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Expansion of (1 + x)/(1 + x + 2x^2).
+10
6
1, 0, -2, 2, 2, -6, 2, 10, -14, -6, 34, -22, -46, 90, 2, -182, 178, 186, -542, 170, 914, -1254, -574, 3082, -1934, -4230, 8098, 362, -16558, 15834, 17282, -48950, 14386, 83514, -112286, -54742, 279314, -169830, -388798, 728458
COMMENTS
Row sums of number triangle A110511. The sequences A110512 and A001607 are conjugated by one of the relations ((-1 + i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 + i*sqrt(7))/2 or ((-1 - i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 - i*sqrt(7))/2. These relations are connected with the Gauss sums; for example, if e := exp(i*2Pi/7) then e + e^2 + e^4 = (-1 + i*sqrt(7))/2 and e^3 + e^5 + e^6 = (-1 - i*sqrt(7))/2 -- for details see Witula's book. We also have a(n+1) = -2* A001607(n), which implies the Binet formula for a(n) (from the respective Binet formula for A001607(n) given in A001607), and A001607(n+1) = a(n) - A001607(n). - Roman Witula, Jul 27 2012 Pisano period lengths: 1, 1, 8, 1, 24, 8, 42, 1, 24, 24, 10, 8, 168, 42, 24, 2, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012
REFERENCES
R. Witula, On some applications of formulas for unimodular complex numbers, Jacek Skalmierski's Press, Gliwice 2011 (in Polish).
FORMULA
a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-2)^(j-k)*C(k, j-k).
G.f.: 2 - x + 2*x^2 + 3*x/Q(0), where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k + 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
MATHEMATICA
CoefficientList[Series[(1 + x)/(1 + x + 2*x^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 29 2017 *)
PROG
(PARI) x='x+O('x^50); Vec((1 + x)/(1 + x + 2*x^2)) \\ G. C. Greubel, Aug 29 2017
a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.
+10
5
1, 1, 0, -1, 0, 3, 4, -1, -8, -5, 12, 23, 0, -45, -44, 47, 136, 43, -228, -313, 144, 771, 484, -1057, -2024, 91, 4140, 3959, -4320, -12237, -3596, 20879, 28072, -13685, -69828, -42457, 97200, 182115, -12284, -376513, -351944, 401083, 1104972, 302807, -1907136, -2512749, 1301524, 6327023, 3723976
COMMENTS
Floretion Algebra Multiplication Program, FAMP Code: ibaseiseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]
FORMULA
G.f.: -(x^2-x+1) / ((x-1)*(2*x^2-x+1)). - Colin Barker, Feb 08 2015
MATHEMATICA
LinearRecurrence[{2, -3, 2}, {1, 1, 0}, 50] (* Harvey P. Dale, Mar 28 2019 *)
PROG
(PARI) Vec(-(x^2-x+1)/((x-1)*(2*x^2-x+1)) + O(x^100)) \\ Colin Barker, Feb 08 2015
a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 3, a(2) = 4.
+10
3
1, 3, 4, 1, -4, -3, 8, 17, 4, -27, -32, 25, 92, 45, -136, -223, 52, 501, 400, -599, -1396, -195, 2600, 2993, -2204, -8187, -3776, 12601, 20156, -5043, -45352, -35263, 55444, 125973, 15088, -236855, -267028, 206685, 740744, 327377, -1154108, -1808859, 499360, 4117081, 3118364, -5115795, -11352520
COMMENTS
Floretion Algebra Multiplication Program, FAMP Code: famseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]
FORMULA
a(n) = 2*a(n-1)-3*a(n-2)+2*a(n-3). G.f.: (1+x+x^2)/((1-x)*(1-x+2*x^2)). [ Colin Barker, Mar 27 2012]
MATHEMATICA
Table[(3 - ((1-I*Sqrt[7])^n + (1+I*Sqrt[7])^n)/2^n)/2 // Simplify, {n, 1, 50}] (* Jean-François Alcover, Jun 04 2017 *)
a(n+3) = a(n) - a(n+1) - a(n+2); a(0) = -5, a(1) = 6, a(2) = 0.
+10
2
-5, 6, 0, -11, 17, -6, -22, 45, -29, -38, 112, -103, -47, 262, -318, 9, 571, -898, 336, 1133, -2367, 1570, 1930, -5867, 5507, 2290, -13664, 16881, -927, -29618, 47426, -18735, -58309, 124470, -84896, -97883, 307249, -294262, -110870, 712381, -895773, 72522, 1535632, -2503927, 1040817, 2998742
COMMENTS
Floretion Algebra Multiplication Program, FAMP Code: 2tesforseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e], 1vesforseq = A000004, ForType: 1A.
FORMULA
G.f. (5-x-x^2)/(x^3-x^2-x-1)
EXAMPLE
This sequence was generated using the same floretion which generated the sequences A105577, A105578, A105579, etc.. However, in this case a force transform was applied. [Specifically, (a(n)) may be seen as the result of a tesfor-transform of the zero-sequence A000004 with respect to the floretion given in the program code.]
MATHEMATICA
Transpose[NestList[Join[Rest[#], ListCorrelate[ {1, -1, -1}, #]]&, {-5, 6, 0}, 50]][[1]] (* Harvey P. Dale, Mar 14 2011 *) CoefficientList[Series[(5-x-x^2)/(x^3-x^2-x-1), {x, 0, 50}], x] (* Harvey P. Dale, Mar 14 2011 *)
Riordan array (1, x/(1-3*x+2*x^2)).
+10
2
1, 0, 1, 0, 3, 1, 0, 7, 6, 1, 0, 15, 23, 9, 1, 0, 31, 72, 48, 12, 1, 0, 63, 201, 198, 82, 15, 1, 0, 127, 522, 699, 420, 125, 18, 1, 0, 255, 1291, 2223, 1795, 765, 177, 21, 1, 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1
FORMULA
Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. Diagonals sums are even-indexed Fibonacci numbers.
G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
T(n, n) = 1.
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=T(2,0)=0, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. (End)
Sum_{k=0..n} (-1)^k*T(n,k) = [n=1] - A009545(n). Sum_{k=0..n} (-2)^k*T(n,k) = [n=1] + A078020(n+1). T(2*n, n+1) = A045741(n+2), n >= 0.
EXAMPLE
Triangle begins:
1;
0, 1;
0, 3, 1;
0, 7, 6, 1;
0, 15, 23, 9, 1;
0, 31, 72, 48, 12, 1;
0, 63, 201, 198, 82, 15, 1;
0, 127, 522, 699, 420, 125, 18, 1;
0, 255, 1291, 2223, 1795, 765, 177, 21, 1;
0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;
0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;
MAPLE
# Uses function PMatrix from A357368.
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==0, 0, 3*T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k]]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 20 2022 *)
PROG
(Magma)
if k lt 0 or k gt n then return 0;
elif k eq n then return 1;
elif k eq 0 then return 0;
else return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k);
end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 20 2022 (SageMath)
if (k<0 or k>n): return 0
elif (k==n): return 1
elif (k==0): return 0
else: return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 20 2022
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