A001614 -id:A001614 - cp's OEIS frontend

A074147 (2n-1) odd numbers followed by 2n even numbers.

1, 2, 4, 3, 5, 7, 6, 8, 10, 12, 9, 11, 13, 15, 17, 14, 16, 18, 20, 22, 24, 19, 21, 23, 25, 27, 29, 31, 26, 28, 30, 32, 34, 36, 38, 40, 33, 35, 37, 39, 41, 43, 45, 47, 49, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 62, 64, 66, 68, 70, 72

Offset: 1

Amarnath Murthy, Aug 28 2002

			As a triangular array:
1,
2, 4,
3, 5, 7,
6, 8, 10, 12,
9, 11, 13, 15, 17,
14, 16, 18, 20, 22, 24,
...

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A074148, A074149 (row sums), A001614.

A074147 := proc(n,k)
    ceil((n-1)^2/2)+1+2*k ;
end proc:
seq(seq( A074147(n,k),k=0..n-1) ,n=1..12) ; # R. J. Mathar, Nov 02 2023

Flatten@Table[Ceiling[(n - 1)^2/2] + 2 k - 1, {n, 12}, {k, n}] (* Ivan Neretin, Dec 15 2016 *)

T(n,k) = A061925(n-1)+2k, 0<=kR. J. Mathar, Jul 17 2007, corrected Nov 02 2023

a(n) = A116941(n-1)+1. - Robert G. Wilson v, Mar 09 2017

A122797 A P_3-stuttered arithmetic progression with a(n+1) = a(n) if n is a triangular number, a(n+1) = a(n) + 1 otherwise.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

P_3(i) = the i-th triangular number.
As a triangle [1; 1,2; 2,3,4; ...], row sums = A064808: (1, 3, 9, 22, 45, 81, ...). - Gary W. Adamson, Aug 10 2007
a(n) = n - A003056(n-1). - Reinhard Zumkeller, Feb 12 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187. [From _Johannes W. Meijer_, Jun 21 2010]

Cf. A001614, A122793, A122794, A122795, A122796, A122798, A122799, A122800.

Cf. A064808.

a122797 n = a122797_list !! (n-1)
a122797_list  = 1 : zipWith (+) a122797_list (map ((1 -) . a010054) [1..])
-- Reinhard Zumkeller, Feb 12 2012

nxt[{n_,a_}]:={n+1,If[OddQ[Sqrt[8n+1]],a,a+1]}; NestList[nxt,{1,1},100][[All,2]] (* Harvey P. Dale, Oct 10 2018 *)

isTriang(n) = {if (! issquare(8*n+1), return (0)); return (1);}
lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! isTriang(i), aa = aa + 1););} \\ Michel Marcus, Apr 01 2013

from math import isqrt
def A122797(n): return n+1-(k:=isqrt(m:=n<<1))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

a(n) = A001614(n) - n + 1.

Definition corrected by Michel Marcus, Apr 01 2013

A049039 Geometric Connell sequence: 1 odd, 2 even, 4 odd, 8 even, ...

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 24, 26, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 121, 123, 125

Offset: 1

James Sellers

Reinhard Zumkeller, Rows n=1..13 of triangle, flattened
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A337300 (partial sums), A043529 (first differences).
Cf. A001614, A033292, A030196, A000295, A050487, A050488.
Cf. A160464, A160465 and A160473. - Johannes W. Meijer, May 24 2009

a049039 n k = a049039_tabl !! (n-1) !! (k-1)
a049039_row n = a049039_tabl !! (n-1)
a049039_tabl = f 1 1 [1..] where
   f k p xs = ys : f (2 * k) (1 - p) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== p) . (`mod` 2)) xs
-- Reinhard Zumkeller, Jan 18 2012, Jul 08 2011

Digits := 100: [seq(2*n-1-floor(evalf(log(n)/log(2))), n=1..100)];

a[0] = 0; a[n_?EvenQ] := a[n] = a[n/2]+n-1; a[n_?OddQ] := a[n] = a[(n-1)/2]+n; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 27 2011, after Ralf Stephan *)

a(n) = n<<1 - 1 - logint(n,2); \\ Kevin Ryde, Feb 12 2022

def A049039(n): return (n<<1)-n.bit_length() # Chai Wah Wu, Aug 01 2022

a(n) = 2n - 1 - floor(log_2(n)).
a(2^n-1) = 2^(n+1) - (n+2) = A000295(n+1), the Eulerian numbers.
a(0)=0, a(2n) = a(n) + 2n - 1, a(2n+1) = a(n) + 2n + 1. - Ralf Stephan, Oct 11 2003

Keyword tabf added by Reinhard Zumkeller, Jan 22 2012

A122793 Connell sum sequence (partial sums of the Connell sequence).

Original entry on oeis.org

1, 3, 7, 12, 19, 28, 38, 50, 64, 80, 97, 116, 137, 160, 185, 211, 239, 269, 301, 335, 371, 408, 447, 488, 531, 576, 623, 672, 722, 774, 828, 884, 942, 1002, 1064, 1128, 1193, 1260, 1329, 1400, 1473, 1548, 1625, 1704, 1785, 1867, 1951, 2037, 2125, 2215, 2307, 2401, 2497, 2595, 2695, 2796, 2899, 3004, 3111, 3220

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

a(n) is the sum of the n highest entries in the projection of the n-th tetrahedron or tetrahedral number (e.g., a(7) = 7+6+6+5+5+5+4+4).

a(n) is a sharp upper bound for the value of a gamma-labeling of a graph with n edges (cf. Bullington).

Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Ian Connell, Elementary Problem E1382, American Mathematical Monthly, v. 66, no. 8 (October, 1959), p. 724.
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A001614, A045928, A045929, A045930.
Cf. A122794, A122795, A122796, A122797, A122798, A122799, A122800.
Cf. A337300 (geometric Connell sums).

from math import isqrt
def A122793(n): return n*(n+1)-(r:=(k:=isqrt(m:=n<<1))+int((m<<2)>(k<<2)*(k+1)+1))*((6*n+1)-r**2)//6 # Chai Wah Wu, Jul 26 2022

a(n) = (n-th triangular number) - n + (n-th partial sum of A122797).

a(n) = n^2 + n - R*((6*n+1)-R^2)/6, where R = round(sqrt(2*n)). - Gerald Hillier, Nov 29 2009

A122798 A P_5-stuttered arithmetic progression with a(n+1) = a(n) if n is a pentagonal number, a(n+1) = a(n)+4 otherwise.

Original entry on oeis.org

1, 1, 5, 9, 13, 13, 17, 21, 25, 29, 33, 37, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 181, 185, 189, 193, 197, 201, 205, 209

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

P_5(i) = the i-th pentagonal number.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A001614, A122793, A122794, A122795, A122796, A122797, A122799, A122800.

nxt[{n_,a_}]:={n+1,If[IntegerQ[(1+Sqrt[24n+25])/6],a,a+4]}; Join[{1}, Transpose[ NestList[nxt,{1,1},60]][[2]]] (* Harvey P. Dale, May 07 2015 *)
nxt[{n_,a_}]:=With[{pn=PolygonalNumber[5,Range[0,30]]},{n+1,If[MemberQ[pn,n],a,a+4]}]; NestList[nxt,{1,1},100][[;;,2]] (* Harvey P. Dale, Sep 28 2023 *)

lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! ispolygonal(i, 5), aa += 4););} \\ Michel Marcus, Apr 01 2013, May 02 2015

a(n) = A045929(n) - n + 1.

Definition corrected by Michel Marcus, Apr 01 2013

A122800 A P_4-stuttered arithmetic progression with a(n+1)=a(n) if n is square, a(n+1)=a(n)+2 otherwise.

Original entry on oeis.org

1, 1, 3, 5, 5, 7, 9, 11, 13, 13, 15, 17, 19, 21, 23, 25, 25, 27, 29, 31, 33, 35, 37, 39, 41, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

P_4(i) = the i-th square.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A001614, A122793, A122794, A122795, A122796, A122797, A122798, A122799.

nxt[{n_,a_}]:={n+1,If[IntegerQ[Sqrt[n+1]],a,a+2]}; NestList[nxt,{0,1},100][[All,2]] (* Harvey P. Dale, Jan 01 2020 *)

lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! issquare(i), aa = aa + 2););} \\ Michel Marcus, Apr 01 2013

a(n) = A045928(n)-n+1.

A190717 Triplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 10, 10, 10, 20, 20, 20, 35, 35, 35, 56, 56, 56, 84, 84, 84, 120, 120, 120, 165, 165, 165, 220, 220, 220, 286, 286, 286, 364, 364, 364, 455, 455, 455, 560, 560, 560, 680, 680, 680, 816, 816, 816, 969, 969, 969

Offset: 0

Johannes W. Meijer, May 18 2011

The Ca1 and Ze3 triangle sums, see A180662 for their definitions, of the triangle A159797 are linear sums of shifted versions of the triplicated tetrahedral numbers, e.g. Ca1(n) = a(n-1) + a(n-2) + 2*a(n-3) + a(n-6).

The Ca1, Ca2, Ze3 and Ze4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), this sequence (triplicated), A190718 (quadruplicated), A049347, A144677.

A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A190717(n),n=0..50);

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,1,1,4,4,4,10,10,10,20},60] (* Harvey P. Dale, Mar 09 2018 *)

a(n) = binomial(floor(n/3)+3,3).
a(n) + a(n-1) + a(n-2) = A144677(n).
a(n) = Sum_{k=0..n} (A144677(n-k)*A049347(k)).
G.f.: 1/((x-1)^4*(x^2+x+1)^3).
Sum_{n>=0} 1/a(n) = 9/2. - Amiram Eldar, Aug 18 2022

A190718 Quadruplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 4, 4, 10, 10, 10, 10, 20, 20, 20, 20, 35, 35, 35, 35, 56, 56, 56, 56, 84, 84, 84, 84, 120, 120, 120, 120, 165, 165, 165, 165, 220, 220, 220, 220, 286, 286, 286, 286, 364, 364, 364, 364, 455, 455, 455, 455

Offset: 0

Johannes W. Meijer, May 18 2011

The Gi1 triangle sums, for the definitions of these and other triangle sums see A180662, of the triangle A159797 are linear sums of shifted versions of the quadruplicated tetrahedral numbers A000292, i.e., Gi1(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-8).

The Gi1 and Gi2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,3,-3,0,0,-3,3,0,0,1,-1).

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), A190717 (triplicated).

A190718:= proc(n) binomial(floor(n/4)+3,3) end:
seq(A190718(n),n=0..52);

LinearRecurrence[{1,0,0,3,-3,0,0,-3,3,0,0,1,-1},{1,1,1,1,4,4,4,4,10,10,10,10,20},60] (* Harvey P. Dale, Oct 20 2012 *)

a(n) = binomial(floor(n/4)+3,3).
a(n-3) + a(n-2) + a(n-1) + a(n) = A144678(n).
a(n) = +a(n-1) +3*a(n-4) -3*a(n-5) -3*a(n-8) +3*a(n-9) +a(n-12) -a(n-13).
G.f.: 1 / ( (1+x)^3*(1+x^2)^3*(x-1)^4 ).
Sum_{n>=0} 1/a(n) = 6. - Amiram Eldar, Aug 18 2022

A122794 Connell (3,2)-sum sequence (partial sums of the (3,2)-Connell sequence).

Original entry on oeis.org

1, 3, 8, 16, 25, 37, 52, 70, 91, 113, 138, 166, 197, 231, 268, 308, 349, 393, 440, 490, 543, 599, 658, 720, 785, 851, 920, 992, 1067, 1145, 1226, 1310, 1397, 1487, 1580, 1676, 1773, 1873, 1976, 2082, 2191, 2303, 2418, 2536, 2657, 2781, 2908, 3038, 3171, 3305, 3442, 3582, 3725, 3871, 4020, 4172, 4327, 4485, 4646, 4810, 4977, 5147, 5320, 5496, 5673, 5853, 6036, 6222, 6411, 6603

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A045928, A001614, A045929, A045930.

Cf. A122793, A122795, A122796, A122797, A122798, A122799, A122800.

a(n) = (n-th triangular number)-n+(n-th partial sum of A122800).

A122795 Connell (5,3)-sum sequence (partial sums of the (5,3)-Connell sequence).

Original entry on oeis.org

1, 3, 10, 22, 39, 57, 80, 108, 141, 179, 222, 270, 319, 373, 432, 496, 565, 639, 718, 802, 891, 985, 1080, 1180, 1285, 1395, 1510, 1630, 1755, 1885, 2020, 2160, 2305, 2455, 2610, 2766, 2927, 3093, 3264, 3440, 3621, 3807, 3998, 4194, 4395, 4601, 4812, 5028, 5249, 5475, 5706, 5938, 6175, 6417, 6664, 6916, 7173, 7435, 7702, 7974, 8251, 8533, 8820, 9112, 9409, 9711, 10018, 10330, 10647, 10969

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A045929, A001614, A045928, A045930.

Cf. A122793, A122794, A122796, A122797, A122798, A122799, A122800.

a(n) = (n-th triangular number)-n+(n-th partial sum of A122798).

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A074147 (2n-1) odd numbers followed by 2n even numbers.

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A122797 A P_3-stuttered arithmetic progression with a(n+1) = a(n) if n is a triangular number, a(n+1) = a(n) + 1 otherwise.

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A049039 Geometric Connell sequence: 1 odd, 2 even, 4 odd, 8 even, ...

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A122793 Connell sum sequence (partial sums of the Connell sequence).

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A122798 A P_5-stuttered arithmetic progression with a(n+1) = a(n) if n is a pentagonal number, a(n+1) = a(n)+4 otherwise.

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A122800 A P_4-stuttered arithmetic progression with a(n+1)=a(n) if n is square, a(n+1)=a(n)+2 otherwise.

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A190717 Triplicated tetrahedral numbers A000292.

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