A045928 -id:A045928 - cp's OEIS frontend

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

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

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.

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).

A122796 Connell (3,5)-sum sequence (partial sums of the (3,5)-Connell sequence).

Original entry on oeis.org

1, 3, 8, 16, 27, 41, 58, 76, 97, 121, 148, 178, 211, 247, 286, 328, 373, 421, 470, 522, 577, 635, 696, 760, 827, 897, 970, 1046, 1125, 1207, 1292, 1380, 1471, 1565, 1660, 1758, 1859, 1963, 2070, 2180, 2293, 2409, 2528, 2650, 2775, 2903, 3034, 3168, 3305, 3445, 3588, 3734, 3883, 4035, 4190, 4346, 4505, 4667, 4832, 5000, 5171, 5345, 5522, 5702, 5885, 6071, 6260, 6452, 6647, 6845

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. A045930, A001614, A045928, A045929.

Cf. A122793, A122794, A122795, A122797, A122798, A122799, A122800.

a(n) = (n-th triangular number)-n+(n-th partial sum of A122799).

A045930 The generalized Connell sequence C_{3,5}.

Original entry on oeis.org

1, 2, 5, 8, 11, 14, 17, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 156, 159, 162, 165, 168, 171, 174, 177, 180

Offset: 1

N. J. A. Sloane

			From _Michel Marcus_, Apr 02 2013: (Start)
As a triangle, sequence begins:
   1;
   2,  5,  8, 11, 14, 17;
  18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48;
  ...
(End)

Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A045928.

A045930:=n->3*n - 2*floor((13 + sqrt(40*n-31))/10); seq(A045930(n), n=1..100); # Wesley Ivan Hurt, Feb 03 2014

Table[3 n - 2*Floor[(13 + Sqrt[40 n - 31])/10], {n, 100}] (* Wesley Ivan Hurt, Feb 03 2014 *)

a(n) = 3*n - 2*floor((13 + sqrt(40*n-31))/10). See general formula in A045928. - Michel Marcus, Apr 02 2013

More terms from jeroen.lahousse(AT)icl.com

A045929 Generalized Connell sequence C_{5,3}.

Original entry on oeis.org

1, 2, 7, 12, 17, 18, 23, 28, 33, 38, 43, 48, 49, 54, 59, 64, 69, 74, 79, 84, 89, 94, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 156, 161, 166, 171, 176, 181, 186, 191, 196, 201, 206, 211, 216, 221, 226, 231, 232, 237, 242, 247, 252, 257, 262, 267, 272, 277

Offset: 1

N. J. A. Sloane

			From _Michel Marcus_, Apr 02 2013: (Start)
As a triangle, sequence begins:
   1;
   2,  7, 12, 17;
  18, 23, 28, 33, 38, 43, 48;
  ...
(End)

Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A045928.

Table[5*n - 4*Floor[(7 + Sqrt[24*n - 23])/6],{n,61}] (* Stefano Spezia, Apr 15 2023 *)

a(n) = 5*n - 4*floor((7 + sqrt(24*n - 23))/6). See general formula in A045928. - Michel Marcus, Apr 02 2013

More terms from jeroen.lahousse(AT)icl.com

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

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

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

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A122796 Connell (3,5)-sum sequence (partial sums of the (3,5)-Connell sequence).

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A045930 The generalized Connell sequence C_{3,5}.

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A045929 Generalized Connell sequence C_{5,3}.

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