A107247 -id:A107247 - cp's OEIS frontend

A107239 Sum of squares of tribonacci numbers (A000073).

0, 0, 1, 2, 6, 22, 71, 240, 816, 2752, 9313, 31514, 106590, 360606, 1219935, 4126960, 13961456, 47231280, 159782161, 540539330, 1828631430, 6186215574, 20927817799, 70798300288, 239508933824, 810252920400, 2741065994769, 9272959837818, 31370198430718

Offset: 0

Jonathan Vos Post, May 17 2005

			a(7) = 71 = 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2 + 7^2

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(3) (1963), 71-74.
Z. Jakubczyk, Advanced Problems and Solutions, Fib. Quart. 51 (3) (2013) 185, H-715.
Eric Weisstein's World of Mathematics, Tribonacci Number
Index entries for linear recurrences with constant coefficients, signature (3,1,3,-7,1,-1,1).

Cf. A000073, A000213, A085697.

Cf. A107240, A107241, A107242, A107243, A107244, A107245, A107246, A107247.

R:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(1-x-x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)*(1-x)) )); // G. C. Greubel, Nov 20 2021

b:= proc(n) option remember; `if`(n<3, [n*(n-1)/2$2],
     (t-> [t, t^2+b(n-1)[2]])(add(b(n-j)[1], j=1..3)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 22 2021

Accumulate[LinearRecurrence[{1,1,1},{0,0,1},30]^2] (* Harvey P. Dale, Sep 11 2011 *)
LinearRecurrence[{3,1,3,-7,1,-1,1}, {0,0,1,2,6,22,71}, 30] (* Ray Chandler, Aug 02 2015 *)

@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 A107231(n): return sum(T(j)^2 for j in (0..n))
[A107239(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021

a(n) = T(0)^2 + T(1)^2 + ... + T(n)^2 where T(n) = A000073(n).
From R. J. Mathar, Aug 19 2008: (Start)
a(n) = Sum_{i=0..n} A085697(i).
G.f.: x^2*(1-x-x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)*(1-x)). (End)
a(n) = 1/4 - (1/11)*Sum_{R = RootOf(_Z^3-_Z^2-_Z-1)} ((3 + 7*_R + 5*_R^2)/(3*_R^2 - 2*_R - 1)*_R^(-n) - (1/44)*Sum{_R = RootOf(Z^3+_Z^2+3*_Z-1)} ((-1 - 2*_R - 9*_R^2)/(3*_R^2 + 2*_R + 3)*_R^(-n). - _Robert Israel, Mar 26 2010
a(n+1) = A000073(n)*A000073(n+1) + ( (A000073(n+1) - A000073(n-1))^2 - 1 )/4 for n>0 [Jakubczyk]. - R. J. Mathar, Dec 19 2013

A107246 Sum of squares of octanacci numbers (Fibonacci 8-step numbers).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, 345952, 1378208, 5490992, 21877296, 87163696, 347276080, 1383600944, 5512434480, 21962292529, 87500852554, 348615720590, 1388934122190, 5533708922574, 22047074027470

Offset: 0

Jonathan Vos Post, May 27 2005

Primes in this sequence include: a(8) = 2, a(17) = 280927. Semiprimes in this sequence include: a(9) = 6 = 2 * 3, a(10) = 22 = 2 * 11, a(11) = 86 = 2 * 43, a(13) = 1366 = 2 * 683, a(14) = 5462 = 2 * 2731, a(24) = 5512110374 = 2 * 2756055187, a(25) = 21961968423 = 3 * 7320656141, a(36) = 88177707994468342 = 2 * 44088853997234171.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4, 8, 14, 30, 60, 120, -266, -24, -38, -32, 120, -22, -50, -64, 136, 16, 30, 22, -68, 0, 10, 18, -28, 0, -6, -8, 14, 0, 0, -2, 2, 0, 0, 1, -1).

Cf. A079262, A107239-A107245, A107247-A107248.

Accumulate[LinearRecurrence[{1,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,1},40]^2] (* Harvey P. Dale, May 25 2014 *)
LinearRecurrence[{3, 2, 4, 8, 14, 30, 60, 120, -266, -24, -38, -32, 120, -22, -50, -64, 136, 16, 30, 22, -68, 0, 10, 18, -28, 0, -6, -8, 14, 0, 0, -2, 2, 0, 0, 1, -1},{0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, 345952, 1378208, 5490992, 21877296, 87163696, 347276080, 1383600944, 5512434480, 21962292529, 87500852554, 348615720590, 1388934122190, 5533708922574, 22047074027470, 87838639467470, 349961474550734, 1394295671696334, 5555069815204303, 22132178477202944, 88177707994792448},31] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_8(0)^2 + F_8(1)^2 + ... F_8(n)^2, where F_8(n) = A079262(n).

Corrected from a(16) on by R. J. Mathar, Aug 11 2009

A107240 Sum of squares of first n tribonacci numbers (A000213).

Original entry on oeis.org

1, 2, 3, 12, 37, 118, 407, 1368, 4617, 15642, 52891, 178916, 605325, 2047726, 6927407, 23435376, 79281105, 268206130, 907335091, 3069492092, 10384017717, 35128880742, 118840150983, 402033352264, 1360069089113, 4601080768074

Offset: 1

Jonathan Vos Post, May 14 2005

			a(6) = 1^2 + 1^2 + 1^2 + 3^2 + 5^2 + 9^2 = 118.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(3) (1963), 71-74.
Eric Weisstein's World of Mathematics, Tribonacci Number.
Index entries for linear recurrences with constant coefficients, signature (3, 1, 3, -7, 1, -1, 1).

Cf. A000213, A001654, A107239, A107241-A107247.

Accumulate[LinearRecurrence[{1,1,1},{1,1,1},30]^2] (* Harvey P. Dale, Nov 11 2011 *)
LinearRecurrence[{3, 1, 3, -7, 1, -1, 1},{1, 2, 3, 12, 37, 118, 407},26] (* Ray Chandler, Aug 02 2015 *)

a(n) = Sum_{i=1..n} A000213(i)^2.

a(n)= 3*a(n-1) +a(n-2) +3*a(n-3) -7*a(n-4) +a(n-5) -a(n-6) +a(n-7). G.f.: (x^3-x^2+3*x-1)*(1+x)^2/((x-1)*(x^3+x^2+3*x-1)*(x^3-x^2-x-1)). - R. J. Mathar, Aug 11 2009

A107243 Sum of squares of pentanacci numbers (A001591).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, 290416, 1122160, 4337009, 16762634, 64787534, 250400910, 967783566, 3740437902, 14456621263, 55874162432, 215950971648, 834640190272, 3225844698176, 12467736540480

Offset: 0

Jonathan Vos Post, May 19 2005

			a(0) = 0 = 0^2 since F_5(0) = A001591(0) = 0.
a(1) = 0 = 0^2 + 0^2
a(2) = 0 = 0^2 + 0^2 + 0^2
a(3) = 0 = 0^2 + 0^2 + 0^2 + 0^2
a(4) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2
a(5) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
a(6) = 6 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
a(7) = 22 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2
a(8) = 86 = 8^2 + 22
a(9) = 342 = 16^2 + 86

W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 3, 7, 14, -32, -2, 6, -4, -6, 10, 1, -1, 0, 1, -1).

Cf. A001591, A107239, A107242, A107244, A107245, A107246, A107247, A107248.

Accumulate[LinearRecurrence[{1,1,1,1,1},{0,0,0,0,1},30]^2] (* Harvey P. Dale, Jan 04 2015 *)
LinearRecurrence[{3, 2, 3, 7, 14, -32, -2, 6, -4, -6, 10, 1, -1, 0, 1, -1},{0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, 290416, 1122160},28] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_5(1)^2 + F_5(1)^2 + F_5(2)^2 + ... F_5(n)^2 where F_5(n) = A001591(n). a(0) = 0, a(n+1) = a(n) + A001591(n)^2.
a(n)= 3*a(n-1) +2*a(n-2) +3*a(n-3) +7*a(n-4) +14*a(n-5) -32*a(n-6) -2*a(n-7) +6*a(n-8) -4*a(n-9) -6*a(n-10) +10*a(n-11) +a(n-12) -a(n-13) +a(n-15) -a(n-16). [R. J. Mathar, Aug 11 2009]
G.f.: x^4*(x^10 +x^9 +x^7 +x^6 -6*x^5 -5*x^4 -3*x^3 -2*x^2 -x +1) / ((x -1)*(x^5 +x^4 +x^3 +3*x^2 +3*x -1)*(x^10 -x^9 -x^7 +x^6 -6*x^5 +3*x^4 +3*x^3 +2*x^2 +x +1)). - Colin Barker, May 08 2013

a(26) and a(27) corrected by R. J. Mathar, Aug 11 2009

A107241 Sum of squares of first n tetranacci numbers (A000288).

Original entry on oeis.org

1, 2, 3, 4, 20, 69, 238, 863, 3264, 12100, 44861, 166662, 619591, 2301800, 8551800, 31774561, 118060082, 438649107, 1629796276, 6055504952, 22499207241, 83595676570, 310599326171, 1154030334396, 4287794153932, 15931278338957

Offset: 1

Jonathan Vos Post, May 14 2005

Tetranacci numbers are also called Fibonacci 4-step numbers. a(n) is prime for n = 2, 3, 8, 26, ... a(n) is semiprime for n = 4, 6, 11, 13, ... a(10) = 12100 = 94^2 + 3264 = 110^2 = 2^2 * 5^2 * 11^2. For Fibonacci numbers (A000045) F(i) we have Sum_{i=1..n} F(i) = F(n)*F(n+1).

			a(1) = 1^2 = 1.
a(2) = 1^2 + 1^2 = 1.
a(3) = 1^2 + 1^2 + 1^2 = 3, prime.
a(4) = 1^2 + 1^2 + 1^2 + 1^2 = 4 = 2^2, semiprime.
a(5) = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 20.
a(6) = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2 = 69 = 3 * 23, semiprime.
a(8) = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2 + 13^2 + 25^2 = 863, prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3,2,2,6,-16,-2,6,-2,2,1,-1).

Cf. A000288, A107239, A107240, A107241, A107242, A107243, A107244, A107245, A107246, A107247, A107248.

T:= proc(n) option remember;
      if n=0 then 0
    elif n<5 then 1
    else add(T(n-j), j=1..4)
      fi; end:
seq( add(T(k)^2, k=1..n), n=1..30); # G. C. Greubel, Dec 18 2019

Accumulate[LinearRecurrence[{1,1,1,1},{1,1,1,1},30]^2] (* Harvey P. Dale, Feb 14 2012 *)
LinearRecurrence[{3,2,2,6,-16,-2,6,-2,2,1,-1}, {1,2,3,4,20,69,238,863,3264, 12100,44861}, 30] (* Ray Chandler, Aug 02 2015 *)
T[n_]:= T[n]= If[n == 0, 0, If[n < 5, 1, Sum[T[n-j], {j,4}]]]; a[n_]:= Sum[T[j]^2, {j,n}]; Table[a[n], {n, 30}] (* G. C. Greubel, Dec 18 2019 *)

@CachedFunction
def T(n):
    if (n==0): return 0
    elif (n<5): return 1
    else: return sum(T(n-j) for j in (1..4))
def a(n): return sum(T(j)^2 for j in (1..n))
[a(n) for n in (1..30)] # G. C. Greubel, Dec 18 2019

a(n) = Sum_{i=1..n} A000288(i)^2.
From R. J. Mathar, Aug 11 2009: (Start)
a(n) = 3*a(n-1) + 2*a(n-2) + 2*a(n-3) + 6*a(n-4) - 16*a(n-5) - 2*a(n-6) + 6*a(n-7) - 2*a(n-8) + 2*a(n-9) + a(n-10) - a(n-11).
G.f.: (1 - x - 5*x^2 - 11*x^3 - 8*x^4 - x^5 - x^6 - 7*x^7 + x^8 + 4*x^9)/((1 - x)*(1 - 3*x - 3*x^2 + x^3 + x^4)(1 + x + 2*x^2 + 2*x^3 - 2*x^4 + x^5 - x^6)). (End)

A107242 Sum of squares of tetranacci numbers (A001630).

Original entry on oeis.org

0, 0, 1, 5, 14, 50, 194, 723, 2659, 9884, 36780, 136636, 507517, 1885793, 7006962, 26034006, 96728470, 359395319, 1335332919, 4961420008, 18434129192, 68491926888, 254481427113, 945524491213, 3513091674982, 13052875206698

Offset: 0

Jonathan Vos Post, May 18 2005

			a(0) = 0 = 0^2,
a(1) = 0 = 0^2 + 0^2
a(2) = 1 = 0^2 + 0^2 + 1^2
a(3) = 5 = 0^2 + 0^2 + 1^2 + 2^2
a(4) = 14 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2
a(5) = 50 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2
a(6) = 194 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2 + 12^2
a(7) = 723 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2 + 12^2 + 23^2
a(8) = 2659 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2 + 12^2 + 23^2 + 44^2

W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.
Eric Weisstein's World of Mathematics, Tetranacci Number.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 2, 6, -16, -2, 6, -2, 2, 1, -1).

Cf. A001630, A107239, A107243, A107244, A107245, A107246, A107247, A107248.

Accumulate[LinearRecurrence[{1,1,1,1},{0,0,1,2},40]^2] (* or *) LinearRecurrence[{3,2,2,6,-16,-2,6,-2,2,1,-1},{0,0,1,5,14,50,194,723,2659,9884,36780},40] (* Harvey P. Dale, Aug 25 2013 *)

a(n) = F_4(1)^2 + F_4(1)^2 + F_4(2)^2 + ... F_4(n)^2 where F_4(n) = A001630(n). a(0) = 0, a(n+1) = a(n) + A001630(n)^2.

a(n)= 3*a(n-1) +2*a(n-2) +2*a(n-3) +6*a(n-4) -16*a(n-5) -2*a(n-6) +6*a(n-7) -2*a(n-8) +2*a(n-9) +a(n-10) -a(n-11). G.f.: x^2*(1+x)*(x^6-x^5-4*x^2+x+1)/((x-1) *(x^4+x^3-3*x^2-3*x+1) *(x^6-x^5+2*x^4-\ 2*x^3-2*x^2-x-1)). [R. J. Mathar, Aug 11 2009]

a(13) and a(23) corrected by R. J. Mathar, Aug 11 2009

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A107239 Sum of squares of tribonacci numbers (A000073).

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A107246 Sum of squares of octanacci numbers (Fibonacci 8-step numbers).

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A107240 Sum of squares of first n tribonacci numbers (A000213).

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A107243 Sum of squares of pentanacci numbers (A001591).

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A107241 Sum of squares of first n tetranacci numbers (A000288).

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A107242 Sum of squares of tetranacci numbers (A001630).

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