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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A337283 a(n) = Sum_{i=0..n} i*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

Original entry on oeis.org

0, 0, 2, 5, 21, 101, 395, 1578, 6186, 23610, 89220, 333431, 1234343, 4536551, 16567157, 60172532, 217524468, 783111476, 2809027334, 10043413545, 35805255545, 127314522569, 451629771519, 1598650868766, 5647706073630, 19916305738030, 70117445671624, 246478579433947, 865201260035147
Offset: 0

Views

Author

N. J. A. Sloane, Sep 12 2020

Keywords

References

  • Raphael Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202.

Links

Crossrefs

Cf. A000073, A085697, A337282, A337284.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2*(1-x^2)*(2-7*x+7*x^2+3*x^3+9*x^4+7*x^5+x^6+x^7+x^8)/((1-x)*(1+x+x^2-x^3)*(1-3*x-x^2-x^3))^2 )); // G. C. Greubel, Nov 20 2021
  • Mathematica
    T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
a[n_]:= a[n]= Sum[j*T[j]^2, {j,0,n}];
Table[a[n], {n,0,30}] (* G. C. Greubel, Nov 20 2021 *)
  • PARI
    concat([0,0], Vec(x^2*(1+x)*(2 -7*x +7*x^2 +3*x^3 +9*x^4 +7*x^5 +x^6 + x^7 +x^8)/((1-x)*(1 +x +x^2 -x^3)^2*(1 -3*x -x^2 -x^3)^2) + O(x^30))) \\ Colin Barker, Sep 19 2020
  • Sage
    @CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A337283(n): return sum( j*T(j)^2 for j in (0..n) )
[A337283(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021

Formula

From Colin Barker, Sep 13 2020: (Start)
G.f.: x^2*(1 + x)*(2 - 7*x + 7*x^2 + 3*x^3 + 9*x^4 + 7*x^5 + x^6 + x^7 + x^8) / ((1 - x)*(1 + x + x^2 - x^3)^2*(1 - 3*x - x^2 - x^3)^2).
a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3) - 35*a(n-4) + 3*a(n-5) + 48*a(n-7) - 11*a(n-8) + 7*a(n-9) - 14*a(n-10) + 2*a(n-11) - a(n-12) + a(n-13) for n>12.
(End)
a(n) = Sum_{j=0..n} j*A085697(j). - G. C. Greubel, Nov 20 2021

A337284 a(n) = Sum_{i=1..n} (i-1)*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

Original entry on oeis.org

0, 1, 3, 15, 79, 324, 1338, 5370, 20858, 79907, 301917, 1127753, 4175945, 15347222, 56045572, 203563012, 735880196, 2649245173, 9502874215, 33976624115, 121128306995, 430701953720, 1527852568478, 5408197139806, 19106052817630, 67376379676855, 237205619596129, 833831061604429, 2926954896983117
Offset: 1

Views

Author

N. J. A. Sloane, Sep 12 2020

Keywords

References

  • R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202. (Note that this paper uses an offset for the tribonacci numbers that is different from that used in A000073).

Links

Crossrefs

Cf. A000073, A085697, A107239, A337282, A337283, A337285.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1-2*x+2*x^2+12*x^3+8*x^5+2*x^6+4*x^7+3*x^8+2*x^9)/((1-x)*(1-2*x-3*x^2-6*x^3+x^4+x^6)^2) )); // G. C. Greubel, Nov 22 2021
  • Mathematica
    T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]];
a[n_]:= a[n]= Sum[(j-1)*T[j]^2, {j,0,n}];
Table[a[n], {n,40}] (* G. C. Greubel, Nov 22 2021 *)
  • Sage
    @CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A337284(n): return sum( (j-1)*T(j)^2 for j in (0..n) )
[A337284(n) for n in (1..40)] # G. C. Greubel, Nov 22 2021

Formula

Schumacher (on page 194) gives two explicit formulas for a(n) in terms of tribonacci numbers.
From Colin Barker, Sep 14 2020: (Start)
G.f.: x^2*(1 - 2*x + 2*x^2 + 12*x^3 + 8*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 2*x^9) / ((1 - x)*(1 + x + x^2 - x^3)^2*(1 - 3*x - x^2 - x^3)^2)
a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3) - 35*a(n-4) + 3*a(n-5) + 48*a(n-7) - 11*a(n-8) + 7*a(n-9) - 14*a(n-10) + 2*a(n-11) - a(n-12) + a(n-13) for n>13.
(End)
a(n) = A337283(n) - A107239(n). - G. C. Greubel, Nov 22 2021

A141583 Squares of tribonacci numbers A000213.

Original entry on oeis.org

1, 1, 1, 9, 25, 81, 289, 961, 3249, 11025, 37249, 126025, 426409, 1442401, 4879681, 16507969, 55845729, 188925025, 639128961, 2162157001, 7314525625, 24744863025, 83711270241, 283193201281, 958035736849, 3241011678961
Offset: 0

Views

Author

R. J. Mathar, Aug 19 2008

Keywords

Comments

Partial sums are in A107240.
a(n) is also the number of total dominating sets in the (n-1)-ladder graph. - Eric W. Weisstein, Apr 10 2018

Links

Crossrefs

Cf. A000213, A085697, A107240.

Programs

  • Magma
    I:=[1,1,1,9,25,81]; [n le 6 select I[n] else 2*Self(n-1) + 3*Self(n-2) + 6*Self(n-3) - Self(n-4) - Self(n-6): n in [1..30]]; // Vincenzo Librandi, Dec 13 2012
  • Mathematica
    CoefficientList[Series[(1+x)^2*(1-3*x+x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
Table[RootSum[-1 - # - #^2 + #^3 &, 2 #^n - 4 #^(n + 1) + 3 #^(n + 2) &]^2/121, {n, 0, 20}] (* Eric W. Weisstein, Apr 10 2018 *)
LinearRecurrence[{2,3,6,-1,0,-1}, {1,1,9,25,81,289}, {0, 20}] (* Eric W. Weisstein, Apr 10 2018 *)
LinearRecurrence[{1,1,1},{1,1,1},40]^2 (* Harvey P. Dale, Aug 01 2021 *)
  • Sage
    @CachedFunction
def T(n): # A000213
    if (n<3): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A141583(n): return T(n)^2
[A141583(n) for n in (0..40)] # G. C. Greubel, Nov 22 2021

Formula

a(n) = (A000213(n))^2.
O.g.f.: (1+x)^2*(1-3*x+x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)).
a(n) = 2*a(n-1) + 3*a(n-2) + 6*a(n-3) - a(n-4) - a(n-6).

A337285 a(n) = Sum_{i=1..n} (i-1)^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

Original entry on oeis.org

0, 1, 5, 41, 297, 1522, 7606, 35830, 159734, 691175, 2911275, 11995471, 48573775, 193800376, 763577276, 2976338876, 11493413820, 44020618429, 167385941185, 632387189285, 2375420846885, 8876467428110, 33013780952786, 122261706093330, 451010242361106, 1657768413841731, 6073328651742855
Offset: 1

Views

Author

N. J. A. Sloane, Sep 12 2020

Keywords

References

  • R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202. (Note that this paper uses an offset for the tribonacci numbers that is different from that used in A000073.)

Links

Crossrefs

Cf. A000073, A085697, A107239, A337282, A337283, A337284, A337286.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1 -2*x+15*x^2+62*x^3-97*x^4+96*x^5+73*x^6-64*x^7-57*x^8-194*x^9-127*x^10-138*x^11 -55*x^12-12*x^13-9*x^14-4*x^15)/((1-x)*(1+x+x^2-x^3)^3*(1-3*x-x^2 -x^3)^3) )); // G. C. Greubel, Nov 22 2021
  • Mathematica
    T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
A337285[n_]:= Sum[j^2*T[j+1]^2, {j,0,n-1}];
Table[A337285[n], {n, 40}] (* G. C. Greubel, Nov 22 2021 *)
  • Sage
    @CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A337285(n): return sum( j^2*T(j+1)^2 for j in (0..n-1) )
[A337285(n) for n in (1..40)] # G. C. Greubel, Nov 22 2021

Formula

From G. C. Greubel, Nov 22 2021: (Start)
a(n) = A337286(n) - 2*A337283(n) + A107239(n).
a(n) = Sum_{j=0..n-1} j^2*A000073(j+1)^2.
G.f.: x^2*(1 -2*x +15*x^2 +62*x^3 -97*x^4 +96*x^5 +73*x^6 -64*x^7 -57*x^8 -194*x^9 -127*x^10 -138*x^11 -55*x^12 -12*x^13 -9*x^14 -4*x^15)/((1-x)*(1 +x +x^2 -x^3)^3*(1 -3*x -x^2 -x^3)^3). (End)

A337286 a(n) = Sum_{i=0..n} i^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

Original entry on oeis.org

0, 0, 4, 13, 77, 477, 2241, 10522, 47386, 204202, 860302, 3546623, 14357567, 57286271, 225714755, 879795380, 3397426356, 13012405492, 49478890936, 186932228945, 702169068945, 2623863676449, 9758799153349, 36140284390030, 133317609306766, 490032600916766, 1795262239190210, 6557012850772931
Offset: 0

Views

Author

N. J. A. Sloane, Sep 12 2020

Keywords

References

  • R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202. (Note that this paper uses an offset for the tribonacci numbers that is different from that used in A000073.)

Links

Crossrefs

Cf. A000073, A085697, A337282, A337283, A337284, A337285.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(4-15*x+22*x^2+83*x^3-90*x^4+11*x^5-128*x^6-207*x^7-224*x^8-233*x^9-162*x^10- 147*x^11-58*x^12-3*x^13-4*x^14-x^15)/((1-x)*(1+x+x^2-x^3)^3*(1-3*x-x^2-x^3)^3) )); // G. C. Greubel, Nov 22 2021
  • Mathematica
    T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
A337286[n_]:= Sum[j^2*T[j]^2, {j,0,n}];
Table[A337286[n], {n, 0, 50}] (* G. C. Greubel, Nov 22 2021 *)
  • Sage
    @CachedFunction
def T(n): # A000073
    if (n<2): return 0
    elif (n==2): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A337286(n): return sum( j^2*T(j)^2 for j in (0..n) )
[A337286(n) for n in (0..40)] # G. C. Greubel, Nov 22 2021

Formula

G.f.: x^2*(4 - 15*x + 22*x^2 + 83*x^3 - 90*x^4 + 11*x^5 - 128*x^6 - 207*x^7 - 224*x^8 - 233*x^9 - 162*x^10 - 147*x^11 - 58*x^12 - 3*x^13 - 4*x^14 - x^15)/((1-x)*(1 + x + x^2 - x^3)^3*(1 - 3*x - x^2 - x^3)^3). - G. C. Greubel, Nov 22 2021

A343125 Triangle T(k, n) = (n+3)*(k-n) - 4, k >= 2, 1 <= n <= k-1, read by rows.

Original entry on oeis.org

0, 4, 1, 8, 6, 2, 12, 11, 8, 3, 16, 16, 14, 10, 4, 20, 21, 20, 17, 12, 5, 24, 26, 26, 24, 20, 14, 6, 28, 31, 32, 31, 28, 23, 16, 7, 32, 36, 38, 38, 36, 32, 26, 18, 8, 36, 41, 44, 45, 44, 41, 36, 29, 20, 9, 40, 46, 50, 52, 52, 50, 46, 40, 32, 22, 10
Offset: 2

Views

Author

Russell Jay Hendel, Apr 06 2021

Keywords

Comments

T(k, n) is even if k is odd.
T(k, n) = T(k, n+1) for n = k/2 - 2 if k >= 6 is even.
T(k, n) = T(k, n+2) for n = (k-1)/2 - 2 if k >= 7 is odd.
For fixed n, T(k, n) is linear in k.
The T(k, j) contribute coefficients to a closed formula for the sum of the first n+1 squares of the k-generalized Fibonacci numbers, F(k, j) = A092921(k, j). See A343138 for sums of squares of F(k, j). See the Formula section for closed formula. Although other sequences occur in coefficients in the closed formula for sums of squares, they are linear in nature. All coefficient sequences are mentioned in the arXiv link. The closed formula generalizes results of Schumacher (see References) for the cases k=3 and k=4 with a uniform proof method (see arXiv link).

Examples

			Triangle T(k, n) begins:
   k \ n|  1  2  3  4  5  6  7  8  9  10 11
  ------+----------------------------------
   2    |  0
   3    |  4  1
   4    |  8  6  2
   5    | 12 11  8  3
   6    | 16 16 14 10  4
   7    | 20 21 20 17 12  5
   8    | 24 26 26 24 20 14  6
   9    | 28 31 32 31 28 23 16  7
  10    | 32 36 38 38 36 32 26 18  8
  11    | 36 41 44 45 44 41 36 29 20  9
  12    | 40 46 50 52 52 50 46 40 32 22 10
.
The following are the closed formulas for k = 3, 4 for A(k, n) = Sum_{m=0..n} F(k, m)^2, with F(k, n) = A092921(k, n), the k-generalized Fibonacci numbers, and A(k, n) = A343138(k, n), the sum of squares of F(k, n). These formulas are derived from the closed formula in the formula section. Of course further simplifications are possible. For k = 2, T(2, 1) = 0 so illustrations start with k = 3.
k | Formula
--+--------------------------------------------------------
3 | Sum_{m=0..n} F(3,m)^2 = (1/4)*(2*F(3,n)*F(3,n+2) + 4*F(3,n+1)*F(3,n+2) - (k - 2)*F(3,n)^2 - T(3,1)*F(3,n+1)^2 - T(3,2)*F(3,n+2)^2 + 1).
4 | Sum_{m=0..n} F(3,m)^2 = (1/6)*(-2*F(4,n)*F(4,n+1) + 2*F(4,n)*F(4,n+3) + 4*F(4,n+1)*F(4,n+3) + 6*F(4,n+2)*F(4,n+3) - (k-2)*F(4,n)^2 - T(4,1)*F(4,n+1)^2 - T(4, 2)*F(4,n+2)^2 - T(4,3)*F(4,n+3)^2 + 2).

References

  • Raphael Schumacher, How to Sum the Squares of the Tetranacci Numbers and the Fibonacci m-step Numbers, Fibonacci Quarterly, 57, (2019), 168-175.
  • Raphael Schumacher, Explicit Formulas for Sums Involving the Squares of the First n Tribonacci Numbers, Fibonacci Quarterly, 58 (2020), 194-202.

Links

Crossrefs

Cf. A028557, A085697, A092921, A140672, A343138.

Programs

  • Maple
    T := (k, n) -> (n + 3)*(k - n) - 4:
seq(print(seq(T(k, n), n=1..k-1)), k = 2..12); # Peter Luschny, Apr 02 2021
  • Mathematica
    Table[(n + 3) (k - n) - 4, {k, 2, 12}, {n, k - 1}] // Flatten (* Michael De Vlieger, Apr 06 2021 *)
  • PARI
    T(k,n)=(n + 3)*(k - n) - 4
for(k = 2,12,for(n = 1,k - 1, print1(T(k,n),", ")))
  • Sage
    flatten([[(n+3)*(k-n) -4 for n in (1..k-1)] for k in (2..15)]) # G. C. Greubel, Nov 22 2021

Formula

Let F(k, n) = A092921(k, n), the k-generalized Fibonacci numbers. Let A(k, n) = A343138(k, n) = Sum_{m=0..n} F(k, m)^2, the sum of the first m+1 k-generalized Fibonacci numbers. Then, for k >= 2, a closed formula for A(k, n) is:
A(k, n) = (1/(2*k-2)) * (Sum_{j=0..k-2, m=j+1..k-1} 2*(j+1)*(m-k+1) * F(k, n+j) * F(k, n+m)) - (k-2)*F(k, n)^2 - Sum_{j=1..k}(T(k, j) * F(k, n+j)^2) + (k-2)).
From G. C. Greubel, Nov 22 2021: (Start)
T(2*n-2, n) = A028557(n-2), n >= 2.
T(4*n-6, n) = 2*A140672(n-2), n >= 2. (End)
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