cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A208069 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 72, 81, 12, 26, 256, 240, 144, 144, 16, 42, 676, 704, 576, 216, 256, 20, 68, 1764, 2080, 1936, 1008, 324, 400, 25, 110, 4624, 6216, 6400, 3696, 1764, 432, 625, 30, 178, 12100, 18496, 21904, 12960, 7056, 2688, 576, 900, 36
Offset: 1

Views

Author

R. H. Hardin Feb 23 2012

Keywords

Comments

Table starts
..2...4...6...10....16....26.....42......68.....110......178.......288
..4..16..36..100...256...676...1764....4624...12100....31684.....82944
..6..36..72..240...704..2080...6216...18496...55000...163760....487296
..9..81.144..576..1936..6400..21904...73984..250000...846400...2862864
.12.144.216.1008..3696.12960..48840..179520..657000..2423280...8913456
.16.256.324.1764..7056.26244.108900..435600.1726596..6937956..27751824
.20.400.432.2688.11424.44064.198000..844800.3542544.15213984..64859616
.25.625.576.4096.18496.73984.360000.1638400.7268416.33362176.151585344

Examples

			Some solutions for n=4 k=3
..0..0..1....0..1..0....1..1..0....1..1..0....0..1..0....1..1..0....1..0..0
..0..1..1....1..0..1....1..0..1....1..1..0....0..0..1....1..1..0....0..0..1
..0..0..1....0..1..0....1..0..0....1..1..0....0..1..0....1..0..0....1..0..0
..0..1..0....1..0..0....0..0..1....1..1..0....0..0..1....1..1..0....0..0..1

Links

Crossrefs

Column 1 is A002620(n+2)
Column 2 is A030179(n+2)
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207840
Row 4 is A207924

A208078 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 126, 81, 15, 26, 256, 510, 441, 225, 25, 42, 676, 1968, 2601, 1785, 625, 40, 68, 1764, 7722, 15129, 16065, 7225, 1600, 64, 110, 4624, 30114, 88209, 139605, 99225, 27880, 4096, 104, 178, 12100, 117708, 514089, 1228095
Offset: 1

Views

Author

R. H. Hardin, Feb 23 2012

Keywords

Examples

			Table starts
..2....4......6......10.......16.........26..........42............68
..4...16.....36.....100......256........676........1764..........4624
..6...36....126.....510.....1968.......7722.......30114........117708
..9...81....441....2601....15129......88209......514089.......2996361
.15..225...1785...16065...139605....1228095....10751415......94313535
.25..625...7225...99225..1288225...17098225...224850025....2968615225
.40.1600..27880..572040.11204720..223058440..4412968520...87523832240
.64.4096.107584.3297856.97456384.2909955136.86610135616.2580469555456
...
Some solutions for n=4 k=3
..0..1..0....0..1..1....1..0..0....0..1..0....1..0..0....1..1..1....0..1..1
..1..0..1....1..1..0....1..1..0....0..1..1....0..1..1....1..0..1....1..1..0
..1..0..1....1..0..1....1..1..1....1..1..1....0..1..1....1..1..1....1..1..0
..0..1..1....0..1..1....1..1..1....1..0..0....1..1..1....0..1..1....0..1..1

Links

Crossrefs

Column 1 is A006498(n+2)
Column 2 is A189145(n+2)
Column 3 is A202399(n-2)
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A202954(n-2)

A208287 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 8, 16, 100, 102, 64, 10, 26, 256, 378, 216, 100, 12, 42, 676, 1260, 984, 390, 144, 14, 68, 1764, 4374, 3984, 2090, 636, 196, 16, 110, 4624, 14946, 16872, 9900, 3900, 966, 256, 18, 178, 12100, 51384, 70216, 49130, 21096, 6650, 1392
Offset: 1

Views

Author

R. H. Hardin Feb 25 2012

Keywords

Comments

Table starts
..2...4....6....10....16.....26......42.......68.......110........178
..4..16...36...100...256....676....1764.....4624.....12100......31684
..6..36..102...378..1260...4374...14946....51384....176238.....605022
..8..64..216...984..3984..16872...70216...294192...1229400....5142728
.10.100..390..2090..9900..49130..239490..1175440...5754050...28195750
.12.144..636..3900.21096.119580..665892..3733080..20874900..116842500
.14.196..966..6650.40376.256774.1604862.10095932..63357434..397965218
.16.256.1392.10608.71360.502416.3478160.24229696.168399632.1171405168

Examples

			Some solutions for n=4 k=3
..0..1..0....1..1..1....1..0..0....0..1..0....1..1..1....1..0..0....0..1..1
..1..0..1....1..1..0....0..1..0....0..1..1....1..0..1....0..1..1....1..1..0
..1..1..1....1..1..0....1..0..0....0..1..1....1..1..1....0..1..0....1..1..1
..1..1..1....1..1..0....0..1..0....0..1..1....1..1..1....0..1..1....1..1..0

Links

Crossrefs

Column 2 is A016742
Column 3 is A086113
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A060521

Formula

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 2*n^3 + 6*n^2 - 2*n
k=4: a(n) = (4/3)*n^4 + 10*n^3 + (2/3)*n^2 - 2*n
k=5: a(n) = (5/6)*n^5 + 9*n^4 + (91/6)*n^3 - 9*n^2
k=6: a(n) = (8/15)*n^6 + (23/3)*n^5 + (82/3)*n^4 + (1/3)*n^3 - (178/15)*n^2 + 2*n
k=7: a(n) = (61/180)*n^7 + (121/20)*n^6 + (1157/36)*n^5 + (425/12)*n^4 - (2923/90)*n^3 - (22/15)*n^2 + 2*n

A208369 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 114, 81, 14, 26, 256, 450, 351, 196, 21, 42, 676, 1644, 1953, 1162, 441, 31, 68, 1764, 6186, 9999, 9338, 3633, 961, 46, 110, 4624, 23010, 52821, 67396, 41433, 11067, 2116, 68, 178, 12100, 85992, 275769, 507682, 422541
Offset: 1

Views

Author

R. H. Hardin Feb 25 2012

Keywords

Comments

Table starts
..2....4.....6.....10.......16........26.........42...........68...........110
..4...16....36....100......256.......676.......1764.........4624.........12100
..6...36...114....450.....1644......6186......23010........85992........320742
..9...81...351...1953.....9999.....52821.....275769......1446381.......7572429
.14..196..1162...9338....67396....507682....3759574.....28035840.....208473118
.21..441..3633..41433...422541...4503765...47178453....497691495....5235328875
.31..961.11067.177909..2563359..38542393..570085195...8487780687..126039261499
.46.2116.33994.774134.15730896.334331082.6982331490.146860432968.3080068967794

Examples

			Some solutions for n=4 k=3
..0..1..1....1..1..1....1..1..1....0..1..0....0..1..0....0..1..0....1..1..0
..0..1..1....1..0..1....1..1..1....1..1..1....1..0..1....0..1..0....1..1..0
..0..1..1....1..1..1....1..1..0....1..1..1....1..0..1....0..1..0....1..0..0
..0..1..1....0..1..1....1..0..1....1..1..1....1..0..1....0..1..0....1..0..0

Links

Crossrefs

Column 1 is A038718(n+2)
Column 2 is A207069
Column 3 is A207421
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207718

A208379 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 8, 16, 100, 72, 64, 10, 26, 256, 240, 108, 100, 12, 42, 676, 704, 420, 144, 144, 14, 68, 1764, 2080, 1344, 640, 180, 196, 16, 110, 4624, 6216, 4212, 2176, 900, 216, 256, 18, 178, 12100, 18496, 13860, 7072, 3200, 1200, 252, 324, 20, 288
Offset: 1

Views

Author

R. H. Hardin Feb 25 2012

Keywords

Comments

Table starts
..2...4...6...10...16....26....42.....68.....110.....178......288......466
..4..16..36..100..256...676..1764...4624...12100...31684....82944...217156
..6..36..72..240..704..2080..6216..18496...55000..163760...487296..1450192
..8..64.108..420.1344..4212.13860..44880..144540..468852..1517184..4906980
.10.100.144..640.2176..7072.25200..87040..296560.1028128..3545856.12198016
.12.144.180..900.3200.10660.40740.148240..526900.1931300..7015680.25336420
.14.196.216.1200.4416.14976.60984.231744..851400.3276624.12438144.46737936
.16.256.252.1540.5824.20020.86436.340816.1285900.5170900.20393856.79220932

Examples

			Some solutions for n=4 k=3
..0..1..0....0..1..1....1..1..0....0..0..1....0..0..1....1..0..1....1..0..0
..1..1..0....0..1..1....1..1..0....0..1..1....1..0..1....1..1..0....1..0..0
..1..1..0....0..0..1....1..1..0....0..1..0....0..0..1....0..1..0....1..0..0
..1..0..0....0..0..1....1..1..0....0..1..0....0..0..1....0..1..0....1..0..0

Links

Crossrefs

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A044102(n-1) for n>1
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207840

Formula

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 36*n - 36 for n>1
k=4: a(n) = 20*n^2 + 40*n - 60 for n>1
k=5: a(n) = 96*n^2 - 32*n - 64 for n>1
k=6: a(n) = 364*n^2 - 416*n + 52 for n>1
k=7: a(n) = 84*n^3 + 840*n^2 - 1344*n + 420 for n>1
Empirical for row n:
n=1: a(k)=a(k-1)+a(k-2)
n=2: a(k)=2*a(k-1)+2*a(k-2)-a(k-3)
n=3: a(k)=a(k-1)+4*a(k-2)+5*a(k-3)+2*a(k-4)-a(k-5)+a(k-6) for k>8
n=4: a(k)=a(k-1)+4*a(k-2)+9*a(k-3)+5*a(k-4)-2*a(k-5)+4*a(k-6) for k>8
n=5: a(k)=a(k-1)+4*a(k-2)+13*a(k-3)+8*a(k-4)-3*a(k-5)+9*a(k-6) for k>8
n=6: a(k)=a(k-1)+4*a(k-2)+17*a(k-3)+11*a(k-4)-4*a(k-5)+16*a(k-6) for k>8
n=7: a(k)=a(k-1)+4*a(k-2)+21*a(k-3)+14*a(k-4)-5*a(k-5)+25*a(k-6) for k>8

A208420 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 0 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 102, 81, 13, 26, 256, 378, 261, 169, 18, 42, 676, 1260, 1269, 611, 324, 25, 68, 1764, 4374, 5139, 3835, 1278, 625, 34, 110, 4624, 14946, 22509, 18395, 10098, 2625, 1156, 46, 178, 12100, 51384, 95265, 100113, 55404, 26375
Offset: 1

Views

Author

R. H. Hardin Feb 26 2012

Keywords

Comments

Table starts
..2....4....6....10.....16......26.......42........68........110.........178
..4...16...36...100....256.....676.....1764......4624......12100.......31684
..6...36..102...378...1260....4374....14946.....51384.....176238......605022
..9...81..261..1269...5139...22509....95265....409239....1746639.....7475751
.13..169..611..3835..18395..100113...512525...2702193...14044303....73505289
.18..324.1278.10098..55404..365094..2187162..13759164...84389022...524422458
.25..625.2625.26375.161975.1297475..8948825..67061525..479579725..3521095775
.34.1156.5134.65178.440436.4270706.33334858.295643872.2428615750.20882937190

Examples

			Some solutions for n=4 k=3
..1..0..1....1..0..0....1..1..1....1..1..1....1..1..0....0..1..0....0..1..0
..1..0..0....0..1..1....1..1..0....1..1..1....0..1..0....0..1..1....0..1..1
..0..1..1....1..0..0....1..0..1....1..1..1....1..0..1....1..1..0....1..0..0
..1..0..0....0..1..1....1..1..0....1..1..1....0..1..0....0..1..0....0..1..0

Links

Crossrefs

Column 1 is A171861(n+1)
Column 2 is A207025
Column 3 is A207903
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A060521

A208555 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 90, 81, 12, 26, 256, 330, 225, 144, 16, 42, 676, 1008, 1089, 420, 256, 20, 68, 1764, 3354, 3969, 2508, 784, 400, 25, 110, 4624, 10710, 16641, 10080, 5776, 1260, 625, 30, 178, 12100, 34884, 65025, 50052, 25600, 11020
Offset: 1

Views

Author

R. H. Hardin Feb 28 2012

Keywords

Comments

Table starts
..2...4....6....10.....16.....26......42.......68.......110........178
..4..16...36...100....256....676....1764.....4624.....12100......31684
..6..36...90...330...1008...3354...10710....34884....112530.....364722
..9..81..225..1089...3969..16641...65025...263169...1046529....4198401
.12.144..420..2508..10080..50052..221340..1042416...4742628...21989868
.16.256..784..5776..25600.150544..753424..4129024..21492496..115175824
.20.400.1260.11020..52000.351140.1913940.11836400..67894220..407225740
.25.625.2025.21025.105625.819025.4862025.33930625.214476025.1439823025

Examples

			Some solutions for n=4 k=3
..1..1..0....1..1..0....1..1..0....1..1..1....1..0..1....1..0..1....0..1..1
..1..0..0....0..1..0....0..1..0....1..1..0....0..1..0....1..0..1....0..1..1
..1..0..0....1..0..0....0..1..0....1..0..1....1..0..0....1..0..0....0..1..0
..1..0..0....0..1..0....0..1..0....1..1..0....0..1..0....1..0..0....0..1..0

Links

Crossrefs

Column 1 is A002620(n+2)
Column 2 is A030179(n+2)
Column 3 is A207363
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207454

A208840 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 78, 81, 14, 26, 256, 282, 171, 196, 22, 42, 676, 768, 855, 406, 484, 35, 68, 1764, 2430, 2421, 3010, 990, 1225, 56, 110, 4624, 7086, 9801, 8736, 11242, 2485, 3136, 90, 178, 12100, 21588, 31419, 49126, 33088, 44275
Offset: 1

Views

Author

R. H. Hardin Mar 01 2012

Keywords

Comments

Table starts
..2....4....6.....10.....16.......26.......42........68........110.........178
..4...16...36....100....256......676.....1764......4624......12100.......31684
..6...36...78....282....768.....2430.....7086.....21588......64230......193554
..9...81..171....855...2421.....9801....31419....116919.....394965.....1419849
.14..196..406...3010...8736....49126...169974....833364....3166030....14462714
.22..484..990..11242..33088...272206...992574...6800596...28280758...173714530
.35.1225.2485..44275.131355..1644265..6206445..62470275..277136755..2417186345
.56.3136.6328.179032.533568.10399480.40122936.613538688.2842543480.36689660504

Examples

			Some solutions for n=4 k=3
..1..0..0....1..1..1....1..1..1....1..0..1....0..1..0....0..1..0....0..1..1
..0..1..1....0..1..0....1..1..0....1..0..0....1..1..0....1..0..1....0..1..1
..1..0..0....0..1..0....1..1..1....1..1..1....0..1..0....0..1..0....0..1..1
..0..1..1....0..1..1....1..1..0....1..0..0....0..1..0....1..0..0....1..1..1

Links

Crossrefs

Column 1 is A001611(n+2)
Column 2 is A207436
Column 3 is A208103
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A208689

Formula

Empirical for row n:
n=1: a(k)=a(k-1)+a(k-2)
n=2: a(k)=2*a(k-1)+2*a(k-2)-a(k-3)
n=3: a(k)=2*a(k-1)+4*a(k-2)-3*a(k-3)
n=4: a(k)=2*a(k-1)+7*a(k-2)-6*a(k-3)
n=5: a(k)=2*a(k-1)+12*a(k-2)-11*a(k-3)
n=6: a(k)=2*a(k-1)+20*a(k-2)-19*a(k-3)
n=7: a(k)=2*a(k-1)+33*a(k-2)-32*a(k-3)

A078642 Numbers with two representations as the sum of two Fibonacci numbers.

Original entry on oeis.org

4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972
Offset: 1

Views

Author

Joseph L. Pe, Dec 12 2002

Keywords

Comments

A positive integer n has exactly two representations as the sum of two Fibonacci numbers if and only if n is twice a Fibonacci number and n >= 4. Conjectured by John W. Layman, Dec 20 2002. Proved by Max Alekseyev, Mar 02 2007.
From Max Alekseyev, Mar 02 2007: (Start)
Suppose that the number m has exactly two representations as the sum of two Fibonacci numbers. There are three types of representations possible:
(I) the sum of two equal Fibonacci numbers
(II) the sum of two consecutive Fibonacci numbers
(III) the sum of two distinct non-consecutive Fibonacci numbers
Lemma. The two representations of m > 2 must be of different types.
Proof. Two representations of m > 2 both of type (I) are not possible as 2*F(n) is a strictly increasing function for n >= 2. Similarly, two representations of m both of type (II) are not possible as F(n) + F(n+1) is a strictly increasing function for n >= 0. Finally, two representations of m both of type (III) are not possible as that would violate the property of the Fibonacci numeral system (the uniqueness of representation of all nonnegative integers).
Consider all possible pairs of representation types:
(I) and (II) are possible only for m = 2: 2 = 2*F(1) = 2*F(2) = F(1) + F(2) but m = 2 has more than two different representations.
(II) and (III) are not possible together as that would again violate the property of the Fibonacci numeral system.
Finally, (I) and (III) gives rise to the sequence of a(n) = 2 * F(n) = F(n+1) + F(n-1). QED (End)

Examples

			16 has exactly two representations as the sum of Fibonacci numbers: 16 = 3 + 13 and 16 = 8 + 8. Hence 16 belongs to the sequence.

Links

Crossrefs

Essentially the same as A006355 = number of binary vectors of length n containing no singletons; and as A055389: a(0)=1, then twice the Fibonacci sequence.

Programs

  • Mathematica
    t = Split@ Sort@ Flatten@ Table[Fibonacci[i] + Fibonacci[j], {i, 2, 39}, {j, 2, i}]; Take[ t[[ # ]][[1]] & /@ Select[ Range@Length@t, Length[ t[[ # ]]] > 1 &], 36] (* Robert G. Wilson v *)
  • PARI
    a(n)=([0,1; 1,1]^(n-1)*[4;6])[1,1] \\ Charles R Greathouse IV, Oct 07 2015
  • PARI
    Vec(2*x*(2 + x) / (1 - x - x^2) + O(x^60)) \\ Colin Barker, Jan 29 2017

Formula

a(n) = 2F(n + 2), where F(n) is the n-th Fibonacci number.
a(n) = a(n - 1) + a(n - 2), n > 2 ; a(1) = 4, a(2) = 6 . G.f.: 2x*(2+x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = 2F(n + 2) = F(n) + F(n + 3), where F(1) = F(2) = 1. - Alonso del Arte, Jul 07 2013
a(n) = (2^(-n)*((1-r)^n*(-3+r) + (1+r)^n*(3+r))) / r where r=sqrt(5). - Colin Barker, Jan 29 2017

A090991 Number of meaningful differential operations of the n-th order on the space R^6.

Original entry on oeis.org

6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972
Offset: 1

Views

Author

Branko Malesevic, Feb 29 2004

Keywords

Comments

Apparently a(n) = A054886(n+2) for n=1..1000. - Georg Fischer, Oct 06 2018

Links

Crossrefs

Cf. A054886, A055389, A068922, A090989-A090995.
Essentially the same as A006355, A047992 and A078642.

Programs

  • GAP
    a:=[6,10];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # Muniru A Asiru, Oct 06 2018
  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(3+2*x)/(1-x-x^2) )); // G. C. Greubel, Feb 02 2019
  • Maple
    NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 6; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:
  • Mathematica
    CoefficientList[Series[2*(3+2z)/(1-z-z^2), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
  • PARI
    my(x='x+O('x^40)); Vec(2*x*(3+2*x)/(1-x-x^2)) \\ G. C. Greubel, Feb 02 2019
  • Sage
    (2*(3+2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 02 2019

Formula

a(k+4) = 3*a(k+2) - a(k).
a(k) = 2*Fibonacci(k+3).
From Philippe Deléham, Nov 19 2008: (Start)
a(n) = a(n-1) + a(n-2), n>2, where a(1)=6, a(2)=10.
G.f.: 2*x*(3+2*x)/(1-x-x^2). (End)
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 4. - Stefano Spezia, Apr 18 2022
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