A045930 -id:A045930 - 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

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).

A122799 A P_7-stuttered arithmetic progression with a(n+1)=a(n) if n is not a heptagonal number, a(n+1)=a(n)+2 otherwise.

Original entry on oeis.org

1, 1, 3, 5, 7, 9, 11, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187

Offset: 1

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

P_7(i) = the i-th heptagonal number.

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

Cf. A001614, A122793, A122794, A122795, A122796, A122797, A122798, A122800.

isHeptag(n) = {if (! issquare(40*n+9, &res), return (0)); if ((res + 3) % 10, return (0), return (1));}
lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! isHeptag(i), aa += 2););} \\ Michel Marcus, Apr 01 2013

a(n) = A045930(n)-n+1.

Definition corrected by Michel Marcus, Apr 01 2013

cp's OEIS Frontend

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

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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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A122799 A P_7-stuttered arithmetic progression with a(n+1)=a(n) if n is not a heptagonal number, a(n+1)=a(n)+2 otherwise.

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