June 24 2002 1:31:21.750 PM POLPAK_PRB Tests for POLPAK, which computes the values of certain special functions and polynomials. TEST001 ALIGN_ENUM counts the number of possible alignments of two biological sequences. Alignment enumeration table: 0 1 2 3 4 5 6 7 8 9 10 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3 5 7 9 11 13 15 17 19 21 2 1 5 13 25 41 61 85 113 145 181 221 3 1 7 25 63 129 231 377 575 833 1159 1561 4 1 9 41 129 321 681 1289 2241 3649 5641 8361 5 1 11 61 231 681 1683 3653 7183 13073 22363 36365 6 1 13 85 377 1289 3653 8989 19825 40081 75517 134245 7 1 15 113 575 2241 7183 19825 48639 108545 224143 433905 8 1 17 145 833 3649 13073 40081 108545 265729 598417 1256465 9 1 19 181 1159 5641 22363 75517 224143 598417 1462563 3317445 10 1 21 221 1561 8361 36365 134245 433905 1256465 3317445 8097453 TEST002 BELL computes the Bell numbers. I, BELL(I) 0 1 1 1 2 2 3 5 4 15 5 52 6 203 7 877 8 4140 9 21147 10 115975 TEST003 BENFORD(I) is the Benford probability of the initial digit sequence I. I, BENFORD(I) 1 0.301030 2 0.176091 3 0.124939 4 0.969100E-01 5 0.791813E-01 6 0.669468E-01 7 0.579920E-01 8 0.511525E-01 9 0.457575E-01 TEST004: D_FACTORIAL evaluates the factorial function. X Exact F D_FACTORIAL(X) 1. 1.00000 1.00000 2. 2.00000 2.00000 3. 6.00000 6.00000 4. 24.0000 24.0000 5. 120.000 120.000 6. 720.000 720.000 7. 5040.00 5040.00 8. 40320.0 40320.0 9. 362880. 362880. 10. 0.362880E+07 0.362880E+07 11. 0.399168E+08 0.399168E+08 12. 0.479002E+09 0.479002E+09 13. 0.622702E+10 0.622702E+10 14. 0.871783E+11 0.871783E+11 15. 0.130767E+13 0.130767E+13 16. 0.209228E+14 0.209228E+14 17. 0.355687E+15 0.355687E+15 18. 0.640237E+16 0.640237E+16 19. 0.121645E+18 0.121645E+18 20. 0.243290E+19 0.243290E+19 25. 0.155112E+26 0.155112E+26 30. 0.265253E+33 0.265253E+33 TEST005 BERN computes Bernoulli numbers; BERN2 computes Bernoulli numbers; D_BERN3 computes Bernoulli numbers. I B1 B2 B3 0 1.000000000 1.000000000 1.000000000 1 -0.5000000000 -0.5000000000 -0.5000000000 2 0.1666666716 0.1666666716 0.1666666667 3 0.000000000 0.000000000 0.000000000 4 -0.3333334997E-01 -0.3333333507E-01 -0.3333313873E-01 5 0.000000000 0.000000000 0.000000000 6 0.2380960248E-01 0.2376933210E-01 0.2380950859E-01 7 0.000000000 0.000000000 0.000000000 8 -0.3333383054E-01 -0.3332766145E-01 -0.3333332992E-01 9 0.000000000 0.000000000 0.000000000 10 0.7576222718E-01 0.7575616241E-01 0.7575757413E-01 11 0.000000000 0.000000000 0.000000000 12 -0.2531761825 -0.2531130612 -0.2531135519 13 0.000000000 0.000000000 0.000000000 14 1.167824268 1.166665316 1.166666662 15 0.000000000 0.000000000 0.000000000 16 -7.120332718 -7.092144012 -7.092156861 17 0.000000000 0.000000000 0.000000000 18 55.84498215 54.97105789 54.97117794 19 0.000000000 0.000000000 0.000000000 20 -562.7693481 -529.1223755 -529.1242416 TEST006 BERN_POLY evaluates Bernoulli polynomials; BERN_POLY2 evaluates Bernoulli polynomials. X = 0.200000 I BX BX2 1 -0.30000001 -0.30000000 2 0.66666603E-02 0.66666649E-02 3 0.48000004E-01 0.48000000E-01 4 -0.77333488E-02 -0.77331382E-02 5 -0.23680015E-01 -0.23679806E-01 6 0.69135930E-02 0.69136249E-02 7 0.24908906E-01 0.24908833E-01 8 -0.10150384E-01 -0.10149965E-01 9 -0.45279052E-01 -0.45278205E-01 10 0.23330100E-01 0.23326319E-01 11 0.12605961 0.12605002 12 -0.78197524E-01 -0.78146782E-01 13 -0.49813128 -0.49797890 14 0.36137694 0.36043992 15 2.6520307 2.6487812 TEST007: BETA evaluates the Beta function. X Y Exact F BETA(X) 0.2000 1.0000 5.00000 5.00000 0.4000 1.0000 2.50000 2.50000 0.6000 1.0000 1.66667 1.66667 0.8000 1.0000 1.25000 1.25000 1.0000 0.2000 5.00000 5.00000 1.0000 0.4000 2.50000 2.50000 1.0000 1.0000 1.00000 1.00000 2.0000 2.0000 0.166667 0.166667 3.0000 3.0000 0.333333E-01 0.333333E-01 4.0000 4.0000 0.714286E-02 0.714285E-02 5.0000 5.0000 0.158730E-02 0.158730E-02 6.0000 2.0000 0.238095E-01 0.238095E-01 6.0000 3.0000 0.595238E-02 0.595237E-02 6.0000 4.0000 0.198413E-02 0.198412E-02 6.0000 5.0000 0.793651E-03 0.793650E-03 6.0000 6.0000 0.360750E-03 0.360750E-03 7.0000 7.0000 0.832501E-04 0.832499E-04 TEST008 BP01 evaluates Bernstein polynomials. The Bernstein polynomials of degree 10 at X = 0.300000 0 0.282475E-01 1 0.121061 2 0.233474 3 0.266828 4 0.200121 5 0.102919 6 0.367569E-01 7 0.900169E-02 8 0.144670E-02 9 0.137781E-03 10 0.590490E-05 TEST009 BPAB evaluates Bernstein polynomials. The Bernstein polynomials of degree 10 based on the interval from 0.00000 to 1.00000 evaluated at X = 0.300000 0 0.282475E-01 1 0.121061 2 0.233474 3 0.266828 4 0.200121 5 0.102919 6 0.367569E-01 7 0.900169E-02 8 0.144670E-02 9 0.137781E-03 10 0.590490E-05 TEST010 CARDAN_COEFF returns the coefficients of a Cardan polynomial. CARDAN evaluates a Cardan polynomial directly. We use the parameter S = 1.00000 Table of polynomial coefficients: 0 2. 1 0. 1. 2 -2. 0. 1. 3 0. -3. 0. 1. 4 2. 0. -4. 0. 1. 5 0. 5. 0. -5. 0. 1. 6 -2. 0. 9. 0. -6. 0. 1. 7 0. -7. 0. 14. 0. -7. 0. 1. 8 2. 0. -16. 0. 20. 0. -8. 0. 1. 9 0. 9. 0. -30. 0. 27. 0. -9. 0. 1. 10 -2. 0. 25. 0. -50. 0. 35. 0. -10. 0. 1. Compare CARDAN_COEFF + RPOLY_VAL_HORNER versus CARDAN. Evaluate polynomials at X = 0.250000 We use the parameter S = 0.500000 Order, Horner, Direct 0 2.00000 2.00000 1 0.250000 0.250000 2 -0.937500 -0.937500 3 -0.359375 -0.359375 4 0.378906 0.378906 5 0.274414 0.274414 6 -0.120850 -0.120850 7 -0.167419 -0.167419 8 0.185699E-01 0.185699E-01 9 0.883522E-01 0.883522E-01 10 0.128031E-01 0.128031E-01 TEST011 CATALAN computes Catalan numbers. I, computed Cat(I), correct Cat(I) 0 1 1 1 1 1 2 2 2 3 5 5 4 14 14 5 42 42 6 132 132 7 429 429 8 1430 1430 9 4862 4862 10 16796 16796 TEST012 CATALAN_ROW computes a row of Catalan's triangle. First, compute row 7: 7 1 7 27 75 165 297 429 429 Now compute rows one at a time: 0 1 1 1 1 2 1 2 2 3 1 3 5 5 4 1 4 9 14 14 5 1 5 14 28 42 42 6 1 6 20 48 90 132 132 7 1 7 27 75 165 297 429 429 8 1 8 35 110 275 572 1001 1430 1430 9 1 9 44 154 429 1001 2002 3432 4862 4862 10 1 10 54 208 637 1638 3640 7072 11934 16796 16796 TEST013 CHEBY1 evaluates Cheybyshev polynomials of the first kind. Use X = 0.200000 0 1.00000 1 0.200000 2 -0.920000 3 -0.568000 4 0.692800 5 0.845120 6 -0.354752 7 -0.987021 8 -0.400563E-01 9 0.970998 10 0.428456 TEST014 CHEBY2 evaluates Chebyshev polynomials of the second kind. X = 0.200000 0 1.00000 1 0.400000 2 -0.840000 3 -0.736000 4 0.545600 5 0.954240 6 -0.163904 7 -1.01980 8 -0.244017 9 0.922195 10 0.612895 TEST015 COMBIN evaluates C(N,K). N K CNK 0 0 1.00000 1 0 1.00000 1 1 1.00000 2 0 1.00000 2 1 2.00000 2 2 1.00000 3 0 1.00000 3 1 3.00000 3 2 3.00000 3 3 1.00000 4 0 1.00000 4 1 4.00000 4 2 6.00000 4 3 4.00000 4 4 1.00000 TEST016 COMBIN2 evaluates C(N,K). N K CNK 0 0 1 1 0 1 1 1 1 2 0 1 2 1 2 2 2 1 3 0 1 3 1 3 3 2 3 3 3 1 4 0 1 4 1 4 4 2 6 4 3 4 4 4 1 TEST017 COMB_ROW computes a row of Pascal's triangle. 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1 TEST018 EULER computes Euler numbers; D_EULER2 computes Euler numbers. EULER, D_EULER2: 0 1 1.00000000 1 1 0.00000000 2 5 -1.00000000 3 61 0.00000000 4 1385 5.00000000 5 50521 0.00000000 6 2702765 -61.0000000 7 199360981 0.00000000 8 -2083324335 1385.00000 9 -302010319 0.00000000 10 -1024217983 -50521.0000 11 -151230318 0.00000000 12 -259997456 2702765.00 13 747860808 0.00000000 14 1841997600 -199360981. 15 1075158681 0.00000000 16 -1136173598 0.193915121E+11 TEST019 EULERIAN evaluates Eulerian numbers. 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 4 1 0 0 0 0 1 11 11 1 0 0 0 1 26 66 26 1 0 0 1 57 302 302 57 1 0 1 120 1191 2416 1191 120 1 TEST020 EULER_POLY evaluates Euler polynomials. Evaluate at X = 0.500000 0 1.00000 1 0.277556E-16 2 -0.250000 3 -0.145953E-05 4 0.312497 5 -0.332929E-05 6 -0.953128 7 -0.173264E-05 8 5.41016 9 -0.102449E-05 10 -49.3369 11 0.647439E-06 12 659.855 13 0.522754E-05 14 -12168.0 15 0.218677E-03 TEST021 F_HOFSTADTER evaluates Hofstadter's recursive F function. N F(N) 0 0 1 1 2 1 3 2 4 2 5 3 6 3 7 4 8 4 9 5 10 5 11 6 12 6 13 7 14 7 15 8 16 8 17 9 18 9 19 10 20 10 21 11 22 11 23 12 24 12 25 13 26 13 27 14 28 14 29 15 30 15 TEST022: LOG_FACTORIAL evaluates the logarithm of the factorial function. X Exact F LOG_FACTORIAL(X) 1. 0.00000 0.00000 2. 0.693147 0.693147 3. 1.79176 1.79176 4. 3.17805 3.17805 5. 4.78749 4.78749 6. 6.57925 6.57925 7. 8.52516 8.52516 8. 10.6046 10.6046 9. 12.8018 12.8018 10. 15.1044 15.1044 11. 17.5023 17.5023 12. 19.9872 19.9872 13. 22.5522 22.5522 14. 25.1912 25.1912 15. 27.8993 27.8993 16. 30.6719 30.6719 17. 33.5051 33.5051 18. 36.3954 36.3954 19. 39.3399 39.3399 20. 42.3356 42.3356 25. 58.0036 58.0036 30. 74.6582 74.6582 TEST023 FIBONACCI_DIRECT evalutes a Fibonacci number directly. 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 20 6765 TEST024 FIBONACCI_FLOOR computes the largest Fibonacci number less than or equal to a given positive integer. N Fibonacci Index 1 1 2 2 2 3 3 3 4 4 3 4 5 5 5 6 5 5 7 5 5 8 8 6 9 8 6 10 8 6 11 8 6 12 8 6 13 13 7 14 13 7 15 13 7 16 13 7 17 13 7 18 13 7 19 13 7 20 13 7 TEST025 FIBONACCI_RECURSIVE computes the Fibonacci sequence. 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 20 6765 TEST026 G_HOFSTADTER evaluates Hofstadter's recursive G function. N G(N) 0 0 1 1 2 1 3 2 4 3 5 3 6 4 7 4 8 5 9 6 10 6 11 7 12 8 13 8 14 9 15 9 16 10 17 11 18 11 19 12 20 12 21 13 22 14 23 14 24 15 25 16 26 16 27 17 28 17 29 18 30 19 TEST027: GAMMA evaluates the Gamma function. X Exact F GAMMA(X) 0.2000 4.59085 4.59084 0.4000 2.21816 2.21816 0.6000 1.48919 1.48919 0.8000 1.16423 1.16423 1.0000 1.00000 1.00000 1.1000 0.951351 0.951351 1.2000 0.918169 0.918169 1.3000 0.897471 0.897471 1.4000 0.887264 0.887264 1.5000 0.886227 0.886227 1.6000 0.893515 0.893515 1.7000 0.908639 0.908639 1.8000 0.931384 0.931384 1.9000 0.961766 0.961766 2.0000 1.00000 1.00000 10.0000 362880. 362880. 20.0000 0.121645E+18 0.121645E+18 30.0000 0.884176E+31 0.884172E+31 TEST028: GAMMA_LOG evaluates the logarithm of the Gamma function. X Exact F GAMMA_LOG(X) 0.2000 1.52406 1.52406 0.4000 0.796678 0.796678 0.6000 0.398234 0.398234 0.8000 0.152060 0.152060 1.0000 0.00000 0.00000 1.1000 -0.498725E-01 -0.498725E-01 1.2000 -0.853741E-01 -0.853741E-01 1.3000 -0.108175 -0.108175 1.4000 -0.119613 -0.119613 1.5000 -0.120782 -0.120782 1.6000 -0.112592 -0.112592 1.7000 -0.958077E-01 -0.958077E-01 1.8000 -0.710839E-01 -0.710839E-01 1.9000 -0.389843E-01 -0.389843E-01 2.0000 0.00000 0.00000 10.0000 12.8018 12.8018 20.0000 39.3399 39.3399 30.0000 71.2570 71.2570 TEST029 GEGENBAUER evaluates Gegenbauer polynomials. Alfa = 0.500000 X = 1.20000 0 1.00000 1 1.20000 2 1.66000 3 2.52000 4 4.04700 5 6.72552 6 11.4236 7 19.6937 8 34.3151 9 60.2754 10 106.544 TEST030 HAIL(I) computes the length of the hail sequence for I, also known as the 3*N+1 sequence. I, HAIL(I) 1 0 2 1 3 7 4 2 5 5 6 8 7 16 8 3 9 19 10 6 11 14 12 9 13 9 14 17 15 17 16 4 17 12 18 20 19 20 20 7 TEST031 H_HOFSTADTER evaluates Hofstadter's recursive H function. N H(N) 0 0 1 1 2 1 3 2 4 3 5 4 6 4 7 5 8 5 9 6 10 7 11 7 12 8 13 9 14 10 15 10 16 11 17 12 18 13 19 13 20 14 21 14 22 15 23 16 24 17 25 17 26 18 27 18 28 19 29 20 30 20 TEST032 HERMITE evaluates the Hermite polynomials. X = 0.500000 0 1.00000 1 1.00000 2 -1.00000 3 -5.00000 4 1.00000 5 41.0000 6 31.0000 7 -461.000 8 -895.000 9 6481.00 10 22591.0 TEST0325: I_FACTORIAL evaluates the factorial function. X Exact F I_FACTORIAL(X) 1. 1.00000 1 2. 2.00000 2 3. 6.00000 6 4. 24.0000 24 5. 120.000 120 6. 720.000 720 7. 5040.00 5040 8. 40320.0 40320 9. 362880. 362880 10. 0.362880E+07 3628800 11. 0.399168E+08 39916800 12. 0.479002E+09 479001600 13. 0.622702E+10 1932053504 14. 0.871783E+11 1278945280 15. 0.130767E+13 2004310016 16. 0.209228E+14 2004189184 17. 0.355687E+15 -288522240 18. 0.640237E+16 -898433024 19. 0.121645E+18 109641728 20. 0.243290E+19 -2102132736 25. 0.155112E+26 2076180480 30. 0.265253E+33 1409286144 TEST0326: I_FACTORIAL2 evaluates the factorial function. X I_FACTORIAL2(X) 0 1 1 1 2 2 3 3 4 8 5 15 6 48 7 105 8 384 9 945 10 3840 11 10395 12 46080 13 135135 14 645120 15 2027025 16 10321920 TEST033 JACOBI evaluates the Jacobi polynomials. Alfa = -0.500000 Beta = 0.500000 X = 0.500000 0 1.00000 1 0.00000 2 -0.375000 3 -0.312500 4 0.00000 5 0.246094 6 0.225586 7 0.00000 8 -0.196381 9 -0.185471 10 0.00000 TEST034 LAGUERRE_GEN evaluates the generalized Laguerre polynomials. ALFA = 0.100000 X = 0.500000 0 1.00000 1 0.600000 2 0.230000 3 -0.673333E-01 4 -0.289350 5 -0.442469 6 -0.535747 7 -0.578765 8 -0.580771 9 -0.550311 10 -0.495077 TEST035 LAGUERRE_LNM evaluates the associated Laguerre polynomials, M = 1 X = 0.500000 0 0.00000 1 -1.00000 2 -3.00000 3 -13.7500 4 -91.8333 5 -805.938 TEST036 LAGUERRE evaluates the Laguerre polynomials. X = 0.500000 0 1.00000 1 0.500000 2 0.250000 3 -0.875000 4 -7.93750 5 -53.4688 6 -362.984 7 -2612.43 8 -20094.0 9 -164355. 10 -0.141296E+07 TEST037 LEGENDRE_PN evaluates the Legendre polynomials of the first kind, and derivatives. X = 0.500000 0 1.00000 0.00000 1 0.500000 1.00000 2 -0.125000 1.50000 3 -0.437500 0.375000 4 -0.289062 -1.56250 5 0.898437E-01 -2.22656 6 0.323242 -0.574219 7 0.223145 1.97559 8 -0.736389E-01 2.77295 9 -0.267899 0.723724 10 -0.188229 -2.31712 TEST038: LEGENDRE_PNM evaluates associated Legrendre functions. N M X Exact F PNM(X) 1 0 0.0000 0.00000 0.00000 1 0 0.5000 0.500000 0.500000 1 0 0.7071 0.707107 0.707107 1 0 1.0000 1.00000 1.00000 1 1 0.5000 -0.866025 -0.866025 2 0 0.5000 -0.125000 -0.125000 2 1 0.5000 -1.29904 -1.29904 2 2 0.5000 2.25000 2.25000 3 0 0.5000 -0.437500 -0.437500 3 1 0.5000 -0.324759 -0.324759 3 2 0.5000 5.62500 5.62500 3 3 0.5000 -9.74278 -9.74278 4 2 0.5000 4.21875 4.21875 5 2 0.5000 -4.92187 -4.92187 6 3 0.5000 12.7874 12.7874 7 3 0.5000 116.685 116.685 8 4 0.5000 -1050.67 -1050.67 9 4 0.5000 -2078.49 -2078.49 10 5 0.5000 30086.2 30086.2 TEST039 LEGENDRE_QN evaluates Legendre polynomials of the second kind. X = 0.500000 0 0.549306 1 -0.725347 2 -1.36267 3 -1.78756 4 -2.10622 TEST0395 LOCK counts the combinations on a button lock. I, LOCK(I) 0 1 1 1 2 3 3 13 4 75 5 541 6 4683 7 47293 8 545835 9 7087261 10 102247563 TEST040 PENTAGON_NUM computes the pentagonal numbers. 1 1 2 5 3 12 4 22 5 35 6 51 7 70 8 92 9 117 10 145 TEST041 PYRAMID_NUM computes the pyramidal numbers. 1 1 2 4 3 10 4 20 5 35 6 56 7 84 8 120 9 165 10 220 TEST042: R_FACTORIAL evaluates the factorial function. X Exact F R_FACTORIAL(X) 1. 1.00000 1.00000 2. 2.00000 2.00000 3. 6.00000 6.00000 4. 24.0000 24.0000 5. 120.000 120.000 6. 720.000 720.000 7. 5040.00 5040.00 8. 40320.0 40320.0 9. 362880. 362880. 10. 0.362880E+07 0.362880E+07 11. 0.399168E+08 0.399168E+08 12. 0.479002E+09 0.479002E+09 13. 0.622702E+10 0.622702E+10 14. 0.871783E+11 0.871783E+11 15. 0.130767E+13 0.130767E+13 16. 0.209228E+14 0.209228E+14 17. 0.355687E+15 0.355687E+15 18. 0.640237E+16 0.640237E+16 19. 0.121645E+18 0.121645E+18 20. 0.243290E+19 0.243290E+19 25. 0.155112E+26 0.155112E+26 30. 0.265253E+33 0.265253E+33 TEST043 STIRLING1: Stirling numbers of first kind. Get rows 1 through 8 1 1 0 0 0 0 0 0 0 2 -1 1 0 0 0 0 0 0 3 2 -3 1 0 0 0 0 0 4 -6 11 -6 1 0 0 0 0 5 24 -50 35 -10 1 0 0 0 6 -120 274 -225 85 -15 1 0 0 7 720 -1764 1624 -735 175 -21 1 0 8 -5040 13068 -13132 6769 -1960 322 -28 1 TEST044 STIRLING2: Stirling numbers of second kind. Get rows 1 through 8 1 1 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 3 1 3 1 0 0 0 0 0 4 1 7 6 1 0 0 0 0 5 1 15 25 10 1 0 0 0 6 1 31 90 65 15 1 0 0 7 1 63 301 350 140 21 1 0 8 1 127 966 1701 1050 266 28 1 TEST045 TRIANGLE_NUM computes the triangular numbers. 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 45 10 55 TEST046 V_HOFSTADTER evaluates Hofstadter's recursive V function. N V(N) 0 0 1 1 2 1 3 1 4 1 5 2 6 3 7 4 8 5 9 5 10 6 11 6 12 7 13 8 14 8 15 9 16 9 17 10 18 11 19 11 20 11 21 12 22 12 23 13 24 14 25 14 26 15 27 15 28 16 29 17 30 17 TEST047 VIBONACCI computes a Vibonacci sequence. Number of times we compute the series: 3 1 1 1 1 2 1 1 1 3 0 2 0 4 -1 -3 1 5 -1 -5 1 6 0 -2 0 7 1 7 -1 8 -1 5 1 9 0 2 2 10 1 7 -3 11 -1 5 -1 12 0 -2 2 13 -1 -3 3 14 -1 5 -5 15 -2 2 2 16 3 -7 3 17 -5 5 -5 18 2 2 -2 19 -3 -3 7 20 -1 1 -5 TEST048 ZECKENDORF computes the Zeckendorf decomposition of an integer into nonconsecutive Fibonacci numbers. N Sum M Parts 1 1 2 2 3 3 4 3 1 5 5 6 5 1 7 5 2 8 8 9 8 1 10 8 2 11 8 3 12 8 3 1 13 13 14 13 1 15 13 2 16 13 3 17 13 3 1 18 13 5 19 13 5 1 20 13 5 2 21 21 22 21 1 23 21 2 24 21 3 25 21 3 1 26 21 5 27 21 5 1 28 21 5 2 29 21 8 30 21 8 1 31 21 8 2 32 21 8 3 33 21 8 3 1 34 34 35 34 1 36 34 2 37 34 3 38 34 3 1 39 34 5 40 34 5 1 41 34 5 2 42 34 8 43 34 8 1 44 34 8 2 45 34 8 3 46 34 8 3 1 47 34 13 48 34 13 1 49 34 13 2 50 34 13 3 51 34 13 3 1 52 34 13 5 53 34 13 5 1 54 34 13 5 2 55 55 56 55 1 57 55 2 58 55 3 59 55 3 1 60 55 5 61 55 5 1 62 55 5 2 63 55 8 64 55 8 1 65 55 8 2 66 55 8 3 67 55 8 3 1 68 55 13 69 55 13 1 70 55 13 2 71 55 13 3 72 55 13 3 1 73 55 13 5 74 55 13 5 1 75 55 13 5 2 76 55 21 77 55 21 1 78 55 21 2 79 55 21 3 80 55 21 3 1 81 55 21 5 82 55 21 5 1 83 55 21 5 2 84 55 21 8 85 55 21 8 1 86 55 21 8 2 87 55 21 8 3 88 55 21 8 3 1 89 89 90 89 1 91 89 2 92 89 3 93 89 3 1 94 89 5 95 89 5 1 96 89 5 2 97 89 8 98 89 8 1 99 89 8 2 100 89 8 3 POLPAK_PRB Normal end of execution.