A137176 -id:A137176 - cp's OEIS frontend

A136431 Hyperfibonacci square number array a(k,n) = F(n)^(k), read by ascending antidiagonals (k, n >= 0).

0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 7, 7, 5, 0, 1, 5, 11, 14, 12, 8, 0, 1, 6, 16, 25, 26, 20, 13, 0, 1, 7, 22, 41, 51, 46, 33, 21, 0, 1, 8, 29, 63, 92, 97, 79, 54, 34, 0, 1, 9, 37, 92, 155, 189, 176, 133, 88, 55, 0, 1, 10, 46, 129, 247, 344, 365, 309, 221, 143, 89, 0, 1

Offset: 0

Jonathan Vos Post, Apr 01 2008

Main diagonal is A108081. Antidiagonal sums form A027934. - Gerald McGarvey, Oct 01 2008

Seen as triangle read by rows: T(n,0) = 1, T(n,n) = A000045(n) and for 0 < k < n: T(n,k) = T(n-1,k-1) + T(n-1,k). - Reinhard Zumkeller, Jul 16 2013

			The array F(n)^(k) begins:
.....|n=0|n=1|.n=2|.n=3|.n=4.|.n=5.|..n=6.|.n=7..|..n=8..|..n=9..|.n=10..|.in.OEIS
k=0..|.0.|.1.|..1.|..2.|...3.|...5.|....8.|...13.|....21.|....34.|....55.|.A000045
k=1..|.0.|.1.|..2.|..4.|...7.|..12.|...20.|...33.|....54.|....88.|...143.|.A000071
k=2..|.0.|.1.|..3.|..7.|..14.|..26.|...46.|...79.|...133.|...221.|...364.|.A001924
k=3..|.0.|.1.|..4.|.11.|..25.|..51.|...97.|..176.|...309.|...530.|...894.|.A014162
k=4..|.0.|.1.|..5.|.16.|..41.|..92.|..189.|..365.|...674.|..1204.|..2098.|.A014166
k=5..|.0.|.1.|..6.|.22.|..63.|.155.|..344.|..709.|..1383.|..2587.|..4685.|.A053739
k=6..|.0.|.1.|..7.|.29.|..92.|.247.|..591.|.1300.|..2683.|..5270.|..9955.|.A053295
k=7..|.0.|.1.|..8.|.37.|.129.|.376.|..967.|.2267.|..4950.|.10220.|.20175.|.A053296
k=8..|.0.|.1.|..9.|.46.|.175.|.551.|.1518.|.3785.|..8735.|.18955.|.39130.|.A053308
k=9..|.0.|.1.|.10.|.56.|.231.|.782.|.2300.|.6085.|.14820.|.33775.|.72905.|.A053309

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
H. Belbachir, A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, J. Int. Seq. 17 (2014), Article #14.4.3.
Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, arXiv:0803.4388 [math.NT], 2008.
Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, Applied Mathematics and Computation 206(2) (2008), 942-951.
Eric Weisstein's World of Mathematics, Steenrod Algebra

Cf. A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309, A123736, A137176.

a136431 n k = a136431_tabl !! n !! k
a136431_row n = a136431_tabl !! n
a136431_tabl = map fst $ iterate h ([0], 1) where
   h (row, fib) = (zipWith (+) ([0] ++ row) (row ++ [fib]), last row)
-- Reinhard Zumkeller, Jul 16 2013

A136431 := proc(k,n) local x ; coeftayl(x/(1-x-x^2)/(1-x)^k,x=0,n) ; end: for d from 0 to 20 do for n from 0 to d do printf("%d,",A136431(d-n,n)) ; od: od: # R. J. Mathar, Apr 25 2008

t[n_, k_] := CoefficientList[Series[x/(1 - x - x^2)/(1 - x)^k, {x, 0, n + 1}], x][[n + 1]]; Table[ t[n, k - n], {k, 0, 11}, {n, 0, k}] // Flatten
(* To view the table above *) Table[ t[n, k], {k, 0, 9}, {n, 0, 10}] // TableForm

a(k,n) = Apply partial sum operator k times to Fibonacci numbers.

For k > 0 and n > 1, a(k,n) = a(k-1,n) + a(k,n-1). - Gerald McGarvey, Oct 01 2008

A053298 Partial sums of A027964.

Original entry on oeis.org

1, 8, 34, 107, 281, 654, 1397, 2801, 5353, 9859, 17643, 30869, 53062, 89951, 150833, 250780, 414210, 680665, 1114160, 1818310, 2960806, 4813018, 7814074, 12674542, 20544191, 33283434, 53902532, 87272241, 141273663, 228658744

Offset: 0

Barry E. Williams, Mar 04 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1)

Cf. A027964 and A000204.
A column in triangular array A027960.
Cf. A137176 (row k=5).

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x)/((1-x-x^2)*(1-x)^5))); // G. C. Greubel, May 24 2018

LinearRecurrence[{6,-14,15,-5,-4,4,-1},{1,8,34,107,281,654,1397},30] (* Harvey P. Dale, May 09 2018 *)
CoefficientList[Series[(1+2x)/((1-x-x^2)(1-x)^5), {x,0,50}], x] (* G. C. Greubel, May 24 2018 *)

x='x+O('x^30); Vec((1+2*x)/((1-x-x^2)*(1-x)^5)) \\ G. C. Greubel, May 24 2018

a(n) = 3*F(n+10) + F(n+9) - (3*n^4 + 58*n^3 + 489*n^2 + 2234*n + 4752)/24, where F(.) are the Fibonacci numbers (A000045).
a(n) = a(n-1) + a(n-2) + (3*n+4)*C(n+3, 3)/4.
G.f.: (1 + 2*x)/((1 - x - x^2)*(1 - x)^5). - R. J. Mathar, Nov 28 2008

A324242 Incomplete Lucas numbers: irregular triangular array L(n,k) = Sum_{j = 0..k} (n/(n-j)) * binomial(n-j, j), read by rows, with n >= 1 and 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 3, 1, 4, 1, 5, 7, 1, 6, 11, 1, 7, 16, 18, 1, 8, 22, 29, 1, 9, 29, 45, 47, 1, 10, 37, 67, 76, 1, 11, 46, 96, 121, 123, 1, 12, 56, 133, 188, 199, 1, 13, 67, 179, 284, 320, 322, 1, 14, 79, 235, 417, 508, 521, 1, 15, 92, 302, 596, 792, 841, 843, 1, 16, 106, 381, 831, 1209, 1349, 1364

Offset: 1

Petros Hadjicostas, Sep 02 2019

For additional properties of the incomplete Lucas numbers and special cases not listed here, see Filipponi (1996, pp. 45-53).

			Triangle L(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
  1;
  1,  3;
  1,  4;
  1,  5,  7;
  1,  6, 11;
  1,  7, 16,  18;
  1,  8, 22,  29;
  1,  9, 29,  45,  47;
  1, 10, 37,  67,  76;
  1, 11, 46,  96, 121, 123;
  1, 12, 56, 133, 188, 199;
  ...
Row sums are 1, 4, 5, 13, 18, 42, 60, 131, 191, 398, 589, 1186, 1775, 3482, 5257, 10103, 15360, ...

A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Lucas numbers.
Piero Filipponi, Incomplete Fibonacci and Lucas numbers, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 2 (p. 46) that contains the incomplete Lucas numbers.
A. Pintér and H.M. Srivastava, Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.

Cf. A038730, A038792, and A134511 for various versions of the incomplete Fibonacci numbers.

Cf. A000032, A000045, A000124, A000204, A137176.

Flatten[Table[Sum[(n/(n-j))*Binomial[n-j, j],{j,0,k}],{n,1,15},{k,0,Floor[n/2]}]] (* Stefano Spezia, Sep 03 2019 *)

L(n,k) = F(n-1, k-1) + F(n+1, k) for n >= 1 and 0 <= k <= floor(n/2), where F(n,k) = Sum_{j = 0..k} binomial(n-1-j, j) are the incomplete Fibonacci numbers (defined for n >= 1 and 0 <= k <= floor((n-1)/2)).
L(n+2, k+1) = L(n+1, k+1) + L(n,k) for n >= 1 and 0 <= k <= floor((n-1)/2).
L(n,k) = F(n+2,k) - F(n-2, k-2) for n >= 3 and 2 <= k <= floor((n+1)/2).
Special cases: L(n,0) = 1 (n >= 1), L(n,1) = n+1 (n >= 2), L(n,2) = (n^2-n+2)/2 = A000124(n-1) (n >= 4), and L(n, floor(n/2)) = A000204(n) (n >= 1).
Sum of row n = (3 + (-1)^n)*A000204(n)/4 + n*A000045(n)/2.
G.f. for column k >= 1: t^(2*k)*((A000204(2*k) + t*A000204(2*k-1))*(1-t)^(k+1) - t^2*(2-t))/((1-t)^(k+1) * (1-t-t^2)).

A136338 Primes in the array A136431 that are not Fibonacci numbers.

Original entry on oeis.org

7, 11, 29, 37, 41, 67, 79, 97, 137, 191, 211, 277, 379, 631, 709, 821, 947, 967, 991, 1129, 1327, 1597, 1831, 2017, 2081, 2267, 2347, 2557, 2683, 2851, 2927, 3571, 3917, 4561, 4657, 4951, 5051, 5779, 6217, 6329, 6763, 8273, 8647, 8779, 9181, 9871, 10093

Offset: 1

Jonathan Vos Post, Apr 12 2008

A generalization of prime Fibonacci numbers (A005478) are the prime hyperfibonacci numbers (primes in A136431). Referring to the array A(k,n) = Apply partial sum operator k times to Fibonacci numbers, we see that every prime occurs in the n=2 column (as it contains every positive integer).
So excluding n=2 and k=0 (A005478) we have the nontrivially prime hyperfibonacci numbers which are not Fibonacci numbers.
Note that this sequence does not indicate multiplicity (e.g., 7 occurs twice in the valid part of the table).
Continuing the table of primes in the examples, from a computation by Joshua Zucker, we have:
k=1: {7, ...} no more through n = 1000.
k=2: {7, 79, 514201, 14930317, 956722025983, 5527939700884681 4660046610375530219, ...}
k=3: {11, 97, 17519, next value has 60 digits, ...}
k=4: {41, 10093, 16703, 3520457, 591286703533, 6557470285501, 19740274219868101499, ...}
k=5: {709, 8273, 14323, 466004661037329684,1 298611126818977061133263, ...}
k=6: {29, 2683, 23945893, 1835540197, 4052735290427, 27777884012083, ...}
k=7: {37, 967, 2267, 127921, 226007, 62048869, 1131463777, 7540113804271826929, ...}
k=8: {27777538280521, 1409869790947669143312035590804646728957, ...}
k=9: {1033628323428189498226451492123369099, next value has 60 digits, ...}
k=10: {67, 5972304273877744135569337875802249660927, ...}
k=11: {79, 4478413, 19008291293, 61305228407581679, ...}
k=12: {6763, 1982269, 37886753582095837, 2791715456569622316696636389, ...}.

			k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 1000.
k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79, A001924(25) = 514201, ...}
k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97, A014162(16) = 17519}, no more through n = 30.
k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093, A014166(14) = 16703}
k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273, A053739(11) = 14323}, no more through n = 27.
k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683, 23945893(24) = 23945893}, no more through n = 27.
k=7: primes in A053296 = {A053296(3) = 37, A053296(6) = 967, A053296(7) = 2267, A053296(12) = 127921, A053296(13) = 226007}, no more through n = 27.

Cf. A000040, A005478, A136431, A137176.

Cf. A136431.

A136431 := proc(k,n) local x ; coeftayl(x/(1-x-x^2)/(1-x)^k,x=0,n) ; end: A136338 := proc(amax) local a,k,n,a136431; a := [] ; for k from 1 do if A136431(k,3) > amax then break ; fi ; for n from 3 do a136431 := A136431(k,n) ; if a136431 > amax then break ; fi ; if isprime(a136431) and not a136431 in a then a := [op(a),a136431] ; fi ; od: od: sort(a) ; end: A136338(20000) ; # R. J. Mathar, Apr 21 2008

partsumfib(N,s=[],P=[])={ for( n=1+#s,N, s=concat(s,n+1); forstep( i=n,1,-1, isprime( s[i]+= if( i>1, s[i-1], fibonacci(n+2) ) ) & P=setunion(P,[s[i]]) ); print(s); );vecsort(eval(P))} \\ M. F. Hasler

Primes in the hyperfibonacci number array of A136431, excluding the n=2 column (which contains every positive integer).

Revised definition from N. J. A. Sloane, May 09 2008

More terms from R. J. Mathar, Apr 21 2008

A136438 Hypertribonacci number array read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 4, 4, 0, 0, 1, 4, 7, 8, 7, 0, 0, 1, 5, 11, 15, 15, 13, 0, 0, 1, 6, 16, 26, 30, 28, 24, 0, 0, 1, 7, 22, 42, 56, 58, 52, 44, 0, 0, 1, 8, 29, 64, 98, 114, 110, 96, 81, 0, 0, 1, 9, 37, 93, 162, 212, 224, 206, 177, 149

Offset: 1

Jonathan Vos Post, Apr 13 2008

The hypertribonacci numbers are to the hyperfibonacci array of A136431 as the tribonacci numbers A000073 are to the Fibonacci numbers A000045.

			The array a(k,n) begins:
========================================
n=0..|.1.|.2.|...3.|..4.|...5.|....6.|...7..|.....8.|.....9.|....10.|
========================================
k=0..|.0.|.0.|...1.|..2.|...4.|....7.|..13..|....24.|....44.|....81.| A000073
k=1..|.0.|.0.|...2.|..4.|...8.|...15.|..28..|....52.|....96.|...177.| A008937
k=2..|.0.|.0.|...3.|..7.|..15.|...30.|..58..|...110.|...206.|...383.| A062544
k=3..|.0.|.0.|...4.|.11.|..26.|...56.|..114.|...224.|...430.|...813.|
k=4..|.0.|.0.|...5.|.16.|..42.|...98.|..212.|...436.|...866.|..1679.|
k=5..|.0.|.0.|...6.|.22.|..64.|..162.|..374.|...810.|..1676.|..3355.|
k=6..|.0.|.0.|...7.|.29.|..93.|..255.|..629.|..1439.|..3115.|..6470.|
k=7..|.0.|.0.|...8.|.37.|.130.|..385.|.1014.|..2453.|..5568.|.12038.|
k=8..|.0.|.0.|...9.|.46.|.176.|..561.|.1575.|..4028.|..9596.|.21634.|
k=9..|.0.|.0.|..10.|.56.|.232.|..793.|.2368.|..6396.|.15992.|.37626.|
k=10.|.0.|.0.|..11.|.67.|.299.|.1092.|.3460.|..9856.|.25848.|.63474.|
========================================

Columns n=4..6 are A000124, A000125, A055795.

Cf. A000045, A000073, A008937, A062544, A136431, A137176.

\\ create the n X n matrix of nonzero values
hypertribo(n)={ local(M=matrix(n,n)); M[1,]=Vec(1/(1-x-x^2-x^3)+O(x^n));
M[,1]=vector(n,i,1)~; for(i=2,n, for(j=2,n, M[i,j]=M[i-1,j]+M[i,j-1])); M}
{ hypertribo(10) }

a(k,n) = apply partial sum operator k times to tribonacci numbers A000073.

M. F. Hasler notes that the 8th column = vector(25,n,binomial(n+5,6)+binomial(n+5,4)+2*binomial(n+3,1)). R. J. Mathar points out that the repeated partial sums are quickly computed from their o.g.f.s (-1)^(k+1)*x^2/(-1+x+x^2+x^3)/(-1+x)^k, k=1,2,3,...

Examples corrected by R. J. Mathar, Apr 21 2008

cp's OEIS Frontend

A136431 Hyperfibonacci square number array a(k,n) = F(n)^(k), read by ascending antidiagonals (k, n >= 0).

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A053298 Partial sums of A027964.

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A324242 Incomplete Lucas numbers: irregular triangular array L(n,k) = Sum_{j = 0..k} (n/(n-j)) * binomial(n-j, j), read by rows, with n >= 1 and 0 <= k <= floor(n/2).

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A136338 Primes in the array A136431 that are not Fibonacci numbers.

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A136438 Hypertribonacci number array read by antidiagonals.

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