A026383 -id:A026383 - cp's OEIS frontend

A026374 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 7, 17, 17, 7, 1, 1, 9, 30, 45, 30, 9, 1, 1, 10, 39, 75, 75, 39, 10, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 13, 70, 202, 339, 339, 202, 70, 13, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1

Offset: 0

Clark Kimberling

T(n,k) is number of lattice paths from (0,0) to (n,n-2k) using steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: T(4,1)=6 because we have the following paths from (0,0) to (4,2): UUUD, UUH, UUDU, UDUU, HUU and DUUU. Row sums yield A026383. Column 1 is A032766, column 2 is A026381, column 3 is A026382. - Emeric Deutsch, Jan 25 2004

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  4,   4,   1;
  1,  6,  11,   6,    1;
  1,  7,  17,  17,    7,    1;
  1,  9,  30,  45,   30,    9,    1;
  1, 10,  39,  75,   75,   39,   10,    1;
  1, 12,  58, 144,  195,  144,   58,   12,   1;
  1, 13,  70, 202,  339,  339,  202,   70,  13,   1;
  1, 15,  95, 330,  685,  873,  685,  330,  95,  15,  1;
  1, 16, 110, 425, 1015, 1558, 1558, 1015, 425, 110, 16, 1;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Cf. A026383, A051159,A169623, A007318

Cf. A026375 (central terms).

a026374 n k = a026374_tabl !! n !! k
a026374_row n = a026374_tabl !! n
a026374_tabl = [1] : map fst (map snd $ iterate f (1, ([1, 1], [1]))) where
   f (0, (us, vs)) = (1, (zipWith (+) ([0] ++ us) (us ++ [0]), us))
   f (1, (us, vs)) = (0, (zipWith (+) ([0] ++ vs ++ [0]) $
                             zipWith (+) ([0] ++ us) (us ++ [0]), us))
-- Reinhard Zumkeller, Feb 22 2014

p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + 1)^Floor[n/2]];
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
Flatten[a] (* Roger L. Bagula and Gary W. Adamson, Dec 04 2009 *)

T(n, k) = number of integer strings s(0), ..., s(n) such that s(0)=0, s(n) = n-2k, where, for 1 <= i <= n, s(i) is even if i is even and |s(i) - s(i-1)| <= 1.
From Emeric Deutsch, Jan 25 2004: (Start)
T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);
T(2n+1, k) = T(2n, k-1) + T(2n, k).
G.f.: (1 + z + t*z)/(1 - (1+3*t+t^2)*z^2) = 1 + (1+t)*z + (1+3*t+t^2)*z^2+ ... .
Generating polynomial for row 2n is (1 + 3*t + t^2)^n;
Generating polynomial for row 2n+1 it is (1+t)*(1 + 3*t + t^2)^n. (End)
From Emeric Deutsch, Jan 30 2004: (Start)
T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);
T(2n+1, k) = T(2n, k-1) + T(2n, k). (End)

A215788 T(n,k)=Number of permutations of 0..floor((n*k-1)/2) on even squares of an nXk array such that each row, column, diagonal and (downwards) antidiagonal of even squares is increasing.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 5, 2, 1, 1, 1, 1, 5, 12, 10, 4, 1, 1, 1, 1, 5, 42, 29, 25, 4, 1, 1, 1, 1, 14, 110, 262, 189, 50, 8, 1, 1, 1, 1, 14, 462, 932, 2465, 458, 125, 8, 1, 1, 1, 1, 42, 1274, 11694, 26451, 15485, 2988, 250, 16, 1, 1, 1, 1, 42, 6006

Offset: 1

R. H. Hardin Aug 23 2012

Table starts
.1.1.1..1....1......1........1..........1...........1...........1...........1
.1.1.1..1....2......2........5..........5..........14..........14..........42
.1.1.1..2....5.....12.......42........110.........462........1274........6006
.1.1.1..2...10.....29......262........932.......11694.......46988......727846
.1.1.1..4...25....189.....2465......26451......530429.....7027942...187205626
.1.1.1..4...50....458....15485.....234217....14296434...297246092.26970790176
.1.1.1..8..125...2988...146205....6812794...673507749.48337803306
.1.1.1..8..250...7241...918637...60485308.18255280444
.1.1.1.16..625..47241..8674386.1761748159
.1.1.1.16.1250.114482.54503318
.1.1.1.32.3125.746892
.1.1.1.32.6250

			Some solutions for n=7 k=4
..0..x..1..x....0..x..1..x....0..x..1..x....0..x..1..x....0..x..1..x
..x..2..x..3....x..2..x..4....x..2..x..4....x..2..x..3....x..2..x..3
..4..x..5..x....3..x..5..x....3..x..5..x....4..x..5..x....4..x..5..x
..x..6..x..8....x..6..x..8....x..6..x..8....x..6..x..7....x..6..x..7
..7..x..9..x....7..x..9..x....7..x..9..x....8..x..9..x....8..x..9..x
..x.10..x.12....x.10..x.12....x.10..x.11....x.10..x.12....x.10..x.11
.11..x.13..x...11..x.13..x...12..x.13..x...11..x.13..x...12..x.13..x

R. H. Hardin, Table of n, a(n) for n = 1..140

Column 5 is A026383(n-1)
Row 2 is A000108(floor((n-1)/2))
Odd squares: A215870

Empirical for column k:
k=4: a(n) = 2*a(n-2)
k=5: a(n) = 5*a(n-2)
k=6: a(n) = 16*a(n-2) -3*a(n-4)
k=7: a(n) = 61*a(n-2) -99*a(n-4) -2*a(n-6)
k=8: a(n) = 272*a(n-2) -3439*a(n-4) -3336*a(n-6) +140*a(n-8)
k=9: a(n) = 1385*a(n-2) -131648*a(n-4) -318070*a(n-6) -4160916*a(n-8) -1097892*a(n-10) +648*a(n-12)

A072994 Number of solutions to x^n==1 (mod n), 1<=x<=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 8, 3, 2, 1, 8, 5, 2, 9, 4, 1, 4, 1, 16, 1, 2, 1, 12, 1, 2, 3, 16, 1, 12, 1, 4, 3, 2, 1, 16, 7, 10, 1, 8, 1, 18, 5, 8, 3, 2, 1, 16, 1, 2, 9, 32, 1, 4, 1, 8, 1, 4, 1, 24, 1, 2, 5, 4, 1, 12, 1, 32, 27, 2, 1, 24, 1, 2, 1, 8, 1, 12, 1, 4, 3

Offset: 1

Benoit Cloitre, Aug 21 2002

More generally, if the equation a(x)*m=x has solutions, solutions are congruent to m: a(x)*7=x for x=7, 14, 21, 28, 49, 56, 63, 98, 112, ... . There are some composite values of m such that a(x)*m=x has solutions, as m=15. a(n) coincides with A009195(n) at many values of n, but not at n = 20, 30, 40, 42, 52, 60, 66, 68, 70, 78, 80, 84, 90, 100, ... . It seems also that for n large enough sum_{k=1..n} a(k) > n*log(n)*log(log(n)).

Similar (if not the same) coincidences and differences occur between A072995 and A050399. Sequence A072989 lists these indices. - M. F. Hasler, Feb 23 2014

T. D. Noe, Table of n, a(n) for n=1..10000

1, seq(nops(select(t -> t^n mod n = 1, [$1..n-1])),n=2..100); # Robert Israel, Dec 07 2014

f[n_] := (d = If[ OddQ@ n, 1, 2]; d*Length@ Select[ Range[ n/d], PowerMod[#, n, n] == 1 &]); f[1] = f[2] = 1; Array[f, 93] (* or *)
f[n_] := Length@ Select[ Range@ n, PowerMod[#, n, n] == 1 &]; f[n_] := 1 /; n<2; Array[f, 93] (* Robert G. Wilson v, Dec 06 2014 *)

A072994=n->sum(k=1,n,Mod(k,n)^n==1) \\ M. F. Hasler, Feb 23 2014

For n>0, a(A003277(n)) = 1, a(2^n) = 2^(n-1), a(A065119(n)) = A065119(n)/3.

For n>1, a(A026383(n)) = A026383(n)/5.

Corrected by T. D. Noe, May 19 2007

A004663 Powers of 3 written in base 9.

Original entry on oeis.org

1, 3, 10, 30, 100, 300, 1000, 3000, 10000, 30000, 100000, 300000, 1000000, 3000000, 10000000, 30000000, 100000000, 300000000, 1000000000, 3000000000, 10000000000, 30000000000, 100000000000, 300000000000, 1000000000000, 3000000000000, 10000000000000, 30000000000000

Offset: 0

N. J. A. Sloane

Robert Israel, Table of n, a(n) for n = 0..1997
Index entries for linear recurrences with constant coefficients, signature (0,10).

Cf. A026383, A016116.
Cf. A000244, A004656, A004658, A004659, ... : powers of 3 in base 10, 2, 4, 5, ...
Cf. A000079, A004642, ..., A004655: powers of 2 written in base 10, 2, 3, ..., 16.

seq(op([10^i,3*10^i]),i=0..100); # Robert Israel, Jun 25 2018

Table[FromDigits[IntegerDigits[3^n, 9]], {n, 0, 100}] (* G. C. Greubel, Oct 12 2018 *)

a(n)=3^bittest(n,0)*10^(n\2) \\ M. F. Hasler, Jun 25 2018

From Paul Barry, Jul 14 2004: (Start)
G.f.: (1 + 3*x)/(1 - 10*x^2);
a(n) = 2*a(n-1) + 3*a(n-2) + 10^floor((n-2)/2);
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*3^(n-2*k). (End)
a(n) = 3*a(n-1) + ((1 + (-1)^n)/2)*a(n-2) with a(0)=1, a(1)=3. - Taras Goy, Mar 20 2019
E.g.f.: cosh(sqrt(10)*x) + 3*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Mar 31 2023

A162963 a(n) = 5*a(n-2) for n > 2; a(1) = 2, a(2) = 5.

Original entry on oeis.org

2, 5, 10, 25, 50, 125, 250, 625, 1250, 3125, 6250, 15625, 31250, 78125, 156250, 390625, 781250, 1953125, 3906250, 9765625, 19531250, 48828125, 97656250, 244140625, 488281250, 1220703125, 2441406250, 6103515625, 12207031250

Offset: 1

Klaus Brockhaus, Jul 19 2009

Binomial transform is A162770, second binomial transform is A001077 without initial 1, third binomial transform is A162771, fourth binomial transform is A162772, fifth binomial transform is A162773.

Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A000351 (powers of 5), A026383, A001077, A162770, A162771, A162772, A162773.

[ n le 2 select 3*n-1 else 5*Self(n-2): n in [1..29] ];

a(n) = (3-(-1)^n)*5^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(2+5*x)/(1-5*x^2).
a(n) = A026383(n) for n >= 1.

A096881 Expansion of g.f. (1 + 4x)/(1 - 17x^2).

Original entry on oeis.org

1, 4, 17, 68, 289, 1156, 4913, 19652, 83521, 334084, 1419857, 5679428, 24137569, 96550276, 410338673, 1641354692, 6975757441, 27903029764, 118587876497, 474351505988, 2015993900449, 8063975601796, 34271896307633, 137087585230532, 582622237229761, 2330488948919044

Offset: 0

Paul Barry, Jul 14 2004

Index entries for linear recurrences with constant coefficients, signature (0,17).

Cf. A004663, A026383, A016116.

I:=[1,4,17]; [n le 3 select I[n] else 17*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 26 2016

CoefficientList[Series[(1+4x)/(1-17x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {0,17},{1,4},30] (* Harvey P. Dale, Jan 21 2012 *)

Vec((1+4*x)/(1-17*x^2) + O(x^40)) \\ Michel Marcus, Jan 26 2016

a(n) = 3*a(n-1) + 4*a(n-2) + 17^floor((n-2)/2).
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*4^(n-2*k).
a(n) = 17*a(n-2), n>1. - Harvey P. Dale, Jan 21 2012
E.g.f.: cosh(sqrt(17)*x) + 4*sinh(sqrt(17)*x)/sqrt(17). - Stefano Spezia, Mar 31 2023

More terms from Stefano Spezia, Mar 31 2023

A026384 a(n) = Sum_{j=0..i, i=0..n} T(i,j), where T is the array in A026374.

Original entry on oeis.org

1, 3, 8, 18, 43, 93, 218, 468, 1093, 2343, 5468, 11718, 27343, 58593, 136718, 292968, 683593, 1464843, 3417968, 7324218, 17089843, 36621093, 85449218, 183105468, 427246093, 915527343, 2136230468, 4577636718, 10681152343, 22888183593, 53405761718

Offset: 0

Clark Kimberling

Partial sums of A026383. Number of lattice paths from (0,0) that do not go to right of the line x=n, using the steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: a(2)=8 because we have the empty path, U, D, UU, UD, DD, DU and H. - Emeric Deutsch, Feb 18 2004

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Cf. A026383.

I:=[1,3,8]; [n le 3 select I[n] else Self(n-1)+5*Self(n-2)-5*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Aug 09 2017

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+3 od: seq(a[n], n=1..29); # Zerinvary Lajos, Mar 17 2008

CoefficientList[Series[(1 + 2 x) / ((1 - x) (1 - 5 x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 09 2017 *)
LinearRecurrence[{1,5,-5},{1,3,8},40] (* Harvey P. Dale, May 31 2023 *)

Vec((2*x + 1)/(5*x^3 - 5*x^2 - x + 1) + O(x^40)) \\ Colin Barker, Nov 25 2016

G.f.: (1+2*x) / ((1-x)*(1-5*x^2)). - Ralf Stephan, Apr 30 2004
From Colin Barker, Nov 25 2016: (Start)
a(n) = (7*5^(n/2) - 3)/4 for n even.
a(n) = 3*(5^((n+1)/2) - 1)/4 for n odd.
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n>2.
(End)

A096882 Expansion of g.f. (1 + 7x)/(1 - 50x^2).

Original entry on oeis.org

1, 7, 50, 350, 2500, 17500, 125000, 875000, 6250000, 43750000, 312500000, 2187500000, 15625000000, 109375000000, 781250000000, 5468750000000, 39062500000000, 273437500000000, 1953125000000000, 13671875000000000, 97656250000000000, 683593750000000000, 4882812500000000000

Offset: 0

Paul Barry, Jul 14 2004

Index entries for linear recurrences with constant coefficients, signature (0,50).

Cf. A004663, A026383, A016116, A096881.

a[n_]:=Sum[Binomial[Floor[n/2],k]7^(n-2k),{k,0,Floor[n/2]}]; Array[a,25,0] (* Stefano Spezia, Mar 31 2023 *)
LinearRecurrence[{0,50},{1,7},30] (* Harvey P. Dale, Sep 20 2024 *)

a(n) = 6*a(n-1) + 7*a(n-2) + 50^floor((n-2)/2).
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*7^(n-2*k).
E.g.f.: cosh(5*sqrt(2)*x) + 7*sinh(5*sqrt(2)*x)/(5*sqrt(2)). - Stefano Spezia, Mar 31 2023

More terms from Stefano Spezia, Mar 31 2023

A096883 Expansion of (1+10x)/(1-101x^2).

Original entry on oeis.org

1, 10, 101, 1010, 10201, 102010, 1030301, 10303010, 104060401, 1040604010, 10510100501, 105101005010, 1061520150601, 10615201506010, 107213535210701, 1072135352107010, 10828567056280801, 108285670562808010

Offset: 0

Paul Barry, Jul 14 2004

Index entries for linear recurrences with constant coefficients, signature (0,101).

Cf. A004663, A026383, A016116, A096881, A096882.

a(n)=9a(n-1)+10a(n-2)+101^floor((n-2)/2); a(n)=sum{k=0..floor(n/2), binomial(floor(n/2), k)10^(n-2k) }.

A123486 Riordan array (1/(1-2x), x/(1-4x^2)).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 8, 2, 1, 16, 16, 12, 2, 1, 32, 48, 24, 16, 2, 1, 64, 96, 96, 32, 20, 2, 1, 128, 256, 192, 160, 40, 24, 2, 1, 256, 512, 640, 320, 240, 48, 28, 2, 1, 512, 1280, 1280, 1280, 480, 336, 56, 32, 2, 1, 1024, 2560, 3840, 2560, 2240, 672, 448, 64, 36, 2, 1

Offset: 0

Paul Barry, Sep 30 2006

Row sums are A026581. Diagonal sums are A026383.

			Number triangle begins
1;
2, 1;
4, 2, 1;
8, 8, 2, 1;
16, 16, 12, 2, 1;
32, 48, 24, 16, 2, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Table[Binomial[Floor[(n + k)/2], k]*2^(n - k), {n, 0, 49}, {k, 0,
   n}] // Flatten (* G. C. Greubel, Oct 13 2017 *)

for(n=0,10, for(k=0,n, print1(binomial(floor((n+k)/2),k)*2^(n-k), ", "))) \\ G. C. Greubel, Oct 13 2017

Number triangle T(n,k) = C(floor((n+k)/2), k) * 2^(n-k).

T(n,k) = T(n-1,k-1) + 4*T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 20 2014

Terms a(46) onward added by G. C. Greubel, Oct 14 2017

cp's OEIS Frontend

A026374 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.

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A215788 T(n,k)=Number of permutations of 0..floor((n*k-1)/2) on even squares of an nXk array such that each row, column, diagonal and (downwards) antidiagonal of even squares is increasing.

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A072994 Number of solutions to x^n==1 (mod n), 1<=x<=n.

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A004663 Powers of 3 written in base 9.

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A162963 a(n) = 5*a(n-2) for n > 2; a(1) = 2, a(2) = 5.

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A096881 Expansion of g.f. (1 + 4*x)/(1 - 17*x^2).

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A026384 a(n) = Sum_{j=0..i, i=0..n} T(i,j), where T is the array in A026374.

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A096881 Expansion of g.f. (1 + 4x)/(1 - 17x^2).

A096882 Expansion of g.f. (1 + 7x)/(1 - 50x^2).

A123486 Riordan array (1/(1-2x), x/(1-4x^2)).