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
- Raphael Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202.
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (5,-2,-2,-35,3,0,48,-11,7,-14,2,-1,1).
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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
-
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 *)
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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
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@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
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
- 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.)
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (7,-9,-7,-56,96,108,252,-162,-114,-318,126,-16,136,-36,12,-21,3,-1,1).
-
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
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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 *)
-
@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
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
- 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.)
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (7,-9,-7,-56,96,108,252,-162,-114,-318,126,-16,136,-36,12,-21,3,-1,1).
-
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
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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 *)
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@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
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