. C. Greubel, Jul 08 2019
  • Magma
    F:=Fibonacci; [F(n+4) +n*F(n+3) -(4*n+3): n in [1..40]]; // G. C. Greubel, Jul 08 2019
  • Mathematica
    (* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n+4] + n*Fibonacci[n+3] -4*n-3, {n, 40}] (* G. C. Greubel, Jul 08 2019 *)
  • PARI
    vector(40, n, f=fibonacci; f(n+4) +n*f(n+3) -(4*n+3)) \\ G. C. Greubel, Jul 08 2019
  • Sage
    f=fibonacci; [f(n+4) +n*f(n+3) -(4*n+3) for n in (1..40)] # G. C. Greubel, Jul 08 2019

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).
G.f.: x*(1 + 3*x - 2*x^2 - 4*x^3 - 2*x^4)/(1 - 2*x + x^3)^2.
a(n) = Fibonacci(n+4) + n*Fibonacci(n+3) - (4*n + 3). - G. C. Greubel, Jul 08 2019

A213748 Principal diagonal of the convolution array A213747.

Original entry on oeis.org

1, 16, 125, 758, 4071, 20424, 98185, 458506, 2096651, 9436172, 41941005, 184545294, 805298191, 3489644560, 15032352785, 64424443922, 274877775891, 1168230842388, 4947801800725, 20890719879190, 87960928124951
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Links

Crossrefs

Cf. A213747, A213500.

Programs

  • Mathematica
    (See A213747.)
LinearRecurrence[{12,-53,106,-96,32},{1,16,125,758,4071},30] (* Harvey P. Dale, Aug 15 2012 *)

Formula

a(n) = 12*a(n-1) - 53*a(n-2) + 106*a(n-3) - 96*a(n-4) + 32*a(n-5).
G.f.: f(x)/g(x), where f(x) = x*(1 + 4*x - 14*x^2) and g(x) = (1 - 2*x)*(1 - 5*x + 4*x^2)^2.
a(n) = 2 - 2^n + 4^n*(n-1) + n. - Colin Barker, Nov 07 2017

A213749 Antidiagonal sums of the convolution array A213747.

Original entry on oeis.org

1, 9, 46, 180, 603, 1827, 5164, 13878, 35905, 90189, 221274, 532584, 1261687, 2949255, 6815896, 15597738, 35389629, 79691985, 178258150, 396361980, 876609811, 1929380139, 4227858756, 9227469150, 20065550713, 43486544277, 93952410034, 202400334288
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Links

Crossrefs

Cf. A213747, A213500.

Programs

  • Mathematica
    (See A213747.)
LinearRecurrence[{9,-33,63,-66,36,-8},{1,9,46,180,603,1827},30] (* Harvey P. Dale, May 16 2013 *)
  • PARI
    Vec(x*(1 - 2*x^2) / ((1 - x)^3*(1 - 2*x)^3) + O(x^30)) \\ Colin Barker, Oct 30 2017

Formula

a(n) = 9*a(n-1) - 33*a(n-2) + 63*a(n-3) - 66*a(n-4) + 36*a(n-5) - 8*a(n-6).
G.f.: x*(1 - 2*x^2) / ((1 - x)^3*(1 - 2*x)^3).
a(n) = (1/2)*((1+n)*(4-2^(2+n) + n + 2^(1+n)*n)). - Colin Barker, Oct 30 2017

A213754 Principal diagonal of the convolution array A213753.

Original entry on oeis.org

1, 16, 111, 576, 2631, 11292, 46927, 191680, 775599, 3122076, 12531591, 50220912, 201088855, 804798268, 3220143903, 12882607872, 51534757599, 206148206268, 824612224663, 3298489794160, 13194045161031, 52776361000476
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Links

Crossrefs

Cf. A213753, A213500.

Programs

  • Mathematica
    (See A213753.)
  • PARI
    Vec(x*(1 + 5*x - 18*x^2 + 6*x^3 + 12*x^4) / ((1 - x)^3*(1 - 2*x)^2*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 07 2017

Formula

a(n) = -3*2^n + 3*2^(2*n) - n*2^(n+1) - n^2.
a(n) = 11*a(n-1) - 47*a(n-2) + 101*a(n-3) - 116*a(n-4) + 68*a(n-5) - 16*a(n-6) for n>5. Corrected by Colin Barker, Nov 07 2017
G.f.: f(x)/g(x), where f(x) = x*(1 + 5*x - 18*x^2 + 6*x^3 + 12*x^4) and g(x) = (1 - 4*x) * (1 - x)^3 * (1 - 2*x)^2.

A213755 Antidiagonal sums of the convolution array A213753.

Original entry on oeis.org

1, 9, 44, 160, 491, 1355, 3486, 8546, 20245, 46773, 106048, 236980, 523535, 1145935, 2489202, 5372534, 11532633, 24639513, 52426420, 111146280, 234877811, 494924179, 1040183174, 2181033290, 4563397341, 9529452605
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Links

Crossrefs

Cf. A213753, A213500.

Programs

Formula

a(n) = (1/6)*(84 - 21*2^(n+2) + 23*n + 9*n*2^(n+2) - 3*n^2 - 2*n^3).
a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(1 + x - 2*x^2 - 2*x^3) and g(x) = (1 - x)^4 * (1 - 2*x)^2.

A213757 Principal diagonal of the convolution array A213756.

Original entry on oeis.org

1, 14, 65, 214, 597, 1518, 3649, 8462, 19157, 42646, 93777, 204294, 441781, 949598, 2030849, 4324510, 9174069, 19397574, 40893265, 85981910, 180353621, 377485774, 788527425, 1644165294, 3422550037, 7113537398, 14763947729
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Links

Crossrefs

Cf. A213756, A213500.

Programs

Formula

a(n) = -2 + 2^(n+1) - 6*n + n*2^(n+2) - 3*n^2.
a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5).
G.f.: f(x)/g(x), where f(x) = x*(1 + 7*x - 14*x^2) and g(x) = (1 - 2*x)^2 (1 - x)^3.

A213758 Antidiagonal sums of the convolution array A213756.

Original entry on oeis.org

1, 9, 40, 130, 355, 871, 1994, 4360, 9245, 19205, 39356, 79934, 161415, 324755, 651870, 1306596, 2616609, 5237265, 10479280, 20964090, 41934571, 83876479, 167761330, 335532160, 671075045, 1342162141, 2684337764, 5368690550
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Links

Crossrefs

Cf. A213756, A213500.

Programs

Formula

a(n) = (1/6)*(-120 + 15*2^(n+3) - 81*n - 21*n^2 - 4*n^3).
a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
G.f.: f(x)/g(x), where f(x) = x*(1 + 3*x) and g(x) = (1 - 2*x)*(1 - x)^4.
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