A107239 -id:A107239 - cp's OEIS frontend

A085697 a(n) = T(n)^2, where T(n) = A000073(n) is the n-th tribonacci number.

0, 0, 1, 1, 4, 16, 49, 169, 576, 1936, 6561, 22201, 75076, 254016, 859329, 2907025, 9834496, 33269824, 112550881, 380757169, 1288092100, 4357584144, 14741602225, 49870482489, 168710633536, 570743986576, 1930813074369, 6531893843049

Offset: 0

Emanuele Munarini, Jul 18 2003

In general, squaring the terms of a third-order linear recurrence with signature (x,y,z) will result in a sixth-order recurrence with signature (x^2 + y, x^2*y + z*x + y^2, x^3*z + 4*x*y*z - y^3 + 2*z^2, x^2*z^2 - x*y^2*z - z^2*y, z^2*y^2 - z^3*x, -z^4). - Gary Detlefs, Jan 10 2023

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
Index entries for linear recurrences with constant coefficients, signature (2,3,6,-1,0,-1).

Cf. A000073, A057597, A099463, A107239, A113300.

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

LinearRecurrence[{2,3,6,-1,0,-1},{0,0,1,1,4,16},30] (* Harvey P. Dale, Oct 26 2020 *)

t[0]:0$  t[1]:0$  t[2]:1$
t[n]:=t[n-1]+t[n-2]+t[n-3]$
makelist(t[n]^2,n,0,40); /* Emanuele Munarini, Mar 01 2011 */

@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 A085697(n): return T(n)^2
[A085697(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021

G.f.: x^2*( 1-x-x^2-x^3 )/( (1-3*x-x^2-x^3)*(1+x+x^2-x^3) ).
a(n+6) = 2*a(n+5) + 3*a(n+4) + 6*a(n+3) - a(n+2) - a(n).
a(n) = (-A057597(n-2) + 3*A057597(n-1) + 6*A057597(n) + 5*A113300(n-1) - A099463(n-2))/11. - R. J. Mathar, Aug 19 2008

Offset corrected to match A000073 by N. J. A. Sloane, Sep 12 2020

Name corrected to match corrected offset by Michael A. Allen, Jun 10 2021

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

A107245 Sum of squares of heptanacci numbers (Fibonacci 7-step numbers).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, 339616, 1347632, 5347632, 21219888, 84199984, 334092848, 1325649969, 5260075594, 20871578510, 82816815054, 328610657230, 1303901211854, 5173777051854, 20529140314318

Offset: 0

Jonathan Vos Post, May 20 2005

Primes include: a(7) = 2. Semiprimes include a(8) = 6 = 2 * 3, a(9) = 22 = 2 * 11, a(10) = 86 = 2 * 43, a(12) = 1366 = 2 * 683, a(13) = 5462 = 2 * 2731.

			a(0) = 0 = 0^2
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) = 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2
a(5) = 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2
a(6) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2
a(7) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
a(8) = 6 = 0^2 + 0^2 + 0^2+ 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
a(9) = 22 = 0^2 + 0^2 +0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2 = 2*11
a(10) = 86 = 8^2 + 22
a(11) = 342 = 16^2 + 86

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4, 7, 15, 30, 60, -131, -9, -8, 28, -11, -25, -32, 68, 5, 5, -10, 0, 5, 9, -14, 0, -1, 1, 0, 0, -1, 1).

Cf. A066178, A107239-A107244, A107246-A107248.

LinearRecurrence[{3, 2, 4, 7, 15, 30, 60, -131, -9, -8, 28, -11, -25, -32, 68, 5, 5, -10, 0, 5, 9, -14, 0, -1, 1, 0, 0, -1, 1},{0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, 339616, 1347632, 5347632, 21219888, 84199984, 334092848, 1325649969, 5260075594, 20871578510, 82816815054, 328610657230, 1303901211854, 5173777051854},30] (* Ray Chandler, Aug 02 2015 *)
Accumulate[LinearRecurrence[{1,1,1,1,1,1,1},{0,0,0,0,0,0,1},30]^2] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_7(0)^2 + F_7(1)^2 + ... F_7(n)^2, note that F_7(n) = A066178(n) with corrected offset (from leading zeros). a(0) = 0, a(n+1) = a(n) + F_7(n)^2.

a(14) inserted by R. J. Mathar, Aug 11 2009

A107247 Sum of squares of nonacci numbers (Fibonacci 9-step numbers).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 348503, 1390944, 5552544, 22166320, 88491056, 353269040, 1410299184, 5630100784, 22476064048, 89727075632, 358201316657, 1429983219018, 5708667195022, 22789694921422

Offset: 0

Jonathan Vos Post, May 29 2005

Primes in this sequence include: a(9) = 2, which is next? Semiprimes in this sequence include: a(10) = 6 = 2 * 3, a(11) = 22 = 2 * 11, a(12) = 86 = 2 * 43, a(14) = 1366 = 2 * 683, a(15) = 5462 = 2 * 2731, a(17) = 87382 = 2 * 43691, a(18) = 348503 = 37 * 9419, a(28) = 358201316657 = 71 * 5045088967.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4, 8, 15, 31, 62, 124, 248, -522, -24, -38, -32, 120, -26, -68, -138, -160, 392, 16, 30, 22, -68, 0, 16, 50, 58, -124, 0, -6, -8, 14, 0, 0, -6, -12, 18, 0, 0, 1, -1, 0, 0, 0, 1, -1).

Cf. A104144, A107239-A107246, A107248.

Accumulate[LinearRecurrence[{1,1,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,0,1},31]^2]  (* Ray Chandler, Aug 02 2015 *)

a(n) = F_9(0)^2 + F_9(1)^2 + ... F_9(n)^2, where F_9(n) = A104144(n).

A107244 Sum of squares of hexanacci numbers (A001592, Fibonacci 6-step numbers).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, 324528, 1277104, 5025200, 19770800, 77789489, 306071370, 1204272270, 4738336974, 18643463374, 73354544590, 288620849614, 1135607911375, 4468164041216, 17580442344960

Offset: 0

Jonathan Vos Post, May 19 2005

Primes include: a(6) = 2. Semiprimes include a(7) = 6 = 2 * 3, a(8) = 22 = 2 * 11, a(9) = 86 = 2 * 43, a(11) = 1366 = 2 * 683, a(19) = 77789489 = 3989 * 19501, a(23) = 18643463374 = 2 * 9321731687,

			a(0) = 0 = 0^2
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) = 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2
a(5) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2
a(6) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
a(7) = 6 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
a(8) = 22 = 0^2 + 0^2 +0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4, 6, 14, 28, -67, -9, -8, 28, -8, -12, 20, 5, 5, -10, 0, 2, -2, 0, -1, 1).

Cf. A001592, A107239-A107243, A107245-A107248.

Accumulate[LinearRecurrence[{1,1,1,1,1,1},{0,0,0,0,0,1},50]^2] (* Harvey P. Dale, Jan 19 2012 *)
LinearRecurrence[{3, 2, 4, 6, 14, 28, -67, -9, -8, 28, -8, -12, 20, 5, 5, -10, 0, 2, -2, 0, -1, 1},{0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, 324528, 1277104, 5025200, 19770800, 77789489, 306071370, 1204272270},29] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_6(0)^2 + F_6(1)^2 + ... F_6(n)^2, where F_6(n) = A001592(n). a(0) = 0, a(n+1) = a(n) + A001592(n).

a(n)= 3*a(n-1) +2*a(n-2) +4*a(n-3) +6*a(n-4) +14*a(n-5) +28*a(n-6) -67*a(n-7) -9*a(n-8) -8*a(n-9) +28*a(n-10) -8*a(n-11) -12*a(n-12) +20*a(n-13) +5*a(n-14) +5*a(n-15) -10*a(n-16) +2*a(n-18) -2*a(n-19) -a(n-21) +a(n-22). [From 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

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

N. J. A. Sloane, Sep 12 2020

nonn

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

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

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

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 *)

@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

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

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

cp's OEIS Frontend

A085697 a(n) = T(n)^2, where T(n) = A000073(n) is the n-th tribonacci number.

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

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A107245 Sum of squares of heptanacci numbers (Fibonacci 7-step numbers).

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A107247 Sum of squares of nonacci numbers (Fibonacci 9-step numbers).

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A107244 Sum of squares of hexanacci numbers (A001592, Fibonacci 6-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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